Distance on a great circle

Percentage Accurate: 62.6% → 98.7%

Time: 44.5s

Alternatives: 53

Speedup: 1.3×

Specification

Math

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

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: 62.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* lambda2 -0.5)))
        (t_1 (sin (* lambda1 0.5)))
        (t_2 (* (* (* (cos phi2) (cos phi1)) t_1) (cos (* lambda2 -0.5))))
        (t_3
         (+
          (pow
           (-
            (* (cos (* phi2 -0.5)) (sin (* 0.5 phi1)))
            (* (sin (* 0.5 phi2)) (cos (* 0.5 phi1))))
           2.0)
          (*
           (*
            (+
             1.0
             (/
              (* (* (* t_0 (cos phi2)) (cos phi1)) (cos (* lambda1 -0.5)))
              t_2))
            t_2)
           (fma t_1 (cos (* lambda2 0.5)) (* (cos (* lambda1 0.5)) t_0))))))
   (* 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 = sin((lambda2 * -0.5));
	double t_1 = sin((lambda1 * 0.5));
	double t_2 = ((cos(phi2) * cos(phi1)) * t_1) * cos((lambda2 * -0.5));
	double t_3 = pow(((cos((phi2 * -0.5)) * sin((0.5 * phi1))) - (sin((0.5 * phi2)) * cos((0.5 * phi1)))), 2.0) + (((1.0 + ((((t_0 * cos(phi2)) * cos(phi1)) * cos((lambda1 * -0.5))) / t_2)) * t_2) * fma(t_1, cos((lambda2 * 0.5)), (cos((lambda1 * 0.5)) * t_0)));
	return R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
}

Julia

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(lambda2 * -0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[Cos[N[(lambda2 * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[(N[(N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $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[(1.0 + N[(N[(N[(N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * N[(t$95$1 * N[Cos[N[(lambda2 * 0.5), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision] * t$95$0), $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. Initial program 62.6%

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

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

3. Applied rewrites 63.5%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sin-diff N/A

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

   6. lower--.f64 N/A

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

5. Applied rewrites 78.7%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sub-flip N/A

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

   6. mult-flip N/A

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

   7. metadata-eval N/A

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

   8. distribute-rgt-neg-out N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. sin-sum N/A

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

   13. lift-cos.f64 N/A

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

   14. lower-fma.f64 N/A

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

7. Applied rewrites 77.7%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sub-flip N/A

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

   6. mult-flip N/A

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

   7. metadata-eval N/A

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

   8. distribute-rgt-neg-out N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. sin-sum N/A

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

   13. lift-cos.f64 N/A

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

   14. lower-fma.f64 N/A

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

9. Applied rewrites 79.2%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sub-flip N/A

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

    6. mult-flip N/A

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

    7. metadata-eval N/A

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

    8. distribute-rgt-neg-out N/A

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

    9. metadata-eval N/A

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

    10. *-commutative N/A

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

    11. lift-*.f64 N/A

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

    12. sin-sum N/A

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

    13. lift-cos.f64 N/A

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

    14. lower-fma.f64 N/A

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

11. Applied rewrites 78.9%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sub-flip N/A

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

    6. mult-flip N/A

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

    7. metadata-eval N/A

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

    8. distribute-rgt-neg-out N/A

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

    9. metadata-eval N/A

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

    10. *-commutative N/A

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

    11. lift-*.f64 N/A

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

    12. sin-sum N/A

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

    13. lift-cos.f64 N/A

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

    14. lower-fma.f64 N/A

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

13. Applied rewrites 98.7%

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

15. Applied rewrites 98.7%

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

17. Applied rewrites 98.7%

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

Alternative 2: 98.7% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (sin (* lambda1 0.5))
          (cos (* lambda2 0.5))
          (* (cos (* lambda1 0.5)) (sin (* lambda2 -0.5)))))
        (t_1
         (+
          (pow
           (-
            (* (cos (* phi2 -0.5)) (sin (* 0.5 phi1)))
            (* (sin (* 0.5 phi2)) (cos (* 0.5 phi1))))
           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 = fma(sin((lambda1 * 0.5)), cos((lambda2 * 0.5)), (cos((lambda1 * 0.5)) * sin((lambda2 * -0.5))));
	double t_1 = pow(((cos((phi2 * -0.5)) * sin((0.5 * phi1))) - (sin((0.5 * phi2)) * cos((0.5 * phi1)))), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	return R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(sin(Float64(lambda1 * 0.5)), cos(Float64(lambda2 * 0.5)), Float64(cos(Float64(lambda1 * 0.5)) * sin(Float64(lambda2 * -0.5))))
	t_1 = Float64((Float64(Float64(cos(Float64(phi2 * -0.5)) * sin(Float64(0.5 * phi1))) - Float64(sin(Float64(0.5 * phi2)) * cos(Float64(0.5 * phi1)))) ^ 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

Wolfram

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

Derivation

1. Initial program 62.6%

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

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

3. Applied rewrites 63.5%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sin-diff N/A

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

   6. lower--.f64 N/A

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

5. Applied rewrites 78.7%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sub-flip N/A

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

   6. mult-flip N/A

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

   7. metadata-eval N/A

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

   8. distribute-rgt-neg-out N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. sin-sum N/A

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

   13. lift-cos.f64 N/A

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

   14. lower-fma.f64 N/A

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

7. Applied rewrites 77.7%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sub-flip N/A

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

   6. mult-flip N/A

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

   7. metadata-eval N/A

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

   8. distribute-rgt-neg-out N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. sin-sum N/A

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

   13. lift-cos.f64 N/A

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

   14. lower-fma.f64 N/A

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

9. Applied rewrites 79.2%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sub-flip N/A

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

    6. mult-flip N/A

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

    7. metadata-eval N/A

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

    8. distribute-rgt-neg-out N/A

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

    9. metadata-eval N/A

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

    10. *-commutative N/A

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

    11. lift-*.f64 N/A

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

    12. sin-sum N/A

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

    13. lift-cos.f64 N/A

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

    14. lower-fma.f64 N/A

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

11. Applied rewrites 78.9%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sub-flip N/A

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

    6. mult-flip N/A

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

    7. metadata-eval N/A

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

    8. distribute-rgt-neg-out N/A

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

    9. metadata-eval N/A

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

    10. *-commutative N/A

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

    11. lift-*.f64 N/A

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

    12. sin-sum N/A

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

    13. lift-cos.f64 N/A

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

    14. lower-fma.f64 N/A

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

13. Applied rewrites 98.7%

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

Alternative 3: 98.7% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (sin (* lambda1 0.5))
          (cos (* lambda2 0.5))
          (* (cos (* lambda1 0.5)) (sin (* lambda2 -0.5)))))
        (t_1
         (+
          (pow
           (fma
            (sin (* phi1 0.5))
            (cos (* 0.5 phi2))
            (* (sin (* -0.5 phi2)) (cos (* phi1 -0.5))))
           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 = fma(sin((lambda1 * 0.5)), cos((lambda2 * 0.5)), (cos((lambda1 * 0.5)) * sin((lambda2 * -0.5))));
	double t_1 = pow(fma(sin((phi1 * 0.5)), cos((0.5 * phi2)), (sin((-0.5 * phi2)) * cos((phi1 * -0.5)))), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	return R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(sin(Float64(lambda1 * 0.5)), cos(Float64(lambda2 * 0.5)), Float64(cos(Float64(lambda1 * 0.5)) * sin(Float64(lambda2 * -0.5))))
	t_1 = Float64((fma(sin(Float64(phi1 * 0.5)), cos(Float64(0.5 * phi2)), Float64(sin(Float64(-0.5 * phi2)) * cos(Float64(phi1 * -0.5)))) ^ 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

Wolfram

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

Derivation

1. Initial program 62.6%

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

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

3. Applied rewrites 63.5%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sin-diff N/A

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

   6. lower--.f64 N/A

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

5. Applied rewrites 78.7%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sub-flip N/A

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

   6. mult-flip N/A

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

   7. metadata-eval N/A

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

   8. distribute-rgt-neg-out N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. sin-sum N/A

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

   13. lift-cos.f64 N/A

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

   14. lower-fma.f64 N/A

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

7. Applied rewrites 77.7%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sub-flip N/A

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

   6. mult-flip N/A

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

   7. metadata-eval N/A

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

   8. distribute-rgt-neg-out N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. sin-sum N/A

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

   13. lift-cos.f64 N/A

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

   14. lower-fma.f64 N/A

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

9. Applied rewrites 79.2%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sub-flip N/A

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

    6. mult-flip N/A

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

    7. metadata-eval N/A

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

    8. distribute-rgt-neg-out N/A

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

    9. metadata-eval N/A

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

    10. *-commutative N/A

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

    11. lift-*.f64 N/A

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

    12. sin-sum N/A

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

    13. lift-cos.f64 N/A

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

    14. lower-fma.f64 N/A

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

11. Applied rewrites 78.9%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sub-flip N/A

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

    6. mult-flip N/A

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

    7. metadata-eval N/A

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

    8. distribute-rgt-neg-out N/A

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

    9. metadata-eval N/A

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

    10. *-commutative N/A

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

    11. lift-*.f64 N/A

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

    12. sin-sum N/A

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

    13. lift-cos.f64 N/A

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

    14. lower-fma.f64 N/A

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

13. Applied rewrites 98.7%

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

    1. lift--.f64 N/A

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

    2. sub-flip N/A

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

    3. lift-*.f64 N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f64 N/A

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

    9. lift-cos.f64 N/A

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

    10. lift-*.f64 N/A

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

    11. metadata-eval N/A

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

    12. distribute-rgt-neg-out N/A

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

    13. *-commutative N/A

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

    14. lift-*.f64 N/A

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

    15. cos-neg-rev N/A

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

    16. lower-cos.f64 N/A

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

    17. lift-*.f64 N/A

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

    18. distribute-lft-neg-out N/A

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

    19. lower-*.f64 N/A

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

15. Applied rewrites 98.7%

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

    1. lift--.f64 N/A

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

    2. sub-flip N/A

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

    3. lift-*.f64 N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f64 N/A

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

    9. lift-cos.f64 N/A

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

    10. lift-*.f64 N/A

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

    11. metadata-eval N/A

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

    12. distribute-rgt-neg-out N/A

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

    13. *-commutative N/A

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

    14. lift-*.f64 N/A

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

    15. cos-neg-rev N/A

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

    16. lower-cos.f64 N/A

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

    17. lift-*.f64 N/A

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

    18. distribute-lft-neg-out N/A

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

    19. lower-*.f64 N/A

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

17. Applied rewrites 98.7%

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

Alternative 4: 98.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos phi1)
          (*
           (cos phi2)
           (pow
            (fma
             (cos (* 0.5 lambda1))
             (sin (* -0.5 lambda2))
             (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1))))
            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 * lambda1)), sin((-0.5 * lambda2)), (cos((0.5 * lambda2)) * sin((0.5 * lambda1)))), 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 * lambda1)), sin(Float64(-0.5 * lambda2)), Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1)))) ^ 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 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $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 62.6%

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

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

3. Applied rewrites 63.5%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sin-diff N/A

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

   6. lower--.f64 N/A

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

5. Applied rewrites 78.7%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sub-flip N/A

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

   6. mult-flip N/A

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

   7. metadata-eval N/A

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

   8. distribute-rgt-neg-out N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. sin-sum N/A

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

   13. lift-cos.f64 N/A

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

   14. lower-fma.f64 N/A

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

7. Applied rewrites 77.7%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sub-flip N/A

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

   6. mult-flip N/A

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

   7. metadata-eval N/A

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

   8. distribute-rgt-neg-out N/A

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

   9. metadata-eval N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. sin-sum N/A

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

   13. lift-cos.f64 N/A

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

   14. lower-fma.f64 N/A

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

9. Applied rewrites 79.2%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sub-flip N/A

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

    6. mult-flip N/A

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

    7. metadata-eval N/A

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

    8. distribute-rgt-neg-out N/A

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

    9. metadata-eval N/A

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

    10. *-commutative N/A

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

    11. lift-*.f64 N/A

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

    12. sin-sum N/A

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

    13. lift-cos.f64 N/A

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

    14. lower-fma.f64 N/A

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

11. Applied rewrites 78.9%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sub-flip N/A

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

    6. mult-flip N/A

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

    7. metadata-eval N/A

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

    8. distribute-rgt-neg-out N/A

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

    9. metadata-eval N/A

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

    10. *-commutative N/A

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

    11. lift-*.f64 N/A

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

    12. sin-sum N/A

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

    13. lift-cos.f64 N/A

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

    14. lower-fma.f64 N/A

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

13. Applied rewrites 98.7%

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

14. Taylor expanded in lambda1 around inf

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

    1. lower-fma.f64 N/A

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

16. Applied rewrites 98.7%

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

17. Taylor expanded in lambda1 around inf

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

    1. lower-fma.f64 N/A

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

19. Applied rewrites 98.7%

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

Alternative 5: 88.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\\ t_2 := \sin \left(0.5 \cdot \phi_1\right)\\ t_3 := {\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot t\_2 - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\\ t_4 := \cos \phi_1 \cdot \cos \phi_2\\ t_5 := \left(t\_4 \cdot t\_0\right) \cdot t\_0\\ t_6 := t\_3 + t\_5\\ t_7 := {\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(\phi_1 \cdot -0.5\right), \sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + t\_5\\ \mathbf{if}\;\phi_2 \leq -2.45 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_6}}{\sqrt{1 - t\_6}}\right)\\ \mathbf{elif}\;\phi_2 \leq 8 \cdot 10^{-24}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3 + \left(t\_4 \cdot t\_1\right) \cdot t\_1}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {t\_2}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_7}}{\sqrt{1 - t\_7}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (fma
          (sin (* lambda1 0.5))
          (cos (* lambda2 0.5))
          (* (cos (* lambda1 0.5)) (sin (* lambda2 -0.5)))))
        (t_2 (sin (* 0.5 phi1)))
        (t_3
         (pow
          (-
           (* (cos (* phi2 -0.5)) t_2)
           (* (sin (* 0.5 phi2)) (cos (* 0.5 phi1))))
          2.0))
        (t_4 (* (cos phi1) (cos phi2)))
        (t_5 (* (* t_4 t_0) t_0))
        (t_6 (+ t_3 t_5))
        (t_7
         (+
          (pow
           (fma
            (sin (* -0.5 phi2))
            (cos (* phi1 -0.5))
            (* (sin (* phi1 0.5)) (cos (* 0.5 phi2))))
           2.0)
          t_5)))
   (if (<= phi2 -2.45e-8)
     (* R (* 2.0 (atan2 (sqrt t_6) (sqrt (- 1.0 t_6)))))
     (if (<= phi2 8e-24)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ t_3 (* (* t_4 t_1) t_1)))
          (sqrt
           (-
            1.0
            (fma
             (cos phi1)
             (pow
              (fma
               (cos (* 0.5 lambda1))
               (sin (* -0.5 lambda2))
               (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1))))
              2.0)
             (pow t_2 2.0)))))))
       (* R (* 2.0 (atan2 (sqrt t_7) (sqrt (- 1.0 t_7)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = fma(sin((lambda1 * 0.5)), cos((lambda2 * 0.5)), (cos((lambda1 * 0.5)) * sin((lambda2 * -0.5))));
	double t_2 = sin((0.5 * phi1));
	double t_3 = pow(((cos((phi2 * -0.5)) * t_2) - (sin((0.5 * phi2)) * cos((0.5 * phi1)))), 2.0);
	double t_4 = cos(phi1) * cos(phi2);
	double t_5 = (t_4 * t_0) * t_0;
	double t_6 = t_3 + t_5;
	double t_7 = pow(fma(sin((-0.5 * phi2)), cos((phi1 * -0.5)), (sin((phi1 * 0.5)) * cos((0.5 * phi2)))), 2.0) + t_5;
	double tmp;
	if (phi2 <= -2.45e-8) {
		tmp = R * (2.0 * atan2(sqrt(t_6), sqrt((1.0 - t_6))));
	} else if (phi2 <= 8e-24) {
		tmp = R * (2.0 * atan2(sqrt((t_3 + ((t_4 * t_1) * t_1))), sqrt((1.0 - fma(cos(phi1), pow(fma(cos((0.5 * lambda1)), sin((-0.5 * lambda2)), (cos((0.5 * lambda2)) * sin((0.5 * lambda1)))), 2.0), pow(t_2, 2.0))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_7), sqrt((1.0 - t_7))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = fma(sin(Float64(lambda1 * 0.5)), cos(Float64(lambda2 * 0.5)), Float64(cos(Float64(lambda1 * 0.5)) * sin(Float64(lambda2 * -0.5))))
	t_2 = sin(Float64(0.5 * phi1))
	t_3 = Float64(Float64(cos(Float64(phi2 * -0.5)) * t_2) - Float64(sin(Float64(0.5 * phi2)) * cos(Float64(0.5 * phi1)))) ^ 2.0
	t_4 = Float64(cos(phi1) * cos(phi2))
	t_5 = Float64(Float64(t_4 * t_0) * t_0)
	t_6 = Float64(t_3 + t_5)
	t_7 = Float64((fma(sin(Float64(-0.5 * phi2)), cos(Float64(phi1 * -0.5)), Float64(sin(Float64(phi1 * 0.5)) * cos(Float64(0.5 * phi2)))) ^ 2.0) + t_5)
	tmp = 0.0
	if (phi2 <= -2.45e-8)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_6), sqrt(Float64(1.0 - t_6)))));
	elseif (phi2 <= 8e-24)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_3 + Float64(Float64(t_4 * t_1) * t_1))), sqrt(Float64(1.0 - fma(cos(phi1), (fma(cos(Float64(0.5 * lambda1)), sin(Float64(-0.5 * lambda2)), Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1)))) ^ 2.0), (t_2 ^ 2.0)))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_7), sqrt(Float64(1.0 - t_7)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(lambda2 * 0.5), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(lambda2 * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[(N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision] - N[(N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t$95$4 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$3 + t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[(N[Power[N[(N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi1 * -0.5), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$5), $MachinePrecision]}, If[LessEqual[phi2, -2.45e-8], 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[phi2, 8e-24], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$3 + N[(N[(t$95$4 * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$7], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$7), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -2.4500000000000001e-8

   1. Initial program 62.6%

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

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

   3. Applied rewrites 63.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sin-diff N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 78.7%

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

   if -2.4500000000000001e-8 < phi2 < 7.99999999999999939e-24

   1. Initial program 62.6%

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

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

   3. Applied rewrites 63.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sin-diff N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 78.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sub-flip N/A

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

      6. mult-flip N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. sin-sum N/A

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

      13. lift-cos.f64 N/A

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

      14. lower-fma.f64 N/A

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

   7. Applied rewrites 77.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sub-flip N/A

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

      6. mult-flip N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. sin-sum N/A

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

      13. lift-cos.f64 N/A

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

      14. lower-fma.f64 N/A

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

   9. Applied rewrites 79.2%

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

       1. lift-sin.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift--.f64 N/A

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

       4. div-sub N/A

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

       5. sub-flip N/A

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

       6. mult-flip N/A

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

       7. metadata-eval N/A

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

       8. distribute-rgt-neg-out N/A

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

       9. metadata-eval N/A

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

       10. *-commutative N/A

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

       11. lift-*.f64 N/A

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

       12. sin-sum N/A

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

       13. lift-cos.f64 N/A

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

       14. lower-fma.f64 N/A

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

   11. Applied rewrites 78.9%

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

       1. lift-sin.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift--.f64 N/A

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

       4. div-sub N/A

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

       5. sub-flip N/A

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

       6. mult-flip N/A

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

       7. metadata-eval N/A

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

       8. distribute-rgt-neg-out N/A

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

       9. metadata-eval N/A

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

       10. *-commutative N/A

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

       11. lift-*.f64 N/A

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

       12. sin-sum N/A

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

       13. lift-cos.f64 N/A

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

       14. lower-fma.f64 N/A

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

   13. Applied rewrites 98.7%

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

   14. Taylor expanded in phi2 around 0

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

       1. lower--.f64 N/A

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

       2. lower-fma.f64 N/A

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

   16. Applied rewrites 59.9%

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

   if 7.99999999999999939e-24 < phi2

   1. Initial program 62.6%

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

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

   3. Applied rewrites 63.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sin-diff N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 78.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 78.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 78.7%

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

Alternative 6: 88.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}\\ t_1 := t\_0 + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_2\right) \cdot t\_2\\ t_4 := {\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(\phi_1 \cdot -0.5\right), \sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + t\_3\\ t_5 := t\_0 + t\_3\\ \mathbf{if}\;\phi_2 \leq -7.5 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{elif}\;\phi_2 \leq 8 \cdot 10^{-24}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (-
           (* (cos (* phi2 -0.5)) (sin (* 0.5 phi1)))
           (* (sin (* 0.5 phi2)) (cos (* 0.5 phi1))))
          2.0))
        (t_1
         (+
          t_0
          (*
           (cos phi1)
           (pow
            (fma
             (cos (* 0.5 lambda1))
             (sin (* -0.5 lambda2))
             (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1))))
            2.0))))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3 (* (* (* (cos phi1) (cos phi2)) t_2) t_2))
        (t_4
         (+
          (pow
           (fma
            (sin (* -0.5 phi2))
            (cos (* phi1 -0.5))
            (* (sin (* phi1 0.5)) (cos (* 0.5 phi2))))
           2.0)
          t_3))
        (t_5 (+ t_0 t_3)))
   (if (<= phi2 -7.5e-8)
     (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))
     (if (<= phi2 8e-24)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(((cos((phi2 * -0.5)) * sin((0.5 * phi1))) - (sin((0.5 * phi2)) * cos((0.5 * phi1)))), 2.0);
	double t_1 = t_0 + (cos(phi1) * pow(fma(cos((0.5 * lambda1)), sin((-0.5 * lambda2)), (cos((0.5 * lambda2)) * sin((0.5 * lambda1)))), 2.0));
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = ((cos(phi1) * cos(phi2)) * t_2) * t_2;
	double t_4 = pow(fma(sin((-0.5 * phi2)), cos((phi1 * -0.5)), (sin((phi1 * 0.5)) * cos((0.5 * phi2)))), 2.0) + t_3;
	double t_5 = t_0 + t_3;
	double tmp;
	if (phi2 <= -7.5e-8) {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	} else if (phi2 <= 8e-24) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(cos(Float64(phi2 * -0.5)) * sin(Float64(0.5 * phi1))) - Float64(sin(Float64(0.5 * phi2)) * cos(Float64(0.5 * phi1)))) ^ 2.0
	t_1 = Float64(t_0 + Float64(cos(phi1) * (fma(cos(Float64(0.5 * lambda1)), sin(Float64(-0.5 * lambda2)), Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1)))) ^ 2.0)))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_2) * t_2)
	t_4 = Float64((fma(sin(Float64(-0.5 * phi2)), cos(Float64(phi1 * -0.5)), Float64(sin(Float64(phi1 * 0.5)) * cos(Float64(0.5 * phi2)))) ^ 2.0) + t_3)
	t_5 = Float64(t_0 + t_3)
	tmp = 0.0
	if (phi2 <= -7.5e-8)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	elseif (phi2 <= 8e-24)
		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_4), sqrt(Float64(1.0 - t_4)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[(N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[(N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi1 * -0.5), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$0 + t$95$3), $MachinePrecision]}, If[LessEqual[phi2, -7.5e-8], 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, 8e-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], 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]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -7.4999999999999997e-8

   1. Initial program 62.6%

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

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

   3. Applied rewrites 63.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sin-diff N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 78.7%

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

   if -7.4999999999999997e-8 < phi2 < 7.99999999999999939e-24

   1. Initial program 62.6%

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

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

   3. Applied rewrites 63.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sin-diff N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 78.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sub-flip N/A

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

      6. mult-flip N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. sin-sum N/A

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

      13. lift-cos.f64 N/A

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

      14. lower-fma.f64 N/A

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

   7. Applied rewrites 77.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sub-flip N/A

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

      6. mult-flip N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. sin-sum N/A

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

      13. lift-cos.f64 N/A

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

      14. lower-fma.f64 N/A

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

   9. Applied rewrites 79.2%

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

       1. lift-sin.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift--.f64 N/A

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

       4. div-sub N/A

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

       5. sub-flip N/A

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

       6. mult-flip N/A

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

       7. metadata-eval N/A

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

       8. distribute-rgt-neg-out N/A

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

       9. metadata-eval N/A

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

       10. *-commutative N/A

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

       11. lift-*.f64 N/A

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

       12. sin-sum N/A

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

       13. lift-cos.f64 N/A

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

       14. lower-fma.f64 N/A

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

   11. Applied rewrites 78.9%

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

       1. lift-sin.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift--.f64 N/A

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

       4. div-sub N/A

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

       5. sub-flip N/A

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

       6. mult-flip N/A

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

       7. metadata-eval N/A

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

       8. distribute-rgt-neg-out N/A

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

       9. metadata-eval N/A

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

       10. *-commutative N/A

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

       11. lift-*.f64 N/A

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

       12. sin-sum N/A

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

       13. lift-cos.f64 N/A

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

       14. lower-fma.f64 N/A

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

   13. Applied rewrites 98.7%

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

   14. Taylor expanded in phi2 around 0

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

       1. lower-*.f64 N/A

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

       2. lower-cos.f64 N/A

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

       3. lower-pow.f64 N/A

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

   16. Applied rewrites 69.5%

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

   17. Taylor expanded in phi2 around 0

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

       1. lower-*.f64 N/A

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

       2. lower-cos.f64 N/A

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

       3. lower-pow.f64 N/A

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

   19. Applied rewrites 66.1%

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

   if 7.99999999999999939e-24 < phi2

   1. Initial program 62.6%

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

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

   3. Applied rewrites 63.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sin-diff N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 78.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 78.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 78.7%

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

Alternative 7: 87.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sin \left(0.5 \cdot \phi_1\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {t\_1}^{2}\right)\\ t_3 := \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0\\ t_4 := {\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(\phi_1 \cdot -0.5\right), \sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + t\_3\\ t_5 := {\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot t\_1 - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + t\_3\\ \mathbf{if}\;\phi_2 \leq -1.55 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{elif}\;\phi_2 \leq 8.2 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (sin (* 0.5 phi1)))
        (t_2
         (fma
          (cos phi1)
          (pow
           (fma
            (cos (* 0.5 lambda1))
            (sin (* -0.5 lambda2))
            (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1))))
           2.0)
          (pow t_1 2.0)))
        (t_3 (* (* (* (cos phi1) (cos phi2)) t_0) t_0))
        (t_4
         (+
          (pow
           (fma
            (sin (* -0.5 phi2))
            (cos (* phi1 -0.5))
            (* (sin (* phi1 0.5)) (cos (* 0.5 phi2))))
           2.0)
          t_3))
        (t_5
         (+
          (pow
           (-
            (* (cos (* phi2 -0.5)) t_1)
            (* (sin (* 0.5 phi2)) (cos (* 0.5 phi1))))
           2.0)
          t_3)))
   (if (<= phi2 -1.55e-8)
     (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))
     (if (<= phi2 8.2e-25)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
       (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = sin((0.5 * phi1));
	double t_2 = fma(cos(phi1), pow(fma(cos((0.5 * lambda1)), sin((-0.5 * lambda2)), (cos((0.5 * lambda2)) * sin((0.5 * lambda1)))), 2.0), pow(t_1, 2.0));
	double t_3 = ((cos(phi1) * cos(phi2)) * t_0) * t_0;
	double t_4 = pow(fma(sin((-0.5 * phi2)), cos((phi1 * -0.5)), (sin((phi1 * 0.5)) * cos((0.5 * phi2)))), 2.0) + t_3;
	double t_5 = pow(((cos((phi2 * -0.5)) * t_1) - (sin((0.5 * phi2)) * cos((0.5 * phi1)))), 2.0) + t_3;
	double tmp;
	if (phi2 <= -1.55e-8) {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	} else if (phi2 <= 8.2e-25) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sin(Float64(0.5 * phi1))
	t_2 = fma(cos(phi1), (fma(cos(Float64(0.5 * lambda1)), sin(Float64(-0.5 * lambda2)), Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1)))) ^ 2.0), (t_1 ^ 2.0))
	t_3 = Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0)
	t_4 = Float64((fma(sin(Float64(-0.5 * phi2)), cos(Float64(phi1 * -0.5)), Float64(sin(Float64(phi1 * 0.5)) * cos(Float64(0.5 * phi2)))) ^ 2.0) + t_3)
	t_5 = Float64((Float64(Float64(cos(Float64(phi2 * -0.5)) * t_1) - Float64(sin(Float64(0.5 * phi2)) * cos(Float64(0.5 * phi1)))) ^ 2.0) + t_3)
	tmp = 0.0
	if (phi2 <= -1.55e-8)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	elseif (phi2 <= 8.2e-25)
		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(t_4), sqrt(Float64(1.0 - t_4)))));
	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[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[(N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi1 * -0.5), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[N[(N[(N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] - N[(N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$3), $MachinePrecision]}, If[LessEqual[phi2, -1.55e-8], 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, 8.2e-25], 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[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.55e-8

   1. Initial program 62.6%

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

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

   3. Applied rewrites 63.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sin-diff N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 78.7%

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

   if -1.55e-8 < phi2 < 8.19999999999999974e-25

   1. Initial program 62.6%

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

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

   3. Applied rewrites 63.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sin-diff N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 78.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sub-flip N/A

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

      6. mult-flip N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. sin-sum N/A

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

      13. lift-cos.f64 N/A

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

      14. lower-fma.f64 N/A

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

   7. Applied rewrites 77.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sub-flip N/A

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

      6. mult-flip N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. sin-sum N/A

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

      13. lift-cos.f64 N/A

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

      14. lower-fma.f64 N/A

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

   9. Applied rewrites 79.2%

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

       1. lift-sin.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift--.f64 N/A

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

       4. div-sub N/A

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

       5. sub-flip N/A

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

       6. mult-flip N/A

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

       7. metadata-eval N/A

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

       8. distribute-rgt-neg-out N/A

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

       9. metadata-eval N/A

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

       10. *-commutative N/A

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

       11. lift-*.f64 N/A

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

       12. sin-sum N/A

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

       13. lift-cos.f64 N/A

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

       14. lower-fma.f64 N/A

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

   11. Applied rewrites 78.9%

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

       1. lift-sin.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift--.f64 N/A

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

       4. div-sub N/A

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

       5. sub-flip N/A

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

       6. mult-flip N/A

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

       7. metadata-eval N/A

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

       8. distribute-rgt-neg-out N/A

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

       9. metadata-eval N/A

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

       10. *-commutative N/A

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

       11. lift-*.f64 N/A

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

       12. sin-sum N/A

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

       13. lift-cos.f64 N/A

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

       14. lower-fma.f64 N/A

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

   13. Applied rewrites 98.7%

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

   14. Taylor expanded in phi2 around 0

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

       1. lower-fma.f64 N/A

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

   16. Applied rewrites 57.8%

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

   17. Taylor expanded in phi2 around 0

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

       1. lower-fma.f64 N/A

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

   19. Applied rewrites 56.9%

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

   if 8.19999999999999974e-25 < phi2

   1. Initial program 62.6%

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

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

   3. Applied rewrites 63.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sin-diff N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 78.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 78.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   9. Applied rewrites 78.7%

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

Alternative 8: 87.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sin \left(0.5 \cdot \phi_1\right)\\ t_2 := {\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot t\_1 - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\\ t_3 := \mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {t\_1}^{2}\right)\\ t_4 := {\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(\phi_1 \cdot -0.5\right), \sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0\\ \mathbf{if}\;\phi_2 \leq -1.55 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{elif}\;\phi_2 \leq 8.2 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (sin (* 0.5 phi1)))
        (t_2
         (+
          (pow
           (-
            (* (cos (* phi2 -0.5)) t_1)
            (* (sin (* 0.5 phi2)) (cos (* 0.5 phi1))))
           2.0)
          (*
           (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5)))))
           (* (cos phi2) (cos phi1)))))
        (t_3
         (fma
          (cos phi1)
          (pow
           (fma
            (cos (* 0.5 lambda1))
            (sin (* -0.5 lambda2))
            (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1))))
           2.0)
          (pow t_1 2.0)))
        (t_4
         (+
          (pow
           (fma
            (sin (* -0.5 phi2))
            (cos (* phi1 -0.5))
            (* (sin (* phi1 0.5)) (cos (* 0.5 phi2))))
           2.0)
          (* (* (* (cos phi1) (cos phi2)) t_0) t_0))))
   (if (<= phi2 -1.55e-8)
     (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
     (if (<= phi2 8.2e-25)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = sin((0.5 * phi1));
	double t_2 = pow(((cos((phi2 * -0.5)) * t_1) - (sin((0.5 * phi2)) * cos((0.5 * phi1)))), 2.0) + ((0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))))) * (cos(phi2) * cos(phi1)));
	double t_3 = fma(cos(phi1), pow(fma(cos((0.5 * lambda1)), sin((-0.5 * lambda2)), (cos((0.5 * lambda2)) * sin((0.5 * lambda1)))), 2.0), pow(t_1, 2.0));
	double t_4 = pow(fma(sin((-0.5 * phi2)), cos((phi1 * -0.5)), (sin((phi1 * 0.5)) * cos((0.5 * phi2)))), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	double tmp;
	if (phi2 <= -1.55e-8) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else if (phi2 <= 8.2e-25) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sin(Float64(0.5 * phi1))
	t_2 = Float64((Float64(Float64(cos(Float64(phi2 * -0.5)) * t_1) - Float64(sin(Float64(0.5 * phi2)) * cos(Float64(0.5 * phi1)))) ^ 2.0) + Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5))))) * Float64(cos(phi2) * cos(phi1))))
	t_3 = fma(cos(phi1), (fma(cos(Float64(0.5 * lambda1)), sin(Float64(-0.5 * lambda2)), Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1)))) ^ 2.0), (t_1 ^ 2.0))
	t_4 = Float64((fma(sin(Float64(-0.5 * phi2)), cos(Float64(phi1 * -0.5)), Float64(sin(Float64(phi1 * 0.5)) * cos(Float64(0.5 * phi2)))) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))
	tmp = 0.0
	if (phi2 <= -1.55e-8)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	elseif (phi2 <= 8.2e-25)
		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_4), sqrt(Float64(1.0 - t_4)))));
	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[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(N[(N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] * t$95$1), $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[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[(N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi1 * -0.5), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[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]}, If[LessEqual[phi2, -1.55e-8], 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, 8.2e-25], 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$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.55e-8

   1. Initial program 62.6%

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

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

   3. Applied rewrites 63.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sin-diff N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 78.7%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 78.7

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

   7. Applied rewrites 76.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. pow2 N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 76.3

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

   9. Applied rewrites 76.3%

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

   if -1.55e-8 < phi2 < 8.19999999999999974e-25

   1. Initial program 62.6%

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

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

   3. Applied rewrites 63.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sin-diff N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 78.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sub-flip N/A

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

      6. mult-flip N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. sin-sum N/A

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

      13. lift-cos.f64 N/A

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

      14. lower-fma.f64 N/A

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

   7. Applied rewrites 77.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sub-flip N/A

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

      6. mult-flip N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-neg-out N/A

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

      9. metadata-eval N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. sin-sum N/A

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

      13. lift-cos.f64 N/A

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

      14. lower-fma.f64 N/A

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

   9. Applied rewrites 79.2%

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

       1. lift-sin.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift--.f64 N/A

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

       4. div-sub N/A

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

       5. sub-flip N/A

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

       6. mult-flip N/A

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

       7. metadata-eval N/A

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

       8. distribute-rgt-neg-out N/A

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

       9. metadata-eval N/A

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

       10. *-commutative N/A

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

       11. lift-*.f64 N/A

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

       12. sin-sum N/A

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

       13. lift-cos.f64 N/A

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

       14. lower-fma.f64 N/A

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

   11. Applied rewrites 78.9%

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

       1. lift-sin.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift--.f64 N/A

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

       4. div-sub N/A

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

       5. sub-flip N/A

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

       6. mult-flip N/A

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

       7. metadata-eval N/A

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

       8. distribute-rgt-neg-out N/A

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

       9. metadata-eval N/A

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

       10. *-commutative N/A

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

       11. lift-*.f64 N/A

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

       12. sin-sum N/A

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

       13. lift-cos.f64 N/A

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

       14. lower-fma.f64 N/A

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

   13. Applied rewrites 98.7%

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

   14. Taylor expanded in phi2 around 0

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

       1. lower-fma.f64 N/A

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

   16. Applied rewrites 57.8%

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

   17. Taylor expanded in phi2 around 0

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

       1. lower-fma.f64 N/A

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

   19. Applied rewrites 56.9%

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

   if 8.19999999999999974e-25 < phi2

   1. Initial program 62.6%

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

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

   3. Applied rewrites 63.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sin-diff N/A

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

      6. lower--.f64 N/A

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

   5. Applied rewrites 78.7%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

   7. Applied rewrites 78.7%

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

      1. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\phi_1 \cdot \frac{-1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\phi_1 \cdot \frac{-1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\phi_1 \cdot \frac{-1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \left(\mathsf{neg}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\phi_1 \cdot \frac{-1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\left(\mathsf{neg}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\phi_1 \cdot \frac{-1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\mathsf{neg}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right)\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(\phi_1 \cdot -0.5\right), \sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(\phi_1 \cdot -0.5\right), \sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 87.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_1\right)\\ t_1 := \mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {t\_0}^{2}\right)\\ t_2 := {\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot t\_0 - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_2 \leq -1.55 \cdot 10^{-8}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_2 \leq 1.1 \cdot 10^{-24}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 phi1)))
        (t_1
         (fma
          (cos phi1)
          (pow
           (fma
            (cos (* 0.5 lambda1))
            (sin (* -0.5 lambda2))
            (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1))))
           2.0)
          (pow t_0 2.0)))
        (t_2
         (+
          (pow
           (-
            (* (cos (* phi2 -0.5)) t_0)
            (* (sin (* 0.5 phi2)) (cos (* 0.5 phi1))))
           2.0)
          (*
           (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5)))))
           (* (cos phi2) (cos phi1)))))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= phi2 -1.55e-8)
     t_3
     (if (<= phi2 1.1e-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 = sin((0.5 * phi1));
	double t_1 = fma(cos(phi1), pow(fma(cos((0.5 * lambda1)), sin((-0.5 * lambda2)), (cos((0.5 * lambda2)) * sin((0.5 * lambda1)))), 2.0), pow(t_0, 2.0));
	double t_2 = pow(((cos((phi2 * -0.5)) * t_0) - (sin((0.5 * phi2)) * cos((0.5 * phi1)))), 2.0) + ((0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))))) * (cos(phi2) * cos(phi1)));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi2 <= -1.55e-8) {
		tmp = t_3;
	} else if (phi2 <= 1.1e-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 = sin(Float64(0.5 * phi1))
	t_1 = fma(cos(phi1), (fma(cos(Float64(0.5 * lambda1)), sin(Float64(-0.5 * lambda2)), Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1)))) ^ 2.0), (t_0 ^ 2.0))
	t_2 = Float64((Float64(Float64(cos(Float64(phi2 * -0.5)) * t_0) - Float64(sin(Float64(0.5 * phi2)) * cos(Float64(0.5 * phi1)))) ^ 2.0) + Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5))))) * Float64(cos(phi2) * cos(phi1))))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi2 <= -1.55e-8)
		tmp = t_3;
	elseif (phi2 <= 1.1e-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[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(N[(N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] * t$95$0), $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[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.55e-8], t$95$3, If[LessEqual[phi2, 1.1e-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 < -1.55e-8 or 1.10000000000000001e-24 < phi2

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-*.f64 78.7

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 76.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      11. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}\right)}}\right) \]

      12. lower-*.f64 76.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}\right)}}\right) \]

   9. Applied rewrites 76.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}\right)}}\right) \]

   if -1.55e-8 < phi2 < 1.10000000000000001e-24

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sub-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. sin-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sub-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right)}}\right) \]

       6. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right)}}\right) \]

       7. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)}}\right) \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)}}\right) \]

       9. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right)}}\right) \]

       10. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right)}}\right) \]

       12. sin-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 87.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_1\right)\\ t_1 := \mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {t\_0}^{2}\right)\\ t_2 := {\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot t\_0 - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_2 \leq -1.55 \cdot 10^{-8}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_2 \leq 1.1 \cdot 10^{-24}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 phi1)))
        (t_1
         (fma
          (cos phi1)
          (pow
           (fma
            (cos (* 0.5 lambda1))
            (sin (* -0.5 lambda2))
            (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1))))
           2.0)
          (pow t_0 2.0)))
        (t_2
         (+
          (pow
           (-
            (* (cos (* phi2 -0.5)) t_0)
            (* (sin (* 0.5 phi2)) (cos (* 0.5 phi1))))
           2.0)
          (*
           (cos phi2)
           (*
            (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5)))))
            (cos phi1)))))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= phi2 -1.55e-8)
     t_3
     (if (<= phi2 1.1e-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 = sin((0.5 * phi1));
	double t_1 = fma(cos(phi1), pow(fma(cos((0.5 * lambda1)), sin((-0.5 * lambda2)), (cos((0.5 * lambda2)) * sin((0.5 * lambda1)))), 2.0), pow(t_0, 2.0));
	double t_2 = pow(((cos((phi2 * -0.5)) * t_0) - (sin((0.5 * phi2)) * cos((0.5 * phi1)))), 2.0) + (cos(phi2) * ((0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))))) * cos(phi1)));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi2 <= -1.55e-8) {
		tmp = t_3;
	} else if (phi2 <= 1.1e-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 = sin(Float64(0.5 * phi1))
	t_1 = fma(cos(phi1), (fma(cos(Float64(0.5 * lambda1)), sin(Float64(-0.5 * lambda2)), Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1)))) ^ 2.0), (t_0 ^ 2.0))
	t_2 = Float64((Float64(Float64(cos(Float64(phi2 * -0.5)) * t_0) - Float64(sin(Float64(0.5 * phi2)) * cos(Float64(0.5 * phi1)))) ^ 2.0) + Float64(cos(phi2) * Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5))))) * cos(phi1))))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi2 <= -1.55e-8)
		tmp = t_3;
	elseif (phi2 <= 1.1e-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[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(N[(N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] * t$95$0), $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[Cos[phi2], $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]}, 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, -1.55e-8], t$95$3, If[LessEqual[phi2, 1.1e-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 < -1.55e-8 or 1.10000000000000001e-24 < phi2

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. associate-*l* N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \color{blue}{\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-*.f64 78.7

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \color{blue}{\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 76.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_2 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. associate-*l* N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      6. pow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)}}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot {\sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}^{2}\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot {\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}^{2}\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}^{2}\right)}}\right) \]

      11. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot {\sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      13. associate-*l* N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}\right)}}\right) \]

      14. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}\right)}}\right) \]

      15. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \left(\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \color{blue}{\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1\right)}\right)}}\right) \]

      16. lower-*.f64 76.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \color{blue}{\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1\right)}\right)}}\right) \]

   9. Applied rewrites 76.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_2 \cdot \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1\right)}\right)}}\right) \]

   if -1.55e-8 < phi2 < 1.10000000000000001e-24

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sub-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. sin-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sub-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right)}}\right) \]

       6. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right)}}\right) \]

       7. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)}}\right) \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)}}\right) \]

       9. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right)}}\right) \]

       10. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right)}}\right) \]

       12. sin-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 75.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}\\ t_1 := \cos \left(0.5 \cdot \phi_1\right)\\ t_2 := \sin \left(0.5 \cdot \phi_2\right)\\ t_3 := \sin \left(0.5 \cdot \phi_1\right)\\ t_4 := \mathsf{fma}\left(\cos \phi_1, t\_0, {t\_3}^{2}\right)\\ t_5 := \mathsf{fma}\left(\cos \phi_2, t\_0, {t\_2}^{2}\right)\\ t_6 := \mathsf{fma}\left(\cos \phi_1, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot t\_3 - t\_1 \cdot t\_2\right)}^{2}\right)\\ t_7 := {\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot t\_3 - t\_2 \cdot t\_1\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ \mathbf{if}\;\phi_2 \leq -2.5 \cdot 10^{+55}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_6}}{\sqrt{1 - t\_6}}\right)\\ \mathbf{elif}\;\phi_2 \leq -2.6 \cdot 10^{-48}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{elif}\;\phi_2 \leq 3.8 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_7}}{\sqrt{1 - t\_7}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (fma
           (cos (* 0.5 lambda1))
           (sin (* -0.5 lambda2))
           (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1))))
          2.0))
        (t_1 (cos (* 0.5 phi1)))
        (t_2 (sin (* 0.5 phi2)))
        (t_3 (sin (* 0.5 phi1)))
        (t_4 (fma (cos phi1) t_0 (pow t_3 2.0)))
        (t_5 (fma (cos phi2) t_0 (pow t_2 2.0)))
        (t_6
         (fma
          (cos phi1)
          (* (- 0.5 (* 0.5 (cos (* 2.0 (* lambda2 -0.5))))) (cos phi2))
          (pow (- (* (cos (* -0.5 phi2)) t_3) (* t_1 t_2)) 2.0)))
        (t_7
         (+
          (pow (- (* (cos (* phi2 -0.5)) t_3) (* t_2 t_1)) 2.0)
          (* (cos phi2) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)))))
   (if (<= phi2 -2.5e+55)
     (* R (* 2.0 (atan2 (sqrt t_6) (sqrt (- 1.0 t_6)))))
     (if (<= phi2 -2.6e-48)
       (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))
       (if (<= phi2 3.8e-8)
         (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
         (* R (* 2.0 (atan2 (sqrt t_7) (sqrt (- 1.0 t_7))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(fma(cos((0.5 * lambda1)), sin((-0.5 * lambda2)), (cos((0.5 * lambda2)) * sin((0.5 * lambda1)))), 2.0);
	double t_1 = cos((0.5 * phi1));
	double t_2 = sin((0.5 * phi2));
	double t_3 = sin((0.5 * phi1));
	double t_4 = fma(cos(phi1), t_0, pow(t_3, 2.0));
	double t_5 = fma(cos(phi2), t_0, pow(t_2, 2.0));
	double t_6 = fma(cos(phi1), ((0.5 - (0.5 * cos((2.0 * (lambda2 * -0.5))))) * cos(phi2)), pow(((cos((-0.5 * phi2)) * t_3) - (t_1 * t_2)), 2.0));
	double t_7 = pow(((cos((phi2 * -0.5)) * t_3) - (t_2 * t_1)), 2.0) + (cos(phi2) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0));
	double tmp;
	if (phi2 <= -2.5e+55) {
		tmp = R * (2.0 * atan2(sqrt(t_6), sqrt((1.0 - t_6))));
	} else if (phi2 <= -2.6e-48) {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	} else if (phi2 <= 3.8e-8) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_7), sqrt((1.0 - t_7))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(Float64(0.5 * lambda1)), sin(Float64(-0.5 * lambda2)), Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1)))) ^ 2.0
	t_1 = cos(Float64(0.5 * phi1))
	t_2 = sin(Float64(0.5 * phi2))
	t_3 = sin(Float64(0.5 * phi1))
	t_4 = fma(cos(phi1), t_0, (t_3 ^ 2.0))
	t_5 = fma(cos(phi2), t_0, (t_2 ^ 2.0))
	t_6 = fma(cos(phi1), Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(lambda2 * -0.5))))) * cos(phi2)), (Float64(Float64(cos(Float64(-0.5 * phi2)) * t_3) - Float64(t_1 * t_2)) ^ 2.0))
	t_7 = Float64((Float64(Float64(cos(Float64(phi2 * -0.5)) * t_3) - Float64(t_2 * t_1)) ^ 2.0) + Float64(cos(phi2) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)))
	tmp = 0.0
	if (phi2 <= -2.5e+55)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_6), sqrt(Float64(1.0 - t_6)))));
	elseif (phi2 <= -2.6e-48)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	elseif (phi2 <= 3.8e-8)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_7), sqrt(Float64(1.0 - t_7)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(lambda2 * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision] - N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[(N[Power[N[(N[(N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision] - N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -2.5e+55], 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[phi2, -2.6e-48], 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, 3.8e-8], 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[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$7], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$7), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if phi2 < -2.50000000000000023e55

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 56.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 56.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \color{blue}{\cos \phi_2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 56.2

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(-0.5 \cdot \lambda_2\right)}^{2} \cdot \color{blue}{\cos \phi_2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \cos \color{blue}{\phi_2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \cos \color{blue}{\phi_2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. sqr-sin-a N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \cos \color{blue}{\phi_2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       9. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \cos \color{blue}{\phi_2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       10. cos-2 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) - \sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       11. cos-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2 + \frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2 + \frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       13. cos-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) - \sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       14. cos-2 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       15. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       16. lower-*.f64 55.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       18. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       19. lift-*.f64 55.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 55.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \color{blue}{\cos \phi_2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]
   14. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \color{blue}{\cos \phi_2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 55.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(-0.5 \cdot \lambda_2\right)}^{2} \cdot \color{blue}{\cos \phi_2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2} \cdot \cos \color{blue}{\phi_2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \cos \color{blue}{\phi_2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. sqr-sin-a N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \cos \color{blue}{\phi_2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       9. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \cos \color{blue}{\phi_2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       10. cos-2 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) - \sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       11. cos-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2 + \frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2 + \frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       13. cos-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) - \sin \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       14. cos-2 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       15. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       16. lower-*.f64 55.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       18. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       19. lift-*.f64 55.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 55.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \cos \phi_2, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \color{blue}{\cos \phi_2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   if -2.50000000000000023e55 < phi2 < -2.59999999999999987e-48

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sub-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. sin-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sub-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right)}}\right) \]

       6. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right)}}\right) \]

       7. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)}}\right) \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)}}\right) \]

       9. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right)}}\right) \]

       10. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right)}}\right) \]

       12. sin-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   if -2.59999999999999987e-48 < phi2 < 3.80000000000000028e-8

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sub-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. sin-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sub-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right)}}\right) \]

       6. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right)}}\right) \]

       7. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)}}\right) \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)}}\right) \]

       9. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right)}}\right) \]

       10. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right)}}\right) \]

       12. sin-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if 3.80000000000000028e-8 < phi2

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 58.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 58.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

       4. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       6. lower--.f64 55.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 55.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 12: 75.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}\\ t_1 := \sin \left(0.5 \cdot \phi_2\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_2, t\_0, {t\_1}^{2}\right)\\ t_3 := \sin \left(0.5 \cdot \phi_1\right)\\ t_4 := {\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot t\_3 - t\_1 \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_5 := \mathsf{fma}\left(\cos \phi_1, t\_0, {t\_3}^{2}\right)\\ \mathbf{if}\;\phi_2 \leq -2.6 \cdot 10^{-48}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{elif}\;\phi_2 \leq 3.8 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (fma
           (cos (* 0.5 lambda1))
           (sin (* -0.5 lambda2))
           (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1))))
          2.0))
        (t_1 (sin (* 0.5 phi2)))
        (t_2 (fma (cos phi2) t_0 (pow t_1 2.0)))
        (t_3 (sin (* 0.5 phi1)))
        (t_4
         (+
          (pow (- (* (cos (* phi2 -0.5)) t_3) (* t_1 (cos (* 0.5 phi1)))) 2.0)
          (* (cos phi2) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))))
        (t_5 (fma (cos phi1) t_0 (pow t_3 2.0))))
   (if (<= phi2 -2.6e-48)
     (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
     (if (<= phi2 3.8e-8)
       (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))
       (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(fma(cos((0.5 * lambda1)), sin((-0.5 * lambda2)), (cos((0.5 * lambda2)) * sin((0.5 * lambda1)))), 2.0);
	double t_1 = sin((0.5 * phi2));
	double t_2 = fma(cos(phi2), t_0, pow(t_1, 2.0));
	double t_3 = sin((0.5 * phi1));
	double t_4 = pow(((cos((phi2 * -0.5)) * t_3) - (t_1 * cos((0.5 * phi1)))), 2.0) + (cos(phi2) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0));
	double t_5 = fma(cos(phi1), t_0, pow(t_3, 2.0));
	double tmp;
	if (phi2 <= -2.6e-48) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else if (phi2 <= 3.8e-8) {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(Float64(0.5 * lambda1)), sin(Float64(-0.5 * lambda2)), Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1)))) ^ 2.0
	t_1 = sin(Float64(0.5 * phi2))
	t_2 = fma(cos(phi2), t_0, (t_1 ^ 2.0))
	t_3 = sin(Float64(0.5 * phi1))
	t_4 = Float64((Float64(Float64(cos(Float64(phi2 * -0.5)) * t_3) - Float64(t_1 * cos(Float64(0.5 * phi1)))) ^ 2.0) + Float64(cos(phi2) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)))
	t_5 = fma(cos(phi1), t_0, (t_3 ^ 2.0))
	tmp = 0.0
	if (phi2 <= -2.6e-48)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	elseif (phi2 <= 3.8e-8)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[(N[(N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision] - N[(t$95$1 * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -2.6e-48], 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, 3.8e-8], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$5], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(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]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -2.59999999999999987e-48

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sub-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. sin-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sub-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right)}}\right) \]

       6. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right)}}\right) \]

       7. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)}}\right) \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)}}\right) \]

       9. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right)}}\right) \]

       10. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right)}}\right) \]

       12. sin-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   if -2.59999999999999987e-48 < phi2 < 3.80000000000000028e-8

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sub-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. sin-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sub-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right)}}\right) \]

       6. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right)}}\right) \]

       7. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)}}\right) \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)}}\right) \]

       9. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right)}}\right) \]

       10. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right)}}\right) \]

       12. sin-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if 3.80000000000000028e-8 < phi2

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 58.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 58.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

       4. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       6. lower--.f64 55.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 55.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 75.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}\\ t_1 := {\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(\phi_1 \cdot -0.5\right), \sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)\\ t_3 := \mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ \mathbf{if}\;\phi_1 \leq -3.4 \cdot 10^{-34}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{elif}\;\phi_1 \leq 3.9 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (fma
           (cos (* 0.5 lambda1))
           (sin (* -0.5 lambda2))
           (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1))))
          2.0))
        (t_1
         (+
          (pow
           (fma
            (sin (* -0.5 phi2))
            (cos (* phi1 -0.5))
            (* (sin (* phi1 0.5)) (cos (* 0.5 phi2))))
           2.0)
          (* (cos phi1) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))))
        (t_2 (fma (cos phi2) t_0 (pow (sin (* 0.5 phi2)) 2.0)))
        (t_3 (fma (cos phi1) t_0 (pow (sin (* 0.5 phi1)) 2.0))))
   (if (<= phi1 -3.4e-34)
     (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
     (if (<= phi1 3.9e-14)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
       (* 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 = pow(fma(cos((0.5 * lambda1)), sin((-0.5 * lambda2)), (cos((0.5 * lambda2)) * sin((0.5 * lambda1)))), 2.0);
	double t_1 = pow(fma(sin((-0.5 * phi2)), cos((phi1 * -0.5)), (sin((phi1 * 0.5)) * cos((0.5 * phi2)))), 2.0) + (cos(phi1) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0));
	double t_2 = fma(cos(phi2), t_0, pow(sin((0.5 * phi2)), 2.0));
	double t_3 = fma(cos(phi1), t_0, pow(sin((0.5 * phi1)), 2.0));
	double tmp;
	if (phi1 <= -3.4e-34) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else if (phi1 <= 3.9e-14) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(Float64(0.5 * lambda1)), sin(Float64(-0.5 * lambda2)), Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1)))) ^ 2.0
	t_1 = Float64((fma(sin(Float64(-0.5 * phi2)), cos(Float64(phi1 * -0.5)), Float64(sin(Float64(phi1 * 0.5)) * cos(Float64(0.5 * phi2)))) ^ 2.0) + Float64(cos(phi1) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)))
	t_2 = fma(cos(phi2), t_0, (sin(Float64(0.5 * phi2)) ^ 2.0))
	t_3 = fma(cos(phi1), t_0, (sin(Float64(0.5 * phi1)) ^ 2.0))
	tmp = 0.0
	if (phi1 <= -3.4e-34)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	elseif (phi1 <= 3.9e-14)
		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(t_1), sqrt(Float64(1.0 - t_1)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi1 * -0.5), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$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[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -3.4e-34], 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, 3.9e-14], 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[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.4000000000000001e-34

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sub-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. sin-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sub-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right)}}\right) \]

       6. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right)}}\right) \]

       7. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)}}\right) \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)}}\right) \]

       9. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right)}}\right) \]

       10. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right)}}\right) \]

       12. sin-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if -3.4000000000000001e-34 < phi1 < 3.8999999999999998e-14

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sub-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. sin-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sub-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right)}}\right) \]

       6. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right)}}\right) \]

       7. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)}}\right) \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)}}\right) \]

       9. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right)}}\right) \]

       10. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right)}}\right) \]

       12. sin-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   if 3.8999999999999998e-14 < phi1

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 59.0

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 59.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

       4. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       6. lower--.f64 55.8

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 55.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   12. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       3. fp-cancel-sub-sign-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \left(\mathsf{neg}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       4. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\left(\mathsf{neg}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\mathsf{neg}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right)\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 55.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(\phi_1 \cdot -0.5\right), \sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]
   14. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\phi_1 \cdot \frac{-1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\phi_1 \cdot \frac{-1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       3. fp-cancel-sub-sign-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\phi_1 \cdot \frac{-1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) + \left(\mathsf{neg}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       4. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\phi_1 \cdot \frac{-1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\color{blue}{\left(\left(\mathsf{neg}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right) + \cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \left(\phi_1 \cdot \frac{-1}{2}\right), \sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\mathsf{neg}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right)\right), \cos \left(\frac{1}{2} \cdot \phi_1\right), \cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 55.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(\phi_1 \cdot -0.5\right), \sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(-0.5 \cdot \phi_2\right), \cos \left(\phi_1 \cdot -0.5\right), \sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 75.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}\\ t_1 := \sin \left(0.5 \cdot \phi_1\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_1, t\_0, {t\_1}^{2}\right)\\ t_3 := \sin \left(0.5 \cdot \phi_2\right)\\ t_4 := {\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot t\_1 - t\_3 \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_1\\ t_5 := \mathsf{fma}\left(\cos \phi_2, t\_0, {t\_3}^{2}\right)\\ \mathbf{if}\;\phi_1 \leq -3.4 \cdot 10^{-34}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{elif}\;\phi_1 \leq 3.9 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (fma
           (cos (* 0.5 lambda1))
           (sin (* -0.5 lambda2))
           (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1))))
          2.0))
        (t_1 (sin (* 0.5 phi1)))
        (t_2 (fma (cos phi1) t_0 (pow t_1 2.0)))
        (t_3 (sin (* 0.5 phi2)))
        (t_4
         (+
          (pow (- (* (cos (* phi2 -0.5)) t_1) (* t_3 (cos (* 0.5 phi1)))) 2.0)
          (* (fma (cos (* -1.0 (- lambda2 lambda1))) -0.5 0.5) (cos phi1))))
        (t_5 (fma (cos phi2) t_0 (pow t_3 2.0))))
   (if (<= phi1 -3.4e-34)
     (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
     (if (<= phi1 3.9e-14)
       (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))
       (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(fma(cos((0.5 * lambda1)), sin((-0.5 * lambda2)), (cos((0.5 * lambda2)) * sin((0.5 * lambda1)))), 2.0);
	double t_1 = sin((0.5 * phi1));
	double t_2 = fma(cos(phi1), t_0, pow(t_1, 2.0));
	double t_3 = sin((0.5 * phi2));
	double t_4 = pow(((cos((phi2 * -0.5)) * t_1) - (t_3 * cos((0.5 * phi1)))), 2.0) + (fma(cos((-1.0 * (lambda2 - lambda1))), -0.5, 0.5) * cos(phi1));
	double t_5 = fma(cos(phi2), t_0, pow(t_3, 2.0));
	double tmp;
	if (phi1 <= -3.4e-34) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else if (phi1 <= 3.9e-14) {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(Float64(0.5 * lambda1)), sin(Float64(-0.5 * lambda2)), Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1)))) ^ 2.0
	t_1 = sin(Float64(0.5 * phi1))
	t_2 = fma(cos(phi1), t_0, (t_1 ^ 2.0))
	t_3 = sin(Float64(0.5 * phi2))
	t_4 = Float64((Float64(Float64(cos(Float64(phi2 * -0.5)) * t_1) - Float64(t_3 * cos(Float64(0.5 * phi1)))) ^ 2.0) + Float64(fma(cos(Float64(-1.0 * Float64(lambda2 - lambda1))), -0.5, 0.5) * cos(phi1)))
	t_5 = fma(cos(phi2), t_0, (t_3 ^ 2.0))
	tmp = 0.0
	if (phi1 <= -3.4e-34)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	elseif (phi1 <= 3.9e-14)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[(N[(N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] - N[(t$95$3 * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[Cos[N[(-1.0 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -3.4e-34], 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, 3.9e-14], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$5], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(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]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.4000000000000001e-34

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sub-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. sin-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sub-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right)}}\right) \]

       6. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right)}}\right) \]

       7. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)}}\right) \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)}}\right) \]

       9. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right)}}\right) \]

       10. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right)}}\right) \]

       12. sin-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if -3.4000000000000001e-34 < phi1 < 3.8999999999999998e-14

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sub-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. sin-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sub-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right)}}\right) \]

       6. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right)}}\right) \]

       7. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)}}\right) \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)}}\right) \]

       9. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right)}}\right) \]

       10. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right)}}\right) \]

       12. sin-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   if 3.8999999999999998e-14 < phi1

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 59.0

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 59.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

       4. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       6. lower--.f64 55.8

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 55.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   13. Applied rewrites 53.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_1}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 53.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_1}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_1}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 69.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sin \left(0.5 \cdot \phi_1\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {t\_1}^{2}\right)\\ t_3 := \sin \left(0.5 \cdot \phi_2\right)\\ t_4 := \cos \left(0.5 \cdot \phi_1\right)\\ t_5 := {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot t\_1 - t\_4 \cdot t\_3\right)}^{2}\\ \mathbf{if}\;\phi_2 \leq -7.5 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot t\_1 - t\_3 \cdot t\_4\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {t\_3}^{2}\right)}}\right)\\ \mathbf{elif}\;\phi_2 \leq 16000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (sin (* 0.5 phi1)))
        (t_2
         (fma
          (cos phi1)
          (pow
           (fma
            (cos (* 0.5 lambda1))
            (sin (* -0.5 lambda2))
            (* (cos (* 0.5 lambda2)) (sin (* 0.5 lambda1))))
           2.0)
          (pow t_1 2.0)))
        (t_3 (sin (* 0.5 phi2)))
        (t_4 (cos (* 0.5 phi1)))
        (t_5 (pow (- (* (cos (* -0.5 phi2)) t_1) (* t_4 t_3)) 2.0)))
   (if (<= phi2 -7.5e-8)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (pow (- (* (cos (* phi2 -0.5)) t_1) (* t_3 t_4)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_0) t_0)))
        (sqrt
         (-
          1.0
          (fma
           (cos phi2)
           (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
           (pow t_3 2.0)))))))
     (if (<= phi2 16000.0)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
       (* 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 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = sin((0.5 * phi1));
	double t_2 = fma(cos(phi1), pow(fma(cos((0.5 * lambda1)), sin((-0.5 * lambda2)), (cos((0.5 * lambda2)) * sin((0.5 * lambda1)))), 2.0), pow(t_1, 2.0));
	double t_3 = sin((0.5 * phi2));
	double t_4 = cos((0.5 * phi1));
	double t_5 = pow(((cos((-0.5 * phi2)) * t_1) - (t_4 * t_3)), 2.0);
	double tmp;
	if (phi2 <= -7.5e-8) {
		tmp = R * (2.0 * atan2(sqrt((pow(((cos((phi2 * -0.5)) * t_1) - (t_3 * t_4)), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt((1.0 - fma(cos(phi2), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow(t_3, 2.0))))));
	} else if (phi2 <= 16000.0) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} 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 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sin(Float64(0.5 * phi1))
	t_2 = fma(cos(phi1), (fma(cos(Float64(0.5 * lambda1)), sin(Float64(-0.5 * lambda2)), Float64(cos(Float64(0.5 * lambda2)) * sin(Float64(0.5 * lambda1)))) ^ 2.0), (t_1 ^ 2.0))
	t_3 = sin(Float64(0.5 * phi2))
	t_4 = cos(Float64(0.5 * phi1))
	t_5 = Float64(Float64(cos(Float64(-0.5 * phi2)) * t_1) - Float64(t_4 * t_3)) ^ 2.0
	tmp = 0.0
	if (phi2 <= -7.5e-8)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((Float64(Float64(cos(Float64(phi2 * -0.5)) * t_1) - Float64(t_3 * t_4)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(Float64(1.0 - fma(cos(phi2), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (t_3 ^ 2.0)))))));
	elseif (phi2 <= 16000.0)
		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(t_5), sqrt(Float64(1.0 - t_5)))));
	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[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] - N[(t$95$4 * t$95$3), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[phi2, -7.5e-8], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[(N[(N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] - N[(t$95$3 * t$95$4), $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[(1.0 - N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[t$95$3, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 16000.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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 phi2 < -7.4999999999999997e-8

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   7. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      4. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      7. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      9. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      10. lower-*.f64 49.7

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   8. Applied rewrites 49.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   if -7.4999999999999997e-8 < phi2 < 16000

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 77.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Applied rewrites 79.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       5. sub-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       6. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       7. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       9. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       10. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       12. sin-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Applied rewrites 78.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} - \frac{\lambda_2}{2}\right)}\right)}}\right) \]

       5. sub-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\frac{\lambda_2}{2}\right)\right)\right)}\right)}}\right) \]

       6. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\color{blue}{\lambda_2 \cdot \frac{1}{2}}\right)\right)\right)\right)}}\right) \]

       7. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \left(\mathsf{neg}\left(\lambda_2 \cdot \color{blue}{\frac{1}{2}}\right)\right)\right)\right)}}\right) \]

       8. distribute-rgt-neg-out N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\lambda_2 \cdot \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)}}\right) \]

       9. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \lambda_2 \cdot \color{blue}{\frac{-1}{2}}\right)\right)}}\right) \]

       10. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right)}}\right) \]

       11. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{-1}{2} \cdot \lambda_2}\right)\right)}}\right) \]

       12. sin-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right)}}\right) \]

       13. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \lambda_2\right)} + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}}\right) \]

       14. lower-fma.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}\right)}}\right) \]

   13. Applied rewrites 98.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\lambda_2 \cdot \frac{1}{2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\lambda_2 \cdot \frac{-1}{2}\right)\right)\right)}}\right) \]

   16. Applied rewrites 57.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\lambda_2 \cdot 0.5\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right), \sin \left(\frac{-1}{2} \cdot \lambda_2\right), \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right) + \cos \left(\frac{1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 56.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right), \sin \left(-0.5 \cdot \lambda_2\right), \cos \left(0.5 \cdot \lambda_2\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if 16000 < phi2

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 56.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 56.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]

   12. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]
   13. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 34.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   16. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}\right) \]

   17. Applied rewrites 34.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 67.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_1\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{if}\;\phi_1 \leq -0.012:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 4.8 \cdot 10^{+27}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (+
          (pow
           (-
            (* (cos (* phi2 -0.5)) (sin (* 0.5 phi1)))
            (* (sin (* 0.5 phi2)) (cos (* 0.5 phi1))))
           2.0)
          (* (fma (cos (* -1.0 (- lambda2 lambda1))) -0.5 0.5) (cos phi1))))
        (t_2 (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))
   (if (<= phi1 -0.012)
     t_2
     (if (<= phi1 4.8e+27)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (+
            (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
            (* (* (* (cos phi1) (cos phi2)) t_0) t_0)))
          (sqrt
           (fma
            (-
             (*
              (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2)))
              0.5)
             0.5)
            (* (cos phi2) (cos phi1))
            (+ 0.5 (* (cos (- phi2 phi1)) 0.5)))))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = pow(((cos((phi2 * -0.5)) * sin((0.5 * phi1))) - (sin((0.5 * phi2)) * cos((0.5 * phi1)))), 2.0) + (fma(cos((-1.0 * (lambda2 - lambda1))), -0.5, 0.5) * cos(phi1));
	double t_2 = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	double tmp;
	if (phi1 <= -0.012) {
		tmp = t_2;
	} else if (phi1 <= 4.8e+27) {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(fma(((fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2))) * 0.5) - 0.5), (cos(phi2) * cos(phi1)), (0.5 + (cos((phi2 - phi1)) * 0.5))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64((Float64(Float64(cos(Float64(phi2 * -0.5)) * sin(Float64(0.5 * phi1))) - Float64(sin(Float64(0.5 * phi2)) * cos(Float64(0.5 * phi1)))) ^ 2.0) + Float64(fma(cos(Float64(-1.0 * Float64(lambda2 - lambda1))), -0.5, 0.5) * cos(phi1)))
	t_2 = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
	tmp = 0.0
	if (phi1 <= -0.012)
		tmp = t_2;
	elseif (phi1 <= 4.8e+27)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(fma(Float64(Float64(fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))) * 0.5) - 0.5), Float64(cos(phi2) * cos(phi1)), Float64(0.5 + Float64(cos(Float64(phi2 - phi1)) * 0.5)))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(N[(N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $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[Cos[N[(-1.0 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Cos[phi1], $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, -0.012], t$95$2, If[LessEqual[phi1, 4.8e+27], 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[(N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(0.5 + N[(N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.012 or 4.79999999999999995e27 < phi1

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 59.0

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 59.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

       4. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       6. lower--.f64 55.8

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 55.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   13. Applied rewrites 53.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_1}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 53.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_1}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_1}\right)}}\right) \]

   if -0.012 < phi1 < 4.79999999999999995e27

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      3. cos-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)} \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      7. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_1 \cdot \sin \lambda_2}\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_1 \cdot \sin \lambda_2}\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      9. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_1} \cdot \sin \lambda_2\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      10. lower-sin.f64 63.2

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \color{blue}{\sin \lambda_2}\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}\right) \]

   5. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)} \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 66.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_2\right)\\ t_1 := \cos \phi_2 \cdot \cos \phi_1\\ t_2 := \cos \left(0.5 \cdot \phi_1\right)\\ t_3 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_4 := \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_3\right) \cdot t\_3\\ t_5 := \sin \left(0.5 \cdot \phi_1\right)\\ t_6 := {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot t\_5 - t\_2 \cdot t\_0\right)}^{2}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-26}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot t\_5 - t\_0 \cdot t\_2\right)}^{2} + t\_4}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), t\_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_2 - \phi_1\right)\right)\right)\right)}}\right)\\ \mathbf{elif}\;t\_3 \leq 0.00075:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_6}}{\sqrt{1 - t\_6}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t\_4}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, t\_1, 0.5 + \mathsf{fma}\left(\sin \phi_2, \sin \phi_1, t\_1\right) \cdot 0.5\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 phi2)))
        (t_1 (* (cos phi2) (cos phi1)))
        (t_2 (cos (* 0.5 phi1)))
        (t_3 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_4 (* (* (* (cos phi1) (cos phi2)) t_3) t_3))
        (t_5 (sin (* 0.5 phi1)))
        (t_6 (pow (- (* (cos (* -0.5 phi2)) t_5) (* t_2 t_0)) 2.0)))
   (if (<= t_3 -5e-26)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ (pow (- (* (cos (* phi2 -0.5)) t_5) (* t_0 t_2)) 2.0) t_4))
        (sqrt
         (-
          1.0
          (fma
           (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5)))))
           t_1
           (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi2 phi1))))))))))))
     (if (<= t_3 0.00075)
       (* R (* 2.0 (atan2 (sqrt t_6) (sqrt (- 1.0 t_6)))))
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) t_4))
          (sqrt
           (fma
            (- (* (cos (- lambda2 lambda1)) 0.5) 0.5)
            t_1
            (+ 0.5 (* (fma (sin phi2) (sin phi1) t_1) 0.5)))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * phi2));
	double t_1 = cos(phi2) * cos(phi1);
	double t_2 = cos((0.5 * phi1));
	double t_3 = sin(((lambda1 - lambda2) / 2.0));
	double t_4 = ((cos(phi1) * cos(phi2)) * t_3) * t_3;
	double t_5 = sin((0.5 * phi1));
	double t_6 = pow(((cos((-0.5 * phi2)) * t_5) - (t_2 * t_0)), 2.0);
	double tmp;
	if (t_3 <= -5e-26) {
		tmp = R * (2.0 * atan2(sqrt((pow(((cos((phi2 * -0.5)) * t_5) - (t_0 * t_2)), 2.0) + t_4)), sqrt((1.0 - fma((0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))))), t_1, (0.5 - (0.5 * cos((2.0 * (0.5 * (phi2 - phi1)))))))))));
	} else if (t_3 <= 0.00075) {
		tmp = R * (2.0 * atan2(sqrt(t_6), sqrt((1.0 - t_6))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + t_4)), sqrt(fma(((cos((lambda2 - lambda1)) * 0.5) - 0.5), t_1, (0.5 + (fma(sin(phi2), sin(phi1), t_1) * 0.5))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * phi2))
	t_1 = Float64(cos(phi2) * cos(phi1))
	t_2 = cos(Float64(0.5 * phi1))
	t_3 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_4 = Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_3) * t_3)
	t_5 = sin(Float64(0.5 * phi1))
	t_6 = Float64(Float64(cos(Float64(-0.5 * phi2)) * t_5) - Float64(t_2 * t_0)) ^ 2.0
	tmp = 0.0
	if (t_3 <= -5e-26)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((Float64(Float64(cos(Float64(phi2 * -0.5)) * t_5) - Float64(t_0 * t_2)) ^ 2.0) + t_4)), sqrt(Float64(1.0 - fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5))))), t_1, Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi2 - phi1))))))))))));
	elseif (t_3 <= 0.00075)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_6), sqrt(Float64(1.0 - t_6)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + t_4)), sqrt(fma(Float64(Float64(cos(Float64(lambda2 - lambda1)) * 0.5) - 0.5), t_1, Float64(0.5 + Float64(fma(sin(phi2), sin(phi1), t_1) * 0.5)))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[Power[N[(N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * t$95$5), $MachinePrecision] - N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$3, -5e-26], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[(N[(N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] * t$95$5), $MachinePrecision] - N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$4), $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] * t$95$1 + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.00075], 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], 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] + t$95$4), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision] * t$95$1 + N[(0.5 + N[(N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -5.00000000000000019e-26

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 63.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_2 \cdot \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_2 - \phi_1\right)\right)\right)\right)}}}\right) \]

   if -5.00000000000000019e-26 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 7.5000000000000002e-4

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 56.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 56.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]

   12. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]
   13. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 34.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   16. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}\right) \]

   17. Applied rewrites 34.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]

   if 7.5000000000000002e-4 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \color{blue}{\cos \left(\phi_2 - \phi_1\right)} \cdot \frac{1}{2}\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \color{blue}{\left(\phi_2 - \phi_1\right)} \cdot \frac{1}{2}\right)}}\right) \]

      3. cos-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{1}{2}\right)}}\right) \]

      4. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      6. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      8. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \color{blue}{\left(\sin \phi_2 \cdot \sin \phi_1 + \cos \phi_1 \cdot \cos \phi_2\right)} \cdot \frac{1}{2}\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \color{blue}{\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \cos \phi_2\right)} \cdot \frac{1}{2}\right)}}\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \mathsf{fma}\left(\color{blue}{\sin \phi_2}, \sin \phi_1, \cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{1}{2}\right)}}\right) \]

      11. lower-sin.f64 63.7

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \mathsf{fma}\left(\sin \phi_2, \color{blue}{\sin \phi_1}, \cos \phi_1 \cdot \cos \phi_2\right) \cdot 0.5\right)}}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \color{blue}{\cos \phi_1 \cdot \cos \phi_2}\right) \cdot \frac{1}{2}\right)}}\right) \]

      13. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \color{blue}{\cos \phi_2 \cdot \cos \phi_1}\right) \cdot \frac{1}{2}\right)}}\right) \]

      14. lift-*.f64 63.7

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \color{blue}{\cos \phi_2 \cdot \cos \phi_1}\right) \cdot 0.5\right)}}\right) \]

   5. Applied rewrites 63.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \color{blue}{\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_2 \cdot \cos \phi_1\right)} \cdot 0.5\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 66.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_2\right)\\ t_1 := \cos \left(0.5 \cdot \phi_1\right)\\ t_2 := \cos \phi_2 \cdot \cos \phi_1\\ t_3 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_4 := \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_3\right) \cdot t\_3\\ t_5 := \sin \left(0.5 \cdot \phi_1\right)\\ t_6 := {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot t\_5 - t\_1 \cdot t\_0\right)}^{2}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-26}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot t\_5 - t\_0 \cdot t\_1\right)}^{2} + t\_4}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot t\_2}}\right)\\ \mathbf{elif}\;t\_3 \leq 0.00075:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_6}}{\sqrt{1 - t\_6}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t\_4}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, t\_2, 0.5 + \mathsf{fma}\left(\sin \phi_2, \sin \phi_1, t\_2\right) \cdot 0.5\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 phi2)))
        (t_1 (cos (* 0.5 phi1)))
        (t_2 (* (cos phi2) (cos phi1)))
        (t_3 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_4 (* (* (* (cos phi1) (cos phi2)) t_3) t_3))
        (t_5 (sin (* 0.5 phi1)))
        (t_6 (pow (- (* (cos (* -0.5 phi2)) t_5) (* t_1 t_0)) 2.0)))
   (if (<= t_3 -5e-26)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ (pow (- (* (cos (* phi2 -0.5)) t_5) (* t_0 t_1)) 2.0) t_4))
        (sqrt
         (-
          (+ 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5)))))
          (*
           (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5)))))
           t_2))))))
     (if (<= t_3 0.00075)
       (* R (* 2.0 (atan2 (sqrt t_6) (sqrt (- 1.0 t_6)))))
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) t_4))
          (sqrt
           (fma
            (- (* (cos (- lambda2 lambda1)) 0.5) 0.5)
            t_2
            (+ 0.5 (* (fma (sin phi2) (sin phi1) t_2) 0.5)))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * phi2));
	double t_1 = cos((0.5 * phi1));
	double t_2 = cos(phi2) * cos(phi1);
	double t_3 = sin(((lambda1 - lambda2) / 2.0));
	double t_4 = ((cos(phi1) * cos(phi2)) * t_3) * t_3;
	double t_5 = sin((0.5 * phi1));
	double t_6 = pow(((cos((-0.5 * phi2)) * t_5) - (t_1 * t_0)), 2.0);
	double tmp;
	if (t_3 <= -5e-26) {
		tmp = R * (2.0 * atan2(sqrt((pow(((cos((phi2 * -0.5)) * t_5) - (t_0 * t_1)), 2.0) + t_4)), sqrt(((0.5 + (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5))))) - ((0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))))) * t_2)))));
	} else if (t_3 <= 0.00075) {
		tmp = R * (2.0 * atan2(sqrt(t_6), sqrt((1.0 - t_6))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + t_4)), sqrt(fma(((cos((lambda2 - lambda1)) * 0.5) - 0.5), t_2, (0.5 + (fma(sin(phi2), sin(phi1), t_2) * 0.5))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * phi2))
	t_1 = cos(Float64(0.5 * phi1))
	t_2 = Float64(cos(phi2) * cos(phi1))
	t_3 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_4 = Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_3) * t_3)
	t_5 = sin(Float64(0.5 * phi1))
	t_6 = Float64(Float64(cos(Float64(-0.5 * phi2)) * t_5) - Float64(t_1 * t_0)) ^ 2.0
	tmp = 0.0
	if (t_3 <= -5e-26)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((Float64(Float64(cos(Float64(phi2 * -0.5)) * t_5) - Float64(t_0 * t_1)) ^ 2.0) + t_4)), sqrt(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5))))) - Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5))))) * t_2))))));
	elseif (t_3 <= 0.00075)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_6), sqrt(Float64(1.0 - t_6)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + t_4)), sqrt(fma(Float64(Float64(cos(Float64(lambda2 - lambda1)) * 0.5) - 0.5), t_2, Float64(0.5 + Float64(fma(sin(phi2), sin(phi1), t_2) * 0.5)))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision] * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[Power[N[(N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * t$95$5), $MachinePrecision] - N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$3, -5e-26], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[(N[(N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] * t$95$5), $MachinePrecision] - N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + t$95$4), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 0.00075], 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], 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] + t$95$4), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision] * t$95$2 + N[(0.5 + N[(N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + t$95$2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -5.00000000000000019e-26

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 63.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\color{blue}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}}\right) \]

   if -5.00000000000000019e-26 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 7.5000000000000002e-4

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 56.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 56.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]

   12. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]
   13. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 34.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   16. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}\right) \]

   17. Applied rewrites 34.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]

   if 7.5000000000000002e-4 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \color{blue}{\cos \left(\phi_2 - \phi_1\right)} \cdot \frac{1}{2}\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \color{blue}{\left(\phi_2 - \phi_1\right)} \cdot \frac{1}{2}\right)}}\right) \]

      3. cos-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{1}{2}\right)}}\right) \]

      4. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      6. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      8. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \color{blue}{\left(\sin \phi_2 \cdot \sin \phi_1 + \cos \phi_1 \cdot \cos \phi_2\right)} \cdot \frac{1}{2}\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \color{blue}{\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \cos \phi_2\right)} \cdot \frac{1}{2}\right)}}\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \mathsf{fma}\left(\color{blue}{\sin \phi_2}, \sin \phi_1, \cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{1}{2}\right)}}\right) \]

      11. lower-sin.f64 63.7

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \mathsf{fma}\left(\sin \phi_2, \color{blue}{\sin \phi_1}, \cos \phi_1 \cdot \cos \phi_2\right) \cdot 0.5\right)}}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \color{blue}{\cos \phi_1 \cdot \cos \phi_2}\right) \cdot \frac{1}{2}\right)}}\right) \]

      13. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \color{blue}{\cos \phi_2 \cdot \cos \phi_1}\right) \cdot \frac{1}{2}\right)}}\right) \]

      14. lift-*.f64 63.7

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \color{blue}{\cos \phi_2 \cdot \cos \phi_1}\right) \cdot 0.5\right)}}\right) \]

   5. Applied rewrites 63.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \color{blue}{\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_2 \cdot \cos \phi_1\right)} \cdot 0.5\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 66.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1}\\ t_3 := \cos \phi_2 \cdot \cos \phi_1\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-26}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_2}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot 0.5 - 0.5, t\_3, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}\right)\\ \mathbf{elif}\;t\_1 \leq 0.00075:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_2}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, t\_3, 0.5 + \mathsf{fma}\left(\sin \phi_2, \sin \phi_1, t\_3\right) \cdot 0.5\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (-
           (* (cos (* -0.5 phi2)) (sin (* 0.5 phi1)))
           (* (cos (* 0.5 phi1)) (sin (* 0.5 phi2))))
          2.0))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2
         (sqrt
          (+
           (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
           (* (* (* (cos phi1) (cos phi2)) t_1) t_1))))
        (t_3 (* (cos phi2) (cos phi1))))
   (if (<= t_1 -5e-26)
     (*
      R
      (*
       2.0
       (atan2
        t_2
        (sqrt
         (fma
          (-
           (*
            (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2)))
            0.5)
           0.5)
          t_3
          (+ 0.5 (* (cos (- phi2 phi1)) 0.5)))))))
     (if (<= t_1 0.00075)
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
       (*
        R
        (*
         2.0
         (atan2
          t_2
          (sqrt
           (fma
            (- (* (cos (- lambda2 lambda1)) 0.5) 0.5)
            t_3
            (+ 0.5 (* (fma (sin phi2) (sin phi1) t_3) 0.5)))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(((cos((-0.5 * phi2)) * sin((0.5 * phi1))) - (cos((0.5 * phi1)) * sin((0.5 * phi2)))), 2.0);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_1) * t_1)));
	double t_3 = cos(phi2) * cos(phi1);
	double tmp;
	if (t_1 <= -5e-26) {
		tmp = R * (2.0 * atan2(t_2, sqrt(fma(((fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2))) * 0.5) - 0.5), t_3, (0.5 + (cos((phi2 - phi1)) * 0.5))))));
	} else if (t_1 <= 0.00075) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = R * (2.0 * atan2(t_2, sqrt(fma(((cos((lambda2 - lambda1)) * 0.5) - 0.5), t_3, (0.5 + (fma(sin(phi2), sin(phi1), t_3) * 0.5))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(cos(Float64(-0.5 * phi2)) * sin(Float64(0.5 * phi1))) - Float64(cos(Float64(0.5 * phi1)) * sin(Float64(0.5 * phi2)))) ^ 2.0
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1)))
	t_3 = Float64(cos(phi2) * cos(phi1))
	tmp = 0.0
	if (t_1 <= -5e-26)
		tmp = Float64(R * Float64(2.0 * atan(t_2, sqrt(fma(Float64(Float64(fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))) * 0.5) - 0.5), t_3, Float64(0.5 + Float64(cos(Float64(phi2 - phi1)) * 0.5)))))));
	elseif (t_1 <= 0.00075)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(t_2, sqrt(fma(Float64(Float64(cos(Float64(lambda2 - lambda1)) * 0.5) - 0.5), t_3, Float64(0.5 + Float64(fma(sin(phi2), sin(phi1), t_3) * 0.5)))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-26], N[(R * N[(2.0 * N[ArcTan[t$95$2 / N[Sqrt[N[(N[(N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision] * t$95$3 + N[(0.5 + N[(N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.00075], 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[t$95$2 / N[Sqrt[N[(N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision] * t$95$3 + N[(0.5 + N[(N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + t$95$3), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -5.00000000000000019e-26

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      3. cos-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)} \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)} \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\cos \lambda_2}, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \color{blue}{\cos \lambda_1}, \sin \lambda_2 \cdot \sin \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      7. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_1 \cdot \sin \lambda_2}\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_1 \cdot \sin \lambda_2}\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      9. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \color{blue}{\sin \lambda_1} \cdot \sin \lambda_2\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      10. lower-sin.f64 63.2

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \color{blue}{\sin \lambda_2}\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}\right) \]

   5. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)} \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}\right) \]

   if -5.00000000000000019e-26 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 7.5000000000000002e-4

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 56.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 56.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]

   12. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]
   13. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 34.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   16. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}\right) \]

   17. Applied rewrites 34.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]

   if 7.5000000000000002e-4 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \color{blue}{\cos \left(\phi_2 - \phi_1\right)} \cdot \frac{1}{2}\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \color{blue}{\left(\phi_2 - \phi_1\right)} \cdot \frac{1}{2}\right)}}\right) \]

      3. cos-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{1}{2}\right)}}\right) \]

      4. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      6. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      8. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \color{blue}{\left(\sin \phi_2 \cdot \sin \phi_1 + \cos \phi_1 \cdot \cos \phi_2\right)} \cdot \frac{1}{2}\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \color{blue}{\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \cos \phi_2\right)} \cdot \frac{1}{2}\right)}}\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \mathsf{fma}\left(\color{blue}{\sin \phi_2}, \sin \phi_1, \cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{1}{2}\right)}}\right) \]

      11. lower-sin.f64 63.7

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \mathsf{fma}\left(\sin \phi_2, \color{blue}{\sin \phi_1}, \cos \phi_1 \cdot \cos \phi_2\right) \cdot 0.5\right)}}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \color{blue}{\cos \phi_1 \cdot \cos \phi_2}\right) \cdot \frac{1}{2}\right)}}\right) \]

      13. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \color{blue}{\cos \phi_2 \cdot \cos \phi_1}\right) \cdot \frac{1}{2}\right)}}\right) \]

      14. lift-*.f64 63.7

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \color{blue}{\cos \phi_2 \cdot \cos \phi_1}\right) \cdot 0.5\right)}}\right) \]

   5. Applied rewrites 63.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \color{blue}{\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_2 \cdot \cos \phi_1\right)} \cdot 0.5\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 65.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ t_1 := \cos \phi_2 \cdot \cos \phi_1\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_2\right) \cdot t\_2}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, t\_1, 0.5 + \mathsf{fma}\left(\sin \phi_2, \sin \phi_1, t\_1\right) \cdot 0.5\right)}}\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-26}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 0.00075:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (-
           (* (cos (* -0.5 phi2)) (sin (* 0.5 phi1)))
           (* (cos (* 0.5 phi1)) (sin (* 0.5 phi2))))
          2.0))
        (t_1 (* (cos phi2) (cos phi1)))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3
         (*
          R
          (*
           2.0
           (atan2
            (sqrt
             (+
              (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
              (* (* (* (cos phi1) (cos phi2)) t_2) t_2)))
            (sqrt
             (fma
              (- (* (cos (- lambda2 lambda1)) 0.5) 0.5)
              t_1
              (+ 0.5 (* (fma (sin phi2) (sin phi1) t_1) 0.5)))))))))
   (if (<= t_2 -5e-26)
     t_3
     (if (<= t_2 0.00075)
       (* 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 = pow(((cos((-0.5 * phi2)) * sin((0.5 * phi1))) - (cos((0.5 * phi1)) * sin((0.5 * phi2)))), 2.0);
	double t_1 = cos(phi2) * cos(phi1);
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_2) * t_2))), sqrt(fma(((cos((lambda2 - lambda1)) * 0.5) - 0.5), t_1, (0.5 + (fma(sin(phi2), sin(phi1), t_1) * 0.5))))));
	double tmp;
	if (t_2 <= -5e-26) {
		tmp = t_3;
	} else if (t_2 <= 0.00075) {
		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 = Float64(Float64(cos(Float64(-0.5 * phi2)) * sin(Float64(0.5 * phi1))) - Float64(cos(Float64(0.5 * phi1)) * sin(Float64(0.5 * phi2)))) ^ 2.0
	t_1 = Float64(cos(phi2) * cos(phi1))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = 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_2) * t_2))), sqrt(fma(Float64(Float64(cos(Float64(lambda2 - lambda1)) * 0.5) - 0.5), t_1, Float64(0.5 + Float64(fma(sin(phi2), sin(phi1), t_1) * 0.5)))))))
	tmp = 0.0
	if (t_2 <= -5e-26)
		tmp = t_3;
	elseif (t_2 <= 0.00075)
		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[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]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = 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$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision] * t$95$1 + N[(0.5 + N[(N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + t$95$1), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-26], t$95$3, If[LessEqual[t$95$2, 0.00075], 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 (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -5.00000000000000019e-26 or 7.5000000000000002e-4 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \color{blue}{\cos \left(\phi_2 - \phi_1\right)} \cdot \frac{1}{2}\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \color{blue}{\left(\phi_2 - \phi_1\right)} \cdot \frac{1}{2}\right)}}\right) \]

      3. cos-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right)} \cdot \frac{1}{2}\right)}}\right) \]

      4. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \left(\color{blue}{\cos \phi_2} \cdot \cos \phi_1 + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \left(\cos \phi_2 \cdot \color{blue}{\cos \phi_1} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      6. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \left(\color{blue}{\cos \phi_1 \cdot \cos \phi_2} + \sin \phi_2 \cdot \sin \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      8. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \color{blue}{\left(\sin \phi_2 \cdot \sin \phi_1 + \cos \phi_1 \cdot \cos \phi_2\right)} \cdot \frac{1}{2}\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \color{blue}{\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \cos \phi_2\right)} \cdot \frac{1}{2}\right)}}\right) \]

      10. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \mathsf{fma}\left(\color{blue}{\sin \phi_2}, \sin \phi_1, \cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{1}{2}\right)}}\right) \]

      11. lower-sin.f64 63.7

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \mathsf{fma}\left(\sin \phi_2, \color{blue}{\sin \phi_1}, \cos \phi_1 \cdot \cos \phi_2\right) \cdot 0.5\right)}}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \color{blue}{\cos \phi_1 \cdot \cos \phi_2}\right) \cdot \frac{1}{2}\right)}}\right) \]

      13. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \color{blue}{\cos \phi_2 \cdot \cos \phi_1}\right) \cdot \frac{1}{2}\right)}}\right) \]

      14. lift-*.f64 63.7

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \color{blue}{\cos \phi_2 \cdot \cos \phi_1}\right) \cdot 0.5\right)}}\right) \]

   5. Applied rewrites 63.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \color{blue}{\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_2 \cdot \cos \phi_1\right)} \cdot 0.5\right)}}\right) \]

   if -5.00000000000000019e-26 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 7.5000000000000002e-4

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 56.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 56.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]

   12. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]
   13. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 34.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   16. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}\right) \]

   17. Applied rewrites 34.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 65.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \cos \left(\phi_2 - \phi_1\right)\\ t_2 := t\_1 \cdot 0.5\\ t_3 := \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\\ t_4 := \mathsf{fma}\left(\left(0.5 - t\_3\right) \cdot \left(t\_1 + \cos \left(\phi_2 + \phi_1\right)\right), 0.5, 0.5 - t\_2\right)\\ t_5 := {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-26}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0}}{\sqrt{\left|\mathsf{fma}\left(t\_3 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + t\_2\right)\right|}}\right)\\ \mathbf{elif}\;t\_0 \leq 0.00075:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (cos (- phi2 phi1)))
        (t_2 (* t_1 0.5))
        (t_3 (* (cos (- lambda2 lambda1)) 0.5))
        (t_4 (fma (* (- 0.5 t_3) (+ t_1 (cos (+ phi2 phi1)))) 0.5 (- 0.5 t_2)))
        (t_5
         (pow
          (-
           (* (cos (* -0.5 phi2)) (sin (* 0.5 phi1)))
           (* (cos (* 0.5 phi1)) (sin (* 0.5 phi2))))
          2.0)))
   (if (<= t_0 -5e-26)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_0) t_0)))
        (sqrt
         (fabs (fma (- t_3 0.5) (* (cos phi2) (cos phi1)) (+ 0.5 t_2)))))))
     (if (<= t_0 0.00075)
       (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))
       (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = cos((phi2 - phi1));
	double t_2 = t_1 * 0.5;
	double t_3 = cos((lambda2 - lambda1)) * 0.5;
	double t_4 = fma(((0.5 - t_3) * (t_1 + cos((phi2 + phi1)))), 0.5, (0.5 - t_2));
	double t_5 = pow(((cos((-0.5 * phi2)) * sin((0.5 * phi1))) - (cos((0.5 * phi1)) * sin((0.5 * phi2)))), 2.0);
	double tmp;
	if (t_0 <= -5e-26) {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(fabs(fma((t_3 - 0.5), (cos(phi2) * cos(phi1)), (0.5 + t_2))))));
	} else if (t_0 <= 0.00075) {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = cos(Float64(phi2 - phi1))
	t_2 = Float64(t_1 * 0.5)
	t_3 = Float64(cos(Float64(lambda2 - lambda1)) * 0.5)
	t_4 = fma(Float64(Float64(0.5 - t_3) * Float64(t_1 + cos(Float64(phi2 + phi1)))), 0.5, Float64(0.5 - t_2))
	t_5 = Float64(Float64(cos(Float64(-0.5 * phi2)) * sin(Float64(0.5 * phi1))) - Float64(cos(Float64(0.5 * phi1)) * sin(Float64(0.5 * phi2)))) ^ 2.0
	tmp = 0.0
	if (t_0 <= -5e-26)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(abs(fma(Float64(t_3 - 0.5), Float64(cos(phi2) * cos(phi1)), Float64(0.5 + t_2)))))));
	elseif (t_0 <= 0.00075)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	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[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(0.5 - t$95$3), $MachinePrecision] * N[(t$95$1 + N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 - t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$0, -5e-26], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[Abs[N[(N[(t$95$3 - 0.5), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(0.5 + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.00075], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$5], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(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]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -5.00000000000000019e-26

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\left|\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)\right|}}}\right) \]

   if -5.00000000000000019e-26 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 7.5000000000000002e-4

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 56.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 56.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]

   12. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]
   13. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 34.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   16. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}\right) \]

   17. Applied rewrites 34.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]

   if 7.5000000000000002e-4 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 58.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right), 0.5, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 59.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right), 0.5, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right), 0.5, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 65.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \cos \left(\phi_2 - \phi_1\right)\\ t_2 := t\_1 \cdot 0.5\\ t_3 := \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\\ t_4 := \mathsf{fma}\left(\left(0.5 - t\_3\right) \cdot \left(t\_1 + \cos \left(\phi_2 + \phi_1\right)\right), 0.5, 0.5 - t\_2\right)\\ t_5 := {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-26}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0}}{\sqrt{\mathsf{fma}\left(t\_3 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + t\_2\right)}}\right)\\ \mathbf{elif}\;t\_0 \leq 0.00075:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (cos (- phi2 phi1)))
        (t_2 (* t_1 0.5))
        (t_3 (* (cos (- lambda2 lambda1)) 0.5))
        (t_4 (fma (* (- 0.5 t_3) (+ t_1 (cos (+ phi2 phi1)))) 0.5 (- 0.5 t_2)))
        (t_5
         (pow
          (-
           (* (cos (* -0.5 phi2)) (sin (* 0.5 phi1)))
           (* (cos (* 0.5 phi1)) (sin (* 0.5 phi2))))
          2.0)))
   (if (<= t_0 -5e-26)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_0) t_0)))
        (sqrt (fma (- t_3 0.5) (* (cos phi2) (cos phi1)) (+ 0.5 t_2))))))
     (if (<= t_0 0.00075)
       (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))
       (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = cos((phi2 - phi1));
	double t_2 = t_1 * 0.5;
	double t_3 = cos((lambda2 - lambda1)) * 0.5;
	double t_4 = fma(((0.5 - t_3) * (t_1 + cos((phi2 + phi1)))), 0.5, (0.5 - t_2));
	double t_5 = pow(((cos((-0.5 * phi2)) * sin((0.5 * phi1))) - (cos((0.5 * phi1)) * sin((0.5 * phi2)))), 2.0);
	double tmp;
	if (t_0 <= -5e-26) {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(fma((t_3 - 0.5), (cos(phi2) * cos(phi1)), (0.5 + t_2)))));
	} else if (t_0 <= 0.00075) {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = cos(Float64(phi2 - phi1))
	t_2 = Float64(t_1 * 0.5)
	t_3 = Float64(cos(Float64(lambda2 - lambda1)) * 0.5)
	t_4 = fma(Float64(Float64(0.5 - t_3) * Float64(t_1 + cos(Float64(phi2 + phi1)))), 0.5, Float64(0.5 - t_2))
	t_5 = Float64(Float64(cos(Float64(-0.5 * phi2)) * sin(Float64(0.5 * phi1))) - Float64(cos(Float64(0.5 * phi1)) * sin(Float64(0.5 * phi2)))) ^ 2.0
	tmp = 0.0
	if (t_0 <= -5e-26)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(fma(Float64(t_3 - 0.5), Float64(cos(phi2) * cos(phi1)), Float64(0.5 + t_2))))));
	elseif (t_0 <= 0.00075)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	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[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(0.5 - t$95$3), $MachinePrecision] * N[(t$95$1 + N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 - t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$0, -5e-26], 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[(t$95$3 - 0.5), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(0.5 + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.00075], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$5], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(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]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -5.00000000000000019e-26

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}\right) \]

   if -5.00000000000000019e-26 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 7.5000000000000002e-4

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 56.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 56.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]

   12. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]
   13. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 34.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   16. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}\right) \]

   17. Applied rewrites 34.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]

   if 7.5000000000000002e-4 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 58.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right), 0.5, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 59.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right), 0.5, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right), 0.5, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 65.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \cos \left(\phi_2 - \phi_1\right)\\ t_2 := \mathsf{fma}\left(\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \left(t\_1 + \cos \left(\phi_2 + \phi_1\right)\right), 0.5, 0.5 - t\_1 \cdot 0.5\right)\\ t_3 := {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ t_4 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;t\_0 \leq -0.002:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_0 \leq 0.00075:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (cos (- phi2 phi1)))
        (t_2
         (fma
          (*
           (- 0.5 (* (cos (- lambda2 lambda1)) 0.5))
           (+ t_1 (cos (+ phi2 phi1))))
          0.5
          (- 0.5 (* t_1 0.5))))
        (t_3
         (pow
          (-
           (* (cos (* -0.5 phi2)) (sin (* 0.5 phi1)))
           (* (cos (* 0.5 phi1)) (sin (* 0.5 phi2))))
          2.0))
        (t_4 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= t_0 -0.002)
     t_4
     (if (<= t_0 0.00075)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       t_4))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = cos((phi2 - phi1));
	double t_2 = fma(((0.5 - (cos((lambda2 - lambda1)) * 0.5)) * (t_1 + cos((phi2 + phi1)))), 0.5, (0.5 - (t_1 * 0.5)));
	double t_3 = pow(((cos((-0.5 * phi2)) * sin((0.5 * phi1))) - (cos((0.5 * phi1)) * sin((0.5 * phi2)))), 2.0);
	double t_4 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (t_0 <= -0.002) {
		tmp = t_4;
	} else if (t_0 <= 0.00075) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = cos(Float64(phi2 - phi1))
	t_2 = fma(Float64(Float64(0.5 - Float64(cos(Float64(lambda2 - lambda1)) * 0.5)) * Float64(t_1 + cos(Float64(phi2 + phi1)))), 0.5, Float64(0.5 - Float64(t_1 * 0.5)))
	t_3 = Float64(Float64(cos(Float64(-0.5 * phi2)) * sin(Float64(0.5 * phi1))) - Float64(cos(Float64(0.5 * phi1)) * sin(Float64(0.5 * phi2)))) ^ 2.0
	t_4 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (t_0 <= -0.002)
		tmp = t_4;
	elseif (t_0 <= 0.00075)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = t_4;
	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[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 + N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 - N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = 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]}, Block[{t$95$4 = 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[t$95$0, -0.002], t$95$4, If[LessEqual[t$95$0, 0.00075], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -2e-3 or 7.5000000000000002e-4 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 58.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right), 0.5, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 59.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right), 0.5, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right), 0.5, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}\right) \]

   if -2e-3 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 7.5000000000000002e-4

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 56.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 56.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]

   12. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]
   13. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   14. Applied rewrites 34.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   16. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}\right) \]

   17. Applied rewrites 34.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 63.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\phi_2 - \phi_1\right)\\ t_1 := t\_0 \cdot 0.5\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_4 := t\_3 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_2\right) \cdot t\_2\\ t_5 := \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\\ t_6 := \mathsf{fma}\left(\left(0.5 - t\_5\right) \cdot \left(t\_0 + \cos \left(\phi_2 + \phi_1\right)\right), 0.5, 0.5 - t\_1\right)\\ \mathbf{if}\;2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}} \leq 0.454:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3 + \left(\cos \phi_1 \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot t\_2}}{\sqrt{\mathsf{fma}\left(t\_5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + t\_1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_6}}{\sqrt{1 - t\_6}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- phi2 phi1)))
        (t_1 (* t_0 0.5))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_4 (+ t_3 (* (* (* (cos phi1) (cos phi2)) t_2) t_2)))
        (t_5 (* (cos (- lambda2 lambda1)) 0.5))
        (t_6
         (fma (* (- 0.5 t_5) (+ t_0 (cos (+ phi2 phi1)))) 0.5 (- 0.5 t_1))))
   (if (<= (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))) 0.454)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_3 (* (* (cos phi1) (sin (* 0.5 (- lambda1 lambda2)))) t_2)))
        (sqrt (fma (- t_5 0.5) (* (cos phi2) (cos phi1)) (+ 0.5 t_1))))))
     (* R (* 2.0 (atan2 (sqrt t_6) (sqrt (- 1.0 t_6))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((phi2 - phi1));
	double t_1 = t_0 * 0.5;
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_4 = t_3 + (((cos(phi1) * cos(phi2)) * t_2) * t_2);
	double t_5 = cos((lambda2 - lambda1)) * 0.5;
	double t_6 = fma(((0.5 - t_5) * (t_0 + cos((phi2 + phi1)))), 0.5, (0.5 - t_1));
	double tmp;
	if ((2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4)))) <= 0.454) {
		tmp = R * (2.0 * atan2(sqrt((t_3 + ((cos(phi1) * sin((0.5 * (lambda1 - lambda2)))) * t_2))), sqrt(fma((t_5 - 0.5), (cos(phi2) * cos(phi1)), (0.5 + t_1)))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_6), sqrt((1.0 - t_6))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(phi2 - phi1))
	t_1 = Float64(t_0 * 0.5)
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_4 = Float64(t_3 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_2) * t_2))
	t_5 = Float64(cos(Float64(lambda2 - lambda1)) * 0.5)
	t_6 = fma(Float64(Float64(0.5 - t_5) * Float64(t_0 + cos(Float64(phi2 + phi1)))), 0.5, Float64(0.5 - t_1))
	tmp = 0.0
	if (Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))) <= 0.454)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_3 + Float64(Float64(cos(phi1) * sin(Float64(0.5 * Float64(lambda1 - lambda2)))) * t_2))), sqrt(fma(Float64(t_5 - 0.5), Float64(cos(phi2) * cos(phi1)), Float64(0.5 + t_1))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_6), sqrt(Float64(1.0 - t_6)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(0.5 - t$95$5), $MachinePrecision] * N[(t$95$0 + N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5 + N[(0.5 - t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.454], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$3 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$5 - 0.5), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(0.5 + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]]]]]]]]

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))))))))) < 0.454000000000000015

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}\right) \]

   4. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      3. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      5. lower--.f64 53.2

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}\right) \]

   6. Applied rewrites 53.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}\right) \]

   if 0.454000000000000015 < (*.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 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 58.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right), 0.5, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 59.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right), 0.5, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \left(\cos \left(\phi_2 - \phi_1\right) + \cos \left(\phi_2 + \phi_1\right)\right), 0.5, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 63.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := \mathsf{fma}\left(\cos \phi_1, t\_1, \frac{1}{{\sin \left(\phi_1 \cdot 0.5\right)}^{-2}}\right)\\ \mathbf{if}\;\phi_1 \leq -160:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}} \cdot \left(2 \cdot R\right)\\ \mathbf{elif}\;\phi_1 \leq 0.0007:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot t\_1}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos phi1)
          (- 0.5 (* 0.5 (cos (* -1.0 (- lambda2 lambda1)))))
          (pow (sin (* 0.5 phi1)) 2.0)))
        (t_1 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_2 (fma (cos phi1) t_1 (/ 1.0 (pow (sin (* phi1 0.5)) -2.0)))))
   (if (<= phi1 -160.0)
     (* (atan2 (sqrt t_0) (sqrt (- 1.0 t_0))) (* 2.0 R))
     (if (<= phi1 0.0007)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (cos phi2) t_1)))
          (sqrt
           (fma
            (- (* (cos (- lambda2 lambda1)) 0.5) 0.5)
            (* (cos phi2) (cos phi1))
            (+ 0.5 (* (cos (- phi2 phi1)) 0.5)))))))
       (* 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 = fma(cos(phi1), (0.5 - (0.5 * cos((-1.0 * (lambda2 - lambda1))))), pow(sin((0.5 * phi1)), 2.0));
	double t_1 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_2 = fma(cos(phi1), t_1, (1.0 / pow(sin((phi1 * 0.5)), -2.0)));
	double tmp;
	if (phi1 <= -160.0) {
		tmp = atan2(sqrt(t_0), sqrt((1.0 - t_0))) * (2.0 * R);
	} else if (phi1 <= 0.0007) {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi2) * t_1))), sqrt(fma(((cos((lambda2 - lambda1)) * 0.5) - 0.5), (cos(phi2) * cos(phi1)), (0.5 + (cos((phi2 - phi1)) * 0.5))))));
	} 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 = fma(cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(-1.0 * Float64(lambda2 - lambda1))))), (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_1 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_2 = fma(cos(phi1), t_1, Float64(1.0 / (sin(Float64(phi1 * 0.5)) ^ -2.0)))
	tmp = 0.0
	if (phi1 <= -160.0)
		tmp = Float64(atan(sqrt(t_0), sqrt(Float64(1.0 - t_0))) * Float64(2.0 * R));
	elseif (phi1 <= 0.0007)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi2) * t_1))), sqrt(fma(Float64(Float64(cos(Float64(lambda2 - lambda1)) * 0.5) - 0.5), Float64(cos(phi2) * cos(phi1)), Float64(0.5 + Float64(cos(Float64(phi2 - phi1)) * 0.5)))))));
	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[(N[Cos[phi1], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(-1.0 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $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[Cos[phi1], $MachinePrecision] * t$95$1 + N[(1.0 / N[Power[N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -160.0], N[(N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.0007], 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$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] - 0.5), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(0.5 + N[(N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision] * 0.5), $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 phi1 < -160

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   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 31.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   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 21.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 21.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 19.5%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right)} \]

   16. Taylor expanded in lambda1 around inf

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)\right) + \color{blue}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]
   17. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2}} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       3. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       7. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       10. lower-*.f64 19.5

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

   18. Applied rewrites 19.5%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

   19. Taylor expanded in lambda1 around inf

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left(\cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)\right) + \color{blue}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}\right)}} \cdot \left(2 \cdot R\right) \]
   20. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2}} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       3. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       7. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       10. lower-*.f64 44.4

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

   21. Applied rewrites 44.4%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

   if -160 < phi1 < 6.99999999999999993e-4

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}\right) \]

   4. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      6. lower--.f64 53.0

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}\right) \]

   6. Applied rewrites 53.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}\right) \]

   if 6.99999999999999993e-4 < phi1

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2\right)\right)\right)\right)}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. pow-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-special-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower-special-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      10. metadata-eval 46.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot 0.5\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot 0.5\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       2. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(-2\right)\right)}\right)}}\right) \]

       3. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2\right)\right)\right)\right)}\right)}}\right) \]

       4. pow-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}\right) \]

       5. lower-special-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}\right) \]

       6. lower-special-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}\right) \]

       7. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}\right) \]

       8. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}\right) \]

       10. metadata-eval 46.8

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot 0.5\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot 0.5\right)}^{-2}}\right)}}\right) \]

   11. Applied rewrites 46.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot 0.5\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot 0.5\right)}^{-2}}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 26: 63.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot t\_1\\ t_3 := \mathsf{fma}\left(\cos \phi_1, t\_1, \frac{1}{{\sin \left(\phi_1 \cdot 0.5\right)}^{-2}}\right)\\ \mathbf{if}\;\phi_1 \leq -160:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}} \cdot \left(2 \cdot R\right)\\ \mathbf{elif}\;\phi_1 \leq 0.0006:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos phi1)
          (- 0.5 (* 0.5 (cos (* -1.0 (- lambda2 lambda1)))))
          (pow (sin (* 0.5 phi1)) 2.0)))
        (t_1 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_2 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (cos phi2) t_1)))
        (t_3 (fma (cos phi1) t_1 (/ 1.0 (pow (sin (* phi1 0.5)) -2.0)))))
   (if (<= phi1 -160.0)
     (* (atan2 (sqrt t_0) (sqrt (- 1.0 t_0))) (* 2.0 R))
     (if (<= phi1 0.0006)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
       (* 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 = fma(cos(phi1), (0.5 - (0.5 * cos((-1.0 * (lambda2 - lambda1))))), pow(sin((0.5 * phi1)), 2.0));
	double t_1 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi2) * t_1);
	double t_3 = fma(cos(phi1), t_1, (1.0 / pow(sin((phi1 * 0.5)), -2.0)));
	double tmp;
	if (phi1 <= -160.0) {
		tmp = atan2(sqrt(t_0), sqrt((1.0 - t_0))) * (2.0 * R);
	} else if (phi1 <= 0.0006) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} 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 = fma(cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(-1.0 * Float64(lambda2 - lambda1))))), (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_1 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_2 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi2) * t_1))
	t_3 = fma(cos(phi1), t_1, Float64(1.0 / (sin(Float64(phi1 * 0.5)) ^ -2.0)))
	tmp = 0.0
	if (phi1 <= -160.0)
		tmp = Float64(atan(sqrt(t_0), sqrt(Float64(1.0 - t_0))) * Float64(2.0 * R));
	elseif (phi1 <= 0.0006)
		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(t_3), sqrt(Float64(1.0 - t_3)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(-1.0 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $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[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * t$95$1 + N[(1.0 / N[Power[N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -160.0], N[(N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.0006], 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[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 < -160

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   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 31.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   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 21.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 21.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 19.5%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right)} \]

   16. Taylor expanded in lambda1 around inf

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)\right) + \color{blue}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]
   17. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2}} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       3. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       7. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       10. lower-*.f64 19.5

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

   18. Applied rewrites 19.5%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

   19. Taylor expanded in lambda1 around inf

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left(\cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)\right) + \color{blue}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}\right)}} \cdot \left(2 \cdot R\right) \]
   20. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2}} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       3. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       7. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       10. lower-*.f64 44.4

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

   21. Applied rewrites 44.4%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

   if -160 < phi1 < 5.99999999999999947e-4

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 52.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 52.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      6. lower--.f64 50.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 50.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   if 5.99999999999999947e-4 < phi1

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2\right)\right)\right)\right)}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. pow-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-special-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower-special-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      10. metadata-eval 46.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot 0.5\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot 0.5\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       2. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(-2\right)\right)}\right)}}\right) \]

       3. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2\right)\right)\right)\right)}\right)}}\right) \]

       4. pow-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}\right) \]

       5. lower-special-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}\right) \]

       6. lower-special-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}\right) \]

       7. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}\right) \]

       8. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}\right) \]

       10. metadata-eval 46.8

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot 0.5\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot 0.5\right)}^{-2}}\right)}}\right) \]

   11. Applied rewrites 46.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot 0.5\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot 0.5\right)}^{-2}}\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 27: 63.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot t\_0\\ t_2 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_2 \leq -7.5 \cdot 10^{-8}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_2 \leq 0.0013:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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 -7.5e-8)
     t_3
     (if (<= phi2 0.0013)
       (* 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 <= -7.5e-8) {
		tmp = t_3;
	} else if (phi2 <= 0.0013) {
		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 <= -7.5e-8)
		tmp = t_3;
	elseif (phi2 <= 0.0013)
		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, -7.5e-8], t$95$3, If[LessEqual[phi2, 0.0013], 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 < -7.4999999999999997e-8 or 0.0012999999999999999 < phi2

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   if -7.4999999999999997e-8 < phi2 < 0.0012999999999999999

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 53.2

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 53.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 51.0

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 51.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 28: 62.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_3 := t\_2 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1\\ t_4 := \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\\ t_5 := \left|\mathsf{fma}\left(\left(0.5 - t\_4\right) \cdot \cos \phi_2, \cos \phi_1, 0.5 - t\_0\right)\right|\\ \mathbf{if}\;2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}} \leq 0.454:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2 + \left(\cos \phi_1 \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot t\_1}}{\sqrt{\mathsf{fma}\left(t\_4 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + t\_0\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos (- phi2 phi1)) 0.5))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_3 (+ t_2 (* (* (* (cos phi1) (cos phi2)) t_1) t_1)))
        (t_4 (* (cos (- lambda2 lambda1)) 0.5))
        (t_5 (fabs (fma (* (- 0.5 t_4) (cos phi2)) (cos phi1) (- 0.5 t_0)))))
   (if (<= (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))) 0.454)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_2 (* (* (cos phi1) (sin (* 0.5 (- lambda1 lambda2)))) t_1)))
        (sqrt (fma (- t_4 0.5) (* (cos phi2) (cos phi1)) (+ 0.5 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 - phi1)) * 0.5;
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_3 = t_2 + (((cos(phi1) * cos(phi2)) * t_1) * t_1);
	double t_4 = cos((lambda2 - lambda1)) * 0.5;
	double t_5 = fabs(fma(((0.5 - t_4) * cos(phi2)), cos(phi1), (0.5 - t_0)));
	double tmp;
	if ((2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3)))) <= 0.454) {
		tmp = R * (2.0 * atan2(sqrt((t_2 + ((cos(phi1) * sin((0.5 * (lambda1 - lambda2)))) * t_1))), sqrt(fma((t_4 - 0.5), (cos(phi2) * cos(phi1)), (0.5 + 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(cos(Float64(phi2 - phi1)) * 0.5)
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_3 = Float64(t_2 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1))
	t_4 = Float64(cos(Float64(lambda2 - lambda1)) * 0.5)
	t_5 = abs(fma(Float64(Float64(0.5 - t_4) * cos(phi2)), cos(phi1), Float64(0.5 - t_0)))
	tmp = 0.0
	if (Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))) <= 0.454)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_2 + Float64(Float64(cos(phi1) * sin(Float64(0.5 * Float64(lambda1 - lambda2)))) * t_1))), sqrt(fma(Float64(t_4 - 0.5), Float64(cos(phi2) * cos(phi1)), Float64(0.5 + 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[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$5 = N[Abs[N[(N[(N[(0.5 - t$95$4), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(0.5 - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.454], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$2 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$4 - 0.5), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(0.5 + t$95$0), $MachinePrecision]), $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 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))))))))) < 0.454000000000000015

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}\right) \]

   4. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      3. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      5. lower--.f64 53.2

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}\right) \]

   6. Applied rewrites 53.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}\right) \]

   if 0.454000000000000015 < (*.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 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 58.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left|\mathsf{fma}\left(\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \cos \phi_2, \cos \phi_1, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)\right|}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 58.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left|\mathsf{fma}\left(\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \cos \phi_2, \cos \phi_1, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)\right|}}{\sqrt{1 - \color{blue}{\left|\mathsf{fma}\left(\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \cos \phi_2, \cos \phi_1, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)\right|}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 29: 62.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_3 := t\_2 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1\\ t_4 := \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\\ t_5 := \left|\mathsf{fma}\left(\left(0.5 - t\_4\right) \cdot \cos \phi_2, \cos \phi_1, 0.5 - t\_0\right)\right|\\ \mathbf{if}\;2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}} \leq 0.454:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2 + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(t\_4 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + t\_0\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos (- phi2 phi1)) 0.5))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_3 (+ t_2 (* (* (* (cos phi1) (cos phi2)) t_1) t_1)))
        (t_4 (* (cos (- lambda2 lambda1)) 0.5))
        (t_5 (fabs (fma (* (- 0.5 t_4) (cos phi2)) (cos phi1) (- 0.5 t_0)))))
   (if (<= (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))) 0.454)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+ t_2 (* (cos phi1) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))))
        (sqrt (fma (- t_4 0.5) (* (cos phi2) (cos phi1)) (+ 0.5 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 - phi1)) * 0.5;
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_3 = t_2 + (((cos(phi1) * cos(phi2)) * t_1) * t_1);
	double t_4 = cos((lambda2 - lambda1)) * 0.5;
	double t_5 = fabs(fma(((0.5 - t_4) * cos(phi2)), cos(phi1), (0.5 - t_0)));
	double tmp;
	if ((2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3)))) <= 0.454) {
		tmp = R * (2.0 * atan2(sqrt((t_2 + (cos(phi1) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0)))), sqrt(fma((t_4 - 0.5), (cos(phi2) * cos(phi1)), (0.5 + 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(cos(Float64(phi2 - phi1)) * 0.5)
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_3 = Float64(t_2 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1))
	t_4 = Float64(cos(Float64(lambda2 - lambda1)) * 0.5)
	t_5 = abs(fma(Float64(Float64(0.5 - t_4) * cos(phi2)), cos(phi1), Float64(0.5 - t_0)))
	tmp = 0.0
	if (Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))) <= 0.454)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_2 + Float64(cos(phi1) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)))), sqrt(fma(Float64(t_4 - 0.5), Float64(cos(phi2) * cos(phi1)), Float64(0.5 + 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[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$5 = N[Abs[N[(N[(N[(0.5 - t$95$4), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(0.5 - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.454], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$2 + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$4 - 0.5), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(0.5 + t$95$0), $MachinePrecision]), $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 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))))))))) < 0.454000000000000015

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}\right) \]

   4. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      6. lower--.f64 53.2

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}\right) \]

   6. Applied rewrites 53.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}\right) \]

   if 0.454000000000000015 < (*.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 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 58.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left|\mathsf{fma}\left(\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \cos \phi_2, \cos \phi_1, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)\right|}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 58.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left|\mathsf{fma}\left(\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \cos \phi_2, \cos \phi_1, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)\right|}}{\sqrt{1 - \color{blue}{\left|\mathsf{fma}\left(\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \cos \phi_2, \cos \phi_1, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)\right|}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 30: 62.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_4 := \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\\ t_5 := \mathsf{fma}\left(0.5 - t\_4, t\_0, 0.5 - t\_1\right)\\ \mathbf{if}\;t\_3 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_2\right) \cdot t\_2 \leq 0.0506:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3 + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(t\_4 - 0.5, t\_0, 0.5 + t\_1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (* (cos (- phi2 phi1)) 0.5))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_4 (* (cos (- lambda2 lambda1)) 0.5))
        (t_5 (fma (- 0.5 t_4) t_0 (- 0.5 t_1))))
   (if (<= (+ t_3 (* (* (* (cos phi1) (cos phi2)) t_2) t_2)) 0.0506)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+ t_3 (* (cos phi1) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))))
        (sqrt (fma (- t_4 0.5) t_0 (+ 0.5 t_1))))))
     (* 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((phi2 - phi1)) * 0.5;
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_4 = cos((lambda2 - lambda1)) * 0.5;
	double t_5 = fma((0.5 - t_4), t_0, (0.5 - t_1));
	double tmp;
	if ((t_3 + (((cos(phi1) * cos(phi2)) * t_2) * t_2)) <= 0.0506) {
		tmp = R * (2.0 * atan2(sqrt((t_3 + (cos(phi1) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0)))), sqrt(fma((t_4 - 0.5), t_0, (0.5 + t_1)))));
	} 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(cos(phi2) * cos(phi1))
	t_1 = Float64(cos(Float64(phi2 - phi1)) * 0.5)
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_4 = Float64(cos(Float64(lambda2 - lambda1)) * 0.5)
	t_5 = fma(Float64(0.5 - t_4), t_0, Float64(0.5 - t_1))
	tmp = 0.0
	if (Float64(t_3 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_2) * t_2)) <= 0.0506)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_3 + Float64(cos(phi1) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)))), sqrt(fma(Float64(t_4 - 0.5), t_0, Float64(0.5 + t_1))))));
	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[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$5 = N[(N[(0.5 - t$95$4), $MachinePrecision] * t$95$0 + N[(0.5 - t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$3 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 0.0506], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[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]], $MachinePrecision] / N[Sqrt[N[(N[(t$95$4 - 0.5), $MachinePrecision] * t$95$0 + N[(0.5 + t$95$1), $MachinePrecision]), $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 2 regimes

2. if (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))) < 0.0505999999999999991

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}\right) \]

   4. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      6. lower--.f64 53.2

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}\right) \]

   6. Applied rewrites 53.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}\right) \]

   if 0.0505999999999999991 < (+.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 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 58.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 58.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 31: 62.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_3 := \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\\ t_4 := \sqrt{\mathsf{fma}\left(t\_3 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + t\_1\right)}\\ \mathbf{if}\;t\_2 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0 \leq 0.0506:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2 + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{t\_4}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\left(0.5 - t\_3\right) \cdot \cos \phi_2, \cos \phi_1, 0.5 - t\_1\right)}}{t\_4} \cdot \left(R + R\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (* (cos (- phi2 phi1)) 0.5))
        (t_2 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_3 (* (cos (- lambda2 lambda1)) 0.5))
        (t_4 (sqrt (fma (- t_3 0.5) (* (cos phi2) (cos phi1)) (+ 0.5 t_1)))))
   (if (<= (+ t_2 (* (* (* (cos phi1) (cos phi2)) t_0) t_0)) 0.0506)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+ t_2 (* (cos phi1) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))))
        t_4)))
     (*
      (atan2
       (sqrt (fma (* (- 0.5 t_3) (cos phi2)) (cos phi1) (- 0.5 t_1)))
       t_4)
      (+ R 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 = cos((phi2 - phi1)) * 0.5;
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_3 = cos((lambda2 - lambda1)) * 0.5;
	double t_4 = sqrt(fma((t_3 - 0.5), (cos(phi2) * cos(phi1)), (0.5 + t_1)));
	double tmp;
	if ((t_2 + (((cos(phi1) * cos(phi2)) * t_0) * t_0)) <= 0.0506) {
		tmp = R * (2.0 * atan2(sqrt((t_2 + (cos(phi1) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0)))), t_4));
	} else {
		tmp = atan2(sqrt(fma(((0.5 - t_3) * cos(phi2)), cos(phi1), (0.5 - t_1))), t_4) * (R + R);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64(cos(Float64(phi2 - phi1)) * 0.5)
	t_2 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_3 = Float64(cos(Float64(lambda2 - lambda1)) * 0.5)
	t_4 = sqrt(fma(Float64(t_3 - 0.5), Float64(cos(phi2) * cos(phi1)), Float64(0.5 + t_1)))
	tmp = 0.0
	if (Float64(t_2 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0)) <= 0.0506)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_2 + Float64(cos(phi1) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)))), t_4)));
	else
		tmp = Float64(atan(sqrt(fma(Float64(Float64(0.5 - t_3) * cos(phi2)), cos(phi1), Float64(0.5 - t_1))), t_4) * Float64(R + 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[(N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision] * 0.5), $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[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(t$95$3 - 0.5), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(0.5 + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$2 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 0.0506], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$2 + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[ArcTan[N[Sqrt[N[(N[(N[(0.5 - t$95$3), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(0.5 - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4], $MachinePrecision] * N[(R + R), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))) < 0.0505999999999999991

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 62.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}}\right) \]

   4. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \frac{1}{2} - \frac{1}{2}, \cos \phi_2 \cdot \cos \phi_1, \frac{1}{2} + \cos \left(\phi_2 - \phi_1\right) \cdot \frac{1}{2}\right)}}\right) \]

      6. lower--.f64 53.2

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}\right) \]

   6. Applied rewrites 53.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}\right) \]

   if 0.0505999999999999991 < (+.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 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 58.2%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \cos \phi_2, \cos \phi_1, 0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}}{\sqrt{\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5 - 0.5, \cos \phi_2 \cdot \cos \phi_1, 0.5 + \cos \left(\phi_2 - \phi_1\right) \cdot 0.5\right)}} \cdot \left(R + R\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 32: 61.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(\cos \phi_1, t\_0, \frac{1}{{\sin \left(\phi_1 \cdot 0.5\right)}^{-2}}\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_2 \leq -2.6 \cdot 10^{-48}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_2 \leq 3.5 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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 (/ 1.0 (pow (sin (* phi1 0.5)) -2.0))))
        (t_2 (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0)))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= phi2 -2.6e-48)
     t_3
     (if (<= phi2 3.5e-7)
       (* 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 = fma(cos(phi1), t_0, (1.0 / pow(sin((phi1 * 0.5)), -2.0)));
	double t_2 = fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi2 <= -2.6e-48) {
		tmp = t_3;
	} else if (phi2 <= 3.5e-7) {
		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 = fma(cos(phi1), t_0, Float64(1.0 / (sin(Float64(phi1 * 0.5)) ^ -2.0)))
	t_2 = fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi2 <= -2.6e-48)
		tmp = t_3;
	elseif (phi2 <= 3.5e-7)
		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[Cos[phi1], $MachinePrecision] * t$95$0 + N[(1.0 / N[Power[N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision], -2.0], $MachinePrecision]), $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, -2.6e-48], t$95$3, If[LessEqual[phi2, 3.5e-7], 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 < -2.59999999999999987e-48 or 3.49999999999999984e-7 < phi2

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   if -2.59999999999999987e-48 < phi2 < 3.49999999999999984e-7

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   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. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2\right)\right)\right)\right)}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. pow-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-special-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower-special-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      10. metadata-eval 46.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot 0.5\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot 0.5\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       2. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(-2\right)\right)}\right)}}\right) \]

       3. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(2\right)\right)\right)\right)}\right)}}\right) \]

       4. pow-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}\right) \]

       5. lower-special-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}\right) \]

       6. lower-special-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}\right) \]

       7. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}\right) \]

       8. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot \frac{1}{2}\right)}^{\left(\mathsf{neg}\left(2\right)\right)}}\right)}}\right) \]

       10. metadata-eval 46.8

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot 0.5\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot 0.5\right)}^{-2}}\right)}}\right) \]

   11. Applied rewrites 46.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot 0.5\right)}^{-2}}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{{\sin \left(\phi_1 \cdot 0.5\right)}^{-2}}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 33: 61.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{if}\;\phi_2 \leq -2.6 \cdot 10^{-48}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 3.5 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\frac{1}{{\left(1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_1 \cdot 0.5\right)\right)\right)\right)}^{-0.5}}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))
   (if (<= phi2 -2.6e-48)
     t_2
     (if (<= phi2 3.5e-7)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (fma (cos phi1) t_0 (pow (sin (* 0.5 phi1)) 2.0)))
          (/
           1.0
           (pow
            (-
             1.0
             (fma
              (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5)))))
              (cos phi1)
              (- 0.5 (* 0.5 (cos (* 2.0 (* phi1 0.5)))))))
            -0.5)))))
       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(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0));
	double t_2 = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	double tmp;
	if (phi2 <= -2.6e-48) {
		tmp = t_2;
	} else if (phi2 <= 3.5e-7) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), t_0, pow(sin((0.5 * phi1)), 2.0))), (1.0 / pow((1.0 - fma((0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))))), cos(phi1), (0.5 - (0.5 * cos((2.0 * (phi1 * 0.5))))))), -0.5))));
	} else {
		tmp = 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(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0))
	t_2 = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
	tmp = 0.0
	if (phi2 <= -2.6e-48)
		tmp = t_2;
	elseif (phi2 <= 3.5e-7)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), t_0, (sin(Float64(0.5 * phi1)) ^ 2.0))), Float64(1.0 / (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(phi1 * 0.5))))))) ^ -0.5)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(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[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -2.6e-48], t$95$2, If[LessEqual[phi2, 3.5e-7], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(1.0 / N[Power[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[(phi1 * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.59999999999999987e-48 or 3.49999999999999984e-7 < phi2

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   if -2.59999999999999987e-48 < phi2 < 3.49999999999999984e-7

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\color{blue}{\frac{1}{{\left(1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_1 \cdot 0.5\right)\right)\right)\right)}^{-0.5}}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 34: 61.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_2 \leq -2.6 \cdot 10^{-48}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_2 \leq 3.5 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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 (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 -2.6e-48)
     t_3
     (if (<= phi2 3.5e-7)
       (* 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 = fma(cos(phi1), t_0, pow(sin((0.5 * phi1)), 2.0));
	double t_2 = fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi2 <= -2.6e-48) {
		tmp = t_3;
	} else if (phi2 <= 3.5e-7) {
		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 = fma(cos(phi1), t_0, (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_2 = fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi2 <= -2.6e-48)
		tmp = t_3;
	elseif (phi2 <= 3.5e-7)
		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[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -2.6e-48], t$95$3, If[LessEqual[phi2, 3.5e-7], 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 < -2.59999999999999987e-48 or 3.49999999999999984e-7 < phi2

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   if -2.59999999999999987e-48 < phi2 < 3.49999999999999984e-7

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 35: 61.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_2\\ t_1 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t\_0\\ \mathbf{if}\;\phi_2 \leq -2.6 \cdot 10^{-48}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{elif}\;\phi_2 \leq 4.2 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{t\_0 + t\_1}}{\sqrt{\left(1 - t\_1\right) - t\_0}} \cdot \left(R + R\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (fma (cos (* -1.0 (- lambda2 lambda1))) -0.5 0.5) (cos phi2)))
        (t_1 (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5))))))
        (t_2
         (fma
          (cos phi1)
          (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
          (pow (sin (* 0.5 phi1)) 2.0)))
        (t_3 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) t_0)))
   (if (<= phi2 -2.6e-48)
     (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
     (if (<= phi2 4.2e-6)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
       (* (atan2 (sqrt (+ t_0 t_1)) (sqrt (- (- 1.0 t_1) t_0))) (+ R R))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos((-1.0 * (lambda2 - lambda1))), -0.5, 0.5) * cos(phi2);
	double t_1 = 0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5))));
	double t_2 = fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow(sin((0.5 * phi1)), 2.0));
	double t_3 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + t_0;
	double tmp;
	if (phi2 <= -2.6e-48) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else if (phi2 <= 4.2e-6) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else {
		tmp = atan2(sqrt((t_0 + t_1)), sqrt(((1.0 - t_1) - t_0))) * (R + R);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(fma(cos(Float64(-1.0 * Float64(lambda2 - lambda1))), -0.5, 0.5) * cos(phi2))
	t_1 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5)))))
	t_2 = fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_3 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + t_0)
	tmp = 0.0
	if (phi2 <= -2.6e-48)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	elseif (phi2 <= 4.2e-6)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	else
		tmp = Float64(atan(sqrt(Float64(t_0 + t_1)), sqrt(Float64(Float64(1.0 - t_1) - t_0))) * Float64(R + R));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[Cos[N[(-1.0 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + t$95$0), $MachinePrecision]}, If[LessEqual[phi2, -2.6e-48], 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[phi2, 4.2e-6], 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[(N[ArcTan[N[Sqrt[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - t$95$1), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(R + R), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -2.59999999999999987e-48

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\pi}, \mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-neg.f64 52.4

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \color{blue}{-\phi_2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 52.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\pi}, \mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-neg.f64 51.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \color{blue}{-\phi_2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 51.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-PI.f64 41.4

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 41.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]

       2. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]

       3. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} - \phi_2\right)\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       5. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       7. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       9. lower-PI.f64 39.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)\right)}}\right) \]

   11. Applied rewrites 39.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}\right)}}\right) \]

   13. Applied rewrites 38.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \color{blue}{\cos \phi_2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)\right)}}\right) \]

   15. Applied rewrites 48.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_2}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \color{blue}{\cos \phi_2}\right)}}\right) \]

   if -2.59999999999999987e-48 < phi2 < 4.1999999999999996e-6

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if 4.1999999999999996e-6 < phi2

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\pi}, \mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-neg.f64 52.4

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \color{blue}{-\phi_2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 52.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\pi}, \mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-neg.f64 51.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \color{blue}{-\phi_2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 51.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-PI.f64 41.4

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 41.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]

       2. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]

       3. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} - \phi_2\right)\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       5. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       7. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       9. lower-PI.f64 39.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)\right)}}\right) \]

   11. Applied rewrites 39.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}\right)}}\right) \]

   13. Applied rewrites 46.2%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_2 + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)\right) - \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_2}} \cdot \left(R + R\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 36: 60.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \mathsf{fma}\left(t\_0, -0.5, 0.5\right) \cdot \cos \phi_2\\ t_2 := \mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_3 := \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}} \cdot \left(2 \cdot R\right)\\ \mathbf{if}\;\phi_1 \leq -160:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_1 \leq 0.0007:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* -1.0 (- lambda2 lambda1))))
        (t_1
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (fma t_0 -0.5 0.5) (cos phi2))))
        (t_2 (fma (cos phi1) (- 0.5 (* 0.5 t_0)) (pow (sin (* 0.5 phi1)) 2.0)))
        (t_3 (* (atan2 (sqrt t_2) (sqrt (- 1.0 t_2))) (* 2.0 R))))
   (if (<= phi1 -160.0)
     t_3
     (if (<= phi1 0.0007)
       (* 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 = cos((-1.0 * (lambda2 - lambda1)));
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (fma(t_0, -0.5, 0.5) * cos(phi2));
	double t_2 = fma(cos(phi1), (0.5 - (0.5 * t_0)), pow(sin((0.5 * phi1)), 2.0));
	double t_3 = atan2(sqrt(t_2), sqrt((1.0 - t_2))) * (2.0 * R);
	double tmp;
	if (phi1 <= -160.0) {
		tmp = t_3;
	} else if (phi1 <= 0.0007) {
		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 = cos(Float64(-1.0 * Float64(lambda2 - lambda1)))
	t_1 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(fma(t_0, -0.5, 0.5) * cos(phi2)))
	t_2 = fma(cos(phi1), Float64(0.5 - Float64(0.5 * t_0)), (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_3 = Float64(atan(sqrt(t_2), sqrt(Float64(1.0 - t_2))) * Float64(2.0 * R))
	tmp = 0.0
	if (phi1 <= -160.0)
		tmp = t_3;
	elseif (phi1 <= 0.0007)
		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[Cos[N[(-1.0 * N[(lambda2 - lambda1), $MachinePrecision]), $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[(t$95$0 * -0.5 + 0.5), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[(0.5 - N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -160.0], t$95$3, If[LessEqual[phi1, 0.0007], 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 < -160 or 6.99999999999999993e-4 < phi1

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   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 31.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   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 21.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 21.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 19.5%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right)} \]

   16. Taylor expanded in lambda1 around inf

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)\right) + \color{blue}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]
   17. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2}} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       3. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       7. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       10. lower-*.f64 19.5

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

   18. Applied rewrites 19.5%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

   19. Taylor expanded in lambda1 around inf

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left(\cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)\right) + \color{blue}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}\right)}} \cdot \left(2 \cdot R\right) \]
   20. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2}} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       3. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       7. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       10. lower-*.f64 44.4

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

   21. Applied rewrites 44.4%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

   if -160 < phi1 < 6.99999999999999993e-4

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\pi}, \mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-neg.f64 52.4

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \color{blue}{-\phi_2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 52.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\pi}, \mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-neg.f64 51.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \color{blue}{-\phi_2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 51.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-PI.f64 41.4

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 41.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]

       2. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]

       3. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} - \phi_2\right)\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       5. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       7. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       9. lower-PI.f64 39.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)\right)}}\right) \]

   11. Applied rewrites 39.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}\right)}}\right) \]

   13. Applied rewrites 38.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \color{blue}{\cos \phi_2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)\right)}}\right) \]

   15. Applied rewrites 48.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_2}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \color{blue}{\cos \phi_2}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 37: 60.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\\ t_1 := \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_2\\ t_2 := \tan^{-1}_* \frac{\sqrt{t\_1 + t\_0}}{\sqrt{\left(1 - t\_0\right) - t\_1}} \cdot \left(R + R\right)\\ t_3 := \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_1 \cdot 0.5\right)\right)\right)\\ \mathbf{if}\;\phi_2 \leq -7.2 \cdot 10^{-8}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 4.2 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5))))))
        (t_1 (* (fma (cos (* -1.0 (- lambda2 lambda1))) -0.5 0.5) (cos phi2)))
        (t_2 (* (atan2 (sqrt (+ t_1 t_0)) (sqrt (- (- 1.0 t_0) t_1))) (+ R R)))
        (t_3
         (fma
          (cos phi1)
          (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
          (- 0.5 (* 0.5 (cos (* 2.0 (* phi1 0.5))))))))
   (if (<= phi2 -7.2e-8)
     t_2
     (if (<= phi2 4.2e-6)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5))));
	double t_1 = fma(cos((-1.0 * (lambda2 - lambda1))), -0.5, 0.5) * cos(phi2);
	double t_2 = atan2(sqrt((t_1 + t_0)), sqrt(((1.0 - t_0) - t_1))) * (R + R);
	double t_3 = fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), (0.5 - (0.5 * cos((2.0 * (phi1 * 0.5))))));
	double tmp;
	if (phi2 <= -7.2e-8) {
		tmp = t_2;
	} else if (phi2 <= 4.2e-6) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5)))))
	t_1 = Float64(fma(cos(Float64(-1.0 * Float64(lambda2 - lambda1))), -0.5, 0.5) * cos(phi2))
	t_2 = Float64(atan(sqrt(Float64(t_1 + t_0)), sqrt(Float64(Float64(1.0 - t_0) - t_1))) * Float64(R + R))
	t_3 = fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(phi1 * 0.5))))))
	tmp = 0.0
	if (phi2 <= -7.2e-8)
		tmp = t_2;
	elseif (phi2 <= 4.2e-6)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = t_2;
	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[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[N[(-1.0 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[N[Sqrt[N[(t$95$1 + t$95$0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - t$95$0), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(R + R), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(phi1 * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -7.2e-8], t$95$2, If[LessEqual[phi2, 4.2e-6], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -7.19999999999999962e-8 or 4.1999999999999996e-6 < phi2

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\pi}, \mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-neg.f64 52.4

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \color{blue}{-\phi_2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 52.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\pi}, \mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-neg.f64 51.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \color{blue}{-\phi_2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 51.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-PI.f64 41.4

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 41.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]

       2. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]

       3. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} - \phi_2\right)\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       5. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       7. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       9. lower-PI.f64 39.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)\right)}}\right) \]

   11. Applied rewrites 39.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}\right)}}\right) \]

   13. Applied rewrites 46.2%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_2 + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)\right) - \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_2}} \cdot \left(R + R\right)} \]

   if -7.19999999999999962e-8 < phi2 < 4.1999999999999996e-6

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. unpow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. sqr-sin-a N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. cos-2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. cos-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{2} \cdot \phi_1 + \frac{1}{2} \cdot \phi_1\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{2} \cdot \phi_1 + \frac{1}{2} \cdot \phi_1\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      10. cos-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      11. cos-2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      13. lower-*.f64 45.7

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

      14. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      15. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      16. lower-*.f64 45.7

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_1 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. Applied rewrites 45.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_1 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       2. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}}\right) \]

       3. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}}\right) \]

       4. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}}\right) \]

       5. sqr-sin-a N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

       7. cos-2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

       8. cos-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{2} \cdot \phi_1 + \frac{1}{2} \cdot \phi_1\right)\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\frac{1}{2} \cdot \phi_1 + \frac{1}{2} \cdot \phi_1\right)\right)}}\right) \]

       10. cos-sum N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \left(\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

       11. cos-2 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

       12. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

       13. lower-*.f64 45.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_1 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

       14. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

       15. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       16. lower-*.f64 45.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_1 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_1 \cdot 0.5\right)\right)\right)}}\right) \]

   11. Applied rewrites 45.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_1 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_1 \cdot 0.5\right)\right)\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 38: 59.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\\ t_1 := \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot t\_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_3 := \mathsf{fma}\left(t\_1, -0.5, 0.5\right) \cdot \cos \phi_2\\ t_4 := \tan^{-1}_* \frac{\sqrt{t\_3 + t\_0}}{\sqrt{\left(1 - t\_0\right) - t\_3}} \cdot \left(R + R\right)\\ \mathbf{if}\;\phi_2 \leq -1.45 \cdot 10^{-8}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;\phi_2 \leq 4.2 \cdot 10^{-6}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5))))))
        (t_1 (cos (* -1.0 (- lambda2 lambda1))))
        (t_2 (fma (cos phi1) (- 0.5 (* 0.5 t_1)) (pow (sin (* 0.5 phi1)) 2.0)))
        (t_3 (* (fma t_1 -0.5 0.5) (cos phi2)))
        (t_4
         (* (atan2 (sqrt (+ t_3 t_0)) (sqrt (- (- 1.0 t_0) t_3))) (+ R R))))
   (if (<= phi2 -1.45e-8)
     t_4
     (if (<= phi2 4.2e-6)
       (* (atan2 (sqrt t_2) (sqrt (- 1.0 t_2))) (* 2.0 R))
       t_4))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5))));
	double t_1 = cos((-1.0 * (lambda2 - lambda1)));
	double t_2 = fma(cos(phi1), (0.5 - (0.5 * t_1)), pow(sin((0.5 * phi1)), 2.0));
	double t_3 = fma(t_1, -0.5, 0.5) * cos(phi2);
	double t_4 = atan2(sqrt((t_3 + t_0)), sqrt(((1.0 - t_0) - t_3))) * (R + R);
	double tmp;
	if (phi2 <= -1.45e-8) {
		tmp = t_4;
	} else if (phi2 <= 4.2e-6) {
		tmp = atan2(sqrt(t_2), sqrt((1.0 - t_2))) * (2.0 * R);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5)))))
	t_1 = cos(Float64(-1.0 * Float64(lambda2 - lambda1)))
	t_2 = fma(cos(phi1), Float64(0.5 - Float64(0.5 * t_1)), (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_3 = Float64(fma(t_1, -0.5, 0.5) * cos(phi2))
	t_4 = Float64(atan(sqrt(Float64(t_3 + t_0)), sqrt(Float64(Float64(1.0 - t_0) - t_3))) * Float64(R + R))
	tmp = 0.0
	if (phi2 <= -1.45e-8)
		tmp = t_4;
	elseif (phi2 <= 4.2e-6)
		tmp = Float64(atan(sqrt(t_2), sqrt(Float64(1.0 - t_2))) * Float64(2.0 * R));
	else
		tmp = t_4;
	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[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(-1.0 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[(0.5 - N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * -0.5 + 0.5), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[ArcTan[N[Sqrt[N[(t$95$3 + t$95$0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - t$95$0), $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(R + R), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.45e-8], t$95$4, If[LessEqual[phi2, 4.2e-6], N[(N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.4500000000000001e-8 or 4.1999999999999996e-6 < phi2

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\pi}, \mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-neg.f64 52.4

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \color{blue}{-\phi_2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 52.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\pi}, \mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-neg.f64 51.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \color{blue}{-\phi_2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 51.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-PI.f64 41.4

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 41.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]

       2. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]

       3. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} - \phi_2\right)\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       5. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       7. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       9. lower-PI.f64 39.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)\right)}}\right) \]

   11. Applied rewrites 39.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}\right)}}\right) \]

   13. Applied rewrites 46.2%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_2 + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)\right) - \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_2}} \cdot \left(R + R\right)} \]

   if -1.4500000000000001e-8 < phi2 < 4.1999999999999996e-6

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   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 31.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   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 21.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 21.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 19.5%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right)} \]

   16. Taylor expanded in lambda1 around inf

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)\right) + \color{blue}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]
   17. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2}} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       3. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       7. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       10. lower-*.f64 19.5

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

   18. Applied rewrites 19.5%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

   19. Taylor expanded in lambda1 around inf

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left(\cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)\right) + \color{blue}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}\right)}} \cdot \left(2 \cdot R\right) \]
   20. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2}} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       3. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       7. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       10. lower-*.f64 44.4

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

   21. Applied rewrites 44.4%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{0.5 - 0.5 \cdot \cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right)}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 39: 58.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_1 \cdot 0.5\right)\right)\\ t_1 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\\ t_2 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\\ t_3 := \mathsf{fma}\left(t\_2, \cos \phi_1, t\_0\right)\\ t_4 := \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_2\\ t_5 := t\_0 + t\_2 \cdot \cos \phi_1\\ \mathbf{if}\;\phi_1 \leq -160:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{elif}\;\phi_1 \leq 0.0007:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{t\_4 + t\_1}}{\sqrt{\left(1 - t\_1\right) - t\_4}} \cdot \left(R + R\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}} \cdot 2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* phi1 0.5))))))
        (t_1 (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5))))))
        (t_2 (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5))))))
        (t_3 (fma t_2 (cos phi1) t_0))
        (t_4 (* (fma (cos (* -1.0 (- lambda2 lambda1))) -0.5 0.5) (cos phi2)))
        (t_5 (+ t_0 (* t_2 (cos phi1)))))
   (if (<= phi1 -160.0)
     (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))
     (if (<= phi1 0.0007)
       (* (atan2 (sqrt (+ t_4 t_1)) (sqrt (- (- 1.0 t_1) t_4))) (+ R R))
       (* (* (atan2 (sqrt t_3) (sqrt (- 1.0 t_3))) 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 * (phi1 * 0.5))));
	double t_1 = 0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5))));
	double t_2 = 0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))));
	double t_3 = fma(t_2, cos(phi1), t_0);
	double t_4 = fma(cos((-1.0 * (lambda2 - lambda1))), -0.5, 0.5) * cos(phi2);
	double t_5 = t_0 + (t_2 * cos(phi1));
	double tmp;
	if (phi1 <= -160.0) {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	} else if (phi1 <= 0.0007) {
		tmp = atan2(sqrt((t_4 + t_1)), sqrt(((1.0 - t_1) - t_4))) * (R + R);
	} else {
		tmp = (atan2(sqrt(t_3), sqrt((1.0 - t_3))) * 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(phi1 * 0.5)))))
	t_1 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5)))))
	t_2 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5)))))
	t_3 = fma(t_2, cos(phi1), t_0)
	t_4 = Float64(fma(cos(Float64(-1.0 * Float64(lambda2 - lambda1))), -0.5, 0.5) * cos(phi2))
	t_5 = Float64(t_0 + Float64(t_2 * cos(phi1)))
	tmp = 0.0
	if (phi1 <= -160.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	elseif (phi1 <= 0.0007)
		tmp = Float64(atan(sqrt(Float64(t_4 + t_1)), sqrt(Float64(Float64(1.0 - t_1) - t_4))) * Float64(R + R));
	else
		tmp = Float64(Float64(atan(sqrt(t_3), sqrt(Float64(1.0 - t_3))) * 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[(phi1 * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Cos[N[(-1.0 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$0 + N[(t$95$2 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -160.0], 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[phi1, 0.0007], N[(N[ArcTan[N[Sqrt[N[(t$95$4 + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - t$95$1), $MachinePrecision] - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(R + R), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -160

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 43.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_1 \cdot 0.5\right)\right)\right) + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 43.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_1 \cdot 0.5\right)\right)\right) + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1}}{\sqrt{1 - \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_1 \cdot 0.5\right)\right)\right) + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1}\right)}}\right) \]

   if -160 < phi1 < 6.99999999999999993e-4

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\pi}, \mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-neg.f64 52.4

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \color{blue}{-\phi_2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 52.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\pi}, \mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-neg.f64 51.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \color{blue}{-\phi_2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 51.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-PI.f64 41.4

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 41.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]

       2. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]

       3. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} - \phi_2\right)\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       5. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       7. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       9. lower-PI.f64 39.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)\right)}}\right) \]

   11. Applied rewrites 39.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}\right)}}\right) \]

   13. Applied rewrites 46.2%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_2 + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)\right) - \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_2}} \cdot \left(R + R\right)} \]

   if 6.99999999999999993e-4 < phi1

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 43.3%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_1 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_1 \cdot 0.5\right)\right)\right)}} \cdot 2\right) \cdot R} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 40: 58.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\\ t_1 := \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_1 \cdot 0.5\right)\right)\right)\\ t_2 := \left(\tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}} \cdot 2\right) \cdot R\\ t_3 := \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -160:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 0.0007:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{t\_3 + t\_0}}{\sqrt{\left(1 - t\_0\right) - t\_3}} \cdot \left(R + R\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5))))))
        (t_1
         (fma
          (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5)))))
          (cos phi1)
          (- 0.5 (* 0.5 (cos (* 2.0 (* phi1 0.5)))))))
        (t_2 (* (* (atan2 (sqrt t_1) (sqrt (- 1.0 t_1))) 2.0) R))
        (t_3 (* (fma (cos (* -1.0 (- lambda2 lambda1))) -0.5 0.5) (cos phi2))))
   (if (<= phi1 -160.0)
     t_2
     (if (<= phi1 0.0007)
       (* (atan2 (sqrt (+ t_3 t_0)) (sqrt (- (- 1.0 t_0) t_3))) (+ R R))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5))));
	double t_1 = fma((0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))))), cos(phi1), (0.5 - (0.5 * cos((2.0 * (phi1 * 0.5))))));
	double t_2 = (atan2(sqrt(t_1), sqrt((1.0 - t_1))) * 2.0) * R;
	double t_3 = fma(cos((-1.0 * (lambda2 - lambda1))), -0.5, 0.5) * cos(phi2);
	double tmp;
	if (phi1 <= -160.0) {
		tmp = t_2;
	} else if (phi1 <= 0.0007) {
		tmp = atan2(sqrt((t_3 + t_0)), sqrt(((1.0 - t_0) - t_3))) * (R + R);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5)))))
	t_1 = 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(phi1 * 0.5))))))
	t_2 = Float64(Float64(atan(sqrt(t_1), sqrt(Float64(1.0 - t_1))) * 2.0) * R)
	t_3 = Float64(fma(cos(Float64(-1.0 * Float64(lambda2 - lambda1))), -0.5, 0.5) * cos(phi2))
	tmp = 0.0
	if (phi1 <= -160.0)
		tmp = t_2;
	elseif (phi1 <= 0.0007)
		tmp = Float64(atan(sqrt(Float64(t_3 + t_0)), sqrt(Float64(Float64(1.0 - t_0) - t_3))) * Float64(R + R));
	else
		tmp = t_2;
	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[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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[(phi1 * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Cos[N[(-1.0 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -160.0], t$95$2, If[LessEqual[phi1, 0.0007], N[(N[ArcTan[N[Sqrt[N[(t$95$3 + t$95$0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - t$95$0), $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(R + R), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -160 or 6.99999999999999993e-4 < phi1

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 43.3%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_1 \cdot 0.5\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\phi_1 \cdot 0.5\right)\right)\right)}} \cdot 2\right) \cdot R} \]

   if -160 < phi1 < 6.99999999999999993e-4

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\pi}, \mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-neg.f64 52.4

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \color{blue}{-\phi_2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 52.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\pi}, \mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-neg.f64 51.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \color{blue}{-\phi_2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 51.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-PI.f64 41.4

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 41.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]

       2. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]

       3. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} - \phi_2\right)\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       5. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       7. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       9. lower-PI.f64 39.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)\right)}}\right) \]

   11. Applied rewrites 39.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}\right)}}\right) \]

   13. Applied rewrites 46.2%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_2 + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)\right) - \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_2}} \cdot \left(R + R\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 41: 57.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\\ t_1 := 1 - t\_0\\ t_2 := \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right)\\ t_3 := t\_2 \cdot \cos \phi_1\\ t_4 := t\_0 + t\_3\\ t_5 := t\_2 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -9.5 \cdot 10^{+22}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{t\_3 + t\_0}}{\sqrt{t\_1 - t\_3}} \cdot \left(R + R\right)\\ \mathbf{elif}\;\phi_1 \leq 0.0007:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{t\_5 + t\_0}}{\sqrt{t\_1 - t\_5}} \cdot \left(R + R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5))))))
        (t_1 (- 1.0 t_0))
        (t_2 (fma (cos (* -1.0 (- lambda2 lambda1))) -0.5 0.5))
        (t_3 (* t_2 (cos phi1)))
        (t_4 (+ t_0 t_3))
        (t_5 (* t_2 (cos phi2))))
   (if (<= phi1 -9.5e+22)
     (* (atan2 (sqrt (+ t_3 t_0)) (sqrt (- t_1 t_3))) (+ R R))
     (if (<= phi1 0.0007)
       (* (atan2 (sqrt (+ t_5 t_0)) (sqrt (- t_1 t_5))) (+ R R))
       (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5))));
	double t_1 = 1.0 - t_0;
	double t_2 = fma(cos((-1.0 * (lambda2 - lambda1))), -0.5, 0.5);
	double t_3 = t_2 * cos(phi1);
	double t_4 = t_0 + t_3;
	double t_5 = t_2 * cos(phi2);
	double tmp;
	if (phi1 <= -9.5e+22) {
		tmp = atan2(sqrt((t_3 + t_0)), sqrt((t_1 - t_3))) * (R + R);
	} else if (phi1 <= 0.0007) {
		tmp = atan2(sqrt((t_5 + t_0)), sqrt((t_1 - t_5))) * (R + R);
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5)))))
	t_1 = Float64(1.0 - t_0)
	t_2 = fma(cos(Float64(-1.0 * Float64(lambda2 - lambda1))), -0.5, 0.5)
	t_3 = Float64(t_2 * cos(phi1))
	t_4 = Float64(t_0 + t_3)
	t_5 = Float64(t_2 * cos(phi2))
	tmp = 0.0
	if (phi1 <= -9.5e+22)
		tmp = Float64(atan(sqrt(Float64(t_3 + t_0)), sqrt(Float64(t_1 - t_3))) * Float64(R + R));
	elseif (phi1 <= 0.0007)
		tmp = Float64(atan(sqrt(Float64(t_5 + t_0)), sqrt(Float64(t_1 - t_5))) * Float64(R + R));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	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[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[N[(-1.0 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -9.5e+22], N[(N[ArcTan[N[Sqrt[N[(t$95$3 + t$95$0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(t$95$1 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(R + R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.0007], N[(N[ArcTan[N[Sqrt[N[(t$95$5 + t$95$0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(t$95$1 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(R + R), $MachinePrecision]), $MachinePrecision], 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]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -9.49999999999999937e22

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 59.0

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 59.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

       4. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       6. lower--.f64 55.8

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 55.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   13. Applied rewrites 46.5%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_1 + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)\right) - \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_1}} \cdot \left(R + R\right)} \]

   if -9.49999999999999937e22 < phi1 < 6.99999999999999993e-4

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\pi}, \mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-neg.f64 52.4

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \color{blue}{-\phi_2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 52.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\pi}, \mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-neg.f64 51.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \color{blue}{-\phi_2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 51.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-PI.f64 41.4

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 41.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]

       2. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]

       3. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} - \phi_2\right)\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       5. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       7. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       9. lower-PI.f64 39.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)\right)}}\right) \]

   11. Applied rewrites 39.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}\right)}}\right) \]

   13. Applied rewrites 46.2%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_2 + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)\right) - \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_2}} \cdot \left(R + R\right)} \]

   if 6.99999999999999993e-4 < phi1

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 59.0

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 59.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

       4. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       6. lower--.f64 55.8

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 55.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   13. Applied rewrites 46.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) + \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_1}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 46.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) + \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_1}}{\sqrt{1 - \color{blue}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right) + \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_1\right)}}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 42: 57.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\\ t_1 := 1 - t\_0\\ t_2 := \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right)\\ t_3 := t\_2 \cdot \cos \phi_1\\ t_4 := \tan^{-1}_* \frac{\sqrt{t\_3 + t\_0}}{\sqrt{t\_1 - t\_3}} \cdot \left(R + R\right)\\ t_5 := t\_2 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -9.5 \cdot 10^{+22}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;\phi_1 \leq 0.0007:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{t\_5 + t\_0}}{\sqrt{t\_1 - t\_5}} \cdot \left(R + R\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5))))))
        (t_1 (- 1.0 t_0))
        (t_2 (fma (cos (* -1.0 (- lambda2 lambda1))) -0.5 0.5))
        (t_3 (* t_2 (cos phi1)))
        (t_4 (* (atan2 (sqrt (+ t_3 t_0)) (sqrt (- t_1 t_3))) (+ R R)))
        (t_5 (* t_2 (cos phi2))))
   (if (<= phi1 -9.5e+22)
     t_4
     (if (<= phi1 0.0007)
       (* (atan2 (sqrt (+ t_5 t_0)) (sqrt (- t_1 t_5))) (+ R R))
       t_4))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5))));
	double t_1 = 1.0 - t_0;
	double t_2 = fma(cos((-1.0 * (lambda2 - lambda1))), -0.5, 0.5);
	double t_3 = t_2 * cos(phi1);
	double t_4 = atan2(sqrt((t_3 + t_0)), sqrt((t_1 - t_3))) * (R + R);
	double t_5 = t_2 * cos(phi2);
	double tmp;
	if (phi1 <= -9.5e+22) {
		tmp = t_4;
	} else if (phi1 <= 0.0007) {
		tmp = atan2(sqrt((t_5 + t_0)), sqrt((t_1 - t_5))) * (R + R);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5)))))
	t_1 = Float64(1.0 - t_0)
	t_2 = fma(cos(Float64(-1.0 * Float64(lambda2 - lambda1))), -0.5, 0.5)
	t_3 = Float64(t_2 * cos(phi1))
	t_4 = Float64(atan(sqrt(Float64(t_3 + t_0)), sqrt(Float64(t_1 - t_3))) * Float64(R + R))
	t_5 = Float64(t_2 * cos(phi2))
	tmp = 0.0
	if (phi1 <= -9.5e+22)
		tmp = t_4;
	elseif (phi1 <= 0.0007)
		tmp = Float64(atan(sqrt(Float64(t_5 + t_0)), sqrt(Float64(t_1 - t_5))) * Float64(R + R));
	else
		tmp = t_4;
	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[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[N[(-1.0 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[ArcTan[N[Sqrt[N[(t$95$3 + t$95$0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(t$95$1 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(R + R), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -9.5e+22], t$95$4, If[LessEqual[phi1, 0.0007], N[(N[ArcTan[N[Sqrt[N[(t$95$5 + t$95$0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(t$95$1 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(R + R), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -9.49999999999999937e22 or 6.99999999999999993e-4 < phi1

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 59.0

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 59.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

       4. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       6. lower--.f64 55.8

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 55.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   13. Applied rewrites 46.5%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_1 + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)\right) - \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_1}} \cdot \left(R + R\right)} \]

   if -9.49999999999999937e22 < phi1 < 6.99999999999999993e-4

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\pi}, \mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-neg.f64 52.4

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \color{blue}{-\phi_2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 52.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. cos-neg-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. sin-+PI/2-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\phi_2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{2} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right) \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\frac{1}{2}} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} + \left(\mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \mathsf{PI}\left(\right), \mathsf{neg}\left(\phi_2\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-PI.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\pi}, \mathsf{neg}\left(\phi_2\right)\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-neg.f64 51.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, \color{blue}{-\phi_2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 51.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(\frac{1}{2}, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-PI.f64 41.4

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 41.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, -\phi_2\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]

       2. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)}\right)}}\right) \]

       3. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)} - \phi_2\right)\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\color{blue}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       5. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       7. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right) - \phi_2\right)\right)}}\right) \]

       9. lower-PI.f64 39.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)\right)}}\right) \]

   11. Applied rewrites 39.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \sin \left(0.5 \cdot \pi - \phi_2\right)}\right)}}\right) \]

   13. Applied rewrites 46.2%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_2 + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)\right) - \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_2}} \cdot \left(R + R\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 43: 49.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_1\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\\ \mathbf{if}\;{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_2\right) \cdot t\_2 \leq 5 \cdot 10^{-28}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, t\_0, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{t\_1 + t\_3}}{\sqrt{\left(1 - t\_3\right) - t\_1}} \cdot \left(R + R\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_1 (* (fma (cos (* -1.0 (- lambda2 lambda1))) -0.5 0.5) (cos phi1)))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3 (- 0.5 (* 0.5 (cos (* 2.0 (* (- phi1 phi2) 0.5)))))))
   (if (<=
        (+
         (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
         (* (* (* (cos phi1) (cos phi2)) t_2) t_2))
        5e-28)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (fma (cos phi1) t_0 (pow (* 0.5 phi1) 2.0)))
        (sqrt (- 1.0 (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0)))))))
     (* (atan2 (sqrt (+ t_1 t_3)) (sqrt (- (- 1.0 t_3) t_1))) (+ R 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(cos((-1.0 * (lambda2 - lambda1))), -0.5, 0.5) * cos(phi1);
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = 0.5 - (0.5 * cos((2.0 * ((phi1 - phi2) * 0.5))));
	double tmp;
	if ((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_2) * t_2)) <= 5e-28) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), t_0, pow((0.5 * phi1), 2.0))), sqrt((1.0 - fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0))))));
	} else {
		tmp = atan2(sqrt((t_1 + t_3)), sqrt(((1.0 - t_3) - t_1))) * (R + R);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_1 = Float64(fma(cos(Float64(-1.0 * Float64(lambda2 - lambda1))), -0.5, 0.5) * cos(phi1))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(phi1 - phi2) * 0.5)))))
	tmp = 0.0
	if (Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_2) * t_2)) <= 5e-28)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), t_0, (Float64(0.5 * phi1) ^ 2.0))), sqrt(Float64(1.0 - fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0)))))));
	else
		tmp = Float64(atan(sqrt(Float64(t_1 + t_3)), sqrt(Float64(Float64(1.0 - t_3) - t_1))) * Float64(R + 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[(N[Cos[N[(-1.0 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 5e-28], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[(0.5 * phi1), $MachinePrecision], 2.0], $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], N[(N[ArcTan[N[Sqrt[N[(t$95$1 + t$95$3), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - t$95$3), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(R + R), $MachinePrecision]), $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))))) < 5.0000000000000002e-28

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   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 31.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   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 21.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 21.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   15. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       7. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       10. lower-*.f64 31.8

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   16. Applied rewrites 31.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   if 5.0000000000000002e-28 < (+.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 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. div-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. sin-diff N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\color{blue}{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 59.0

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Applied rewrites 59.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

       4. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot \frac{-1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right) - \sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

       6. lower--.f64 55.8

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 55.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\cos \left(\phi_2 \cdot -0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   13. Applied rewrites 46.5%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_1 + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)}}{\sqrt{\left(1 - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)\right) - \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_1}} \cdot \left(R + R\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 44: 42.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \lambda_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_2 := \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}} \cdot \left(2 \cdot R\right)\\ \mathbf{if}\;\phi_1 \leq -3.1 \cdot 10^{-34}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 8.6 \cdot 10^{-45}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, t\_0, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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)
          (- 0.5 (* 0.5 (cos lambda1)))
          (pow (sin (* 0.5 phi1)) 2.0)))
        (t_2 (* (atan2 (sqrt t_1) (sqrt (- 1.0 t_1))) (* 2.0 R))))
   (if (<= phi1 -3.1e-34)
     t_2
     (if (<= phi1 8.6e-45)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0)))
          (sqrt (- 1.0 (fma (cos phi1) t_0 (pow (* 0.5 phi1) 2.0)))))))
       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), (0.5 - (0.5 * cos(lambda1))), pow(sin((0.5 * phi1)), 2.0));
	double t_2 = atan2(sqrt(t_1), sqrt((1.0 - t_1))) * (2.0 * R);
	double tmp;
	if (phi1 <= -3.1e-34) {
		tmp = t_2;
	} else if (phi1 <= 8.6e-45) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0))), sqrt((1.0 - fma(cos(phi1), t_0, pow((0.5 * phi1), 2.0))))));
	} else {
		tmp = 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), Float64(0.5 - Float64(0.5 * cos(lambda1))), (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_2 = Float64(atan(sqrt(t_1), sqrt(Float64(1.0 - t_1))) * Float64(2.0 * R))
	tmp = 0.0
	if (phi1 <= -3.1e-34)
		tmp = t_2;
	elseif (phi1 <= 8.6e-45)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0))), sqrt(Float64(1.0 - fma(cos(phi1), t_0, (Float64(0.5 * phi1) ^ 2.0)))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -3.1e-34], t$95$2, If[LessEqual[phi1, 8.6e-45], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[(0.5 * phi1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.0999999999999998e-34 or 8.5999999999999998e-45 < phi1

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   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 31.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   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 21.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 21.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 19.5%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right)} \]

   16. Taylor expanded in lambda2 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right) + \color{blue}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]
   17. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \lambda_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2}} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       3. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \lambda_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       6. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       7. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       8. lower-*.f64 13.4

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \lambda_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

   18. Applied rewrites 13.4%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{0.5 - 0.5 \cdot \cos \lambda_1}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

   19. Taylor expanded in lambda2 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left(\cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right) + \color{blue}{{\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}\right)}} \cdot \left(2 \cdot R\right) \]
   20. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \lambda_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2}} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       3. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \lambda_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       6. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       7. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       8. lower-*.f64 33.2

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \lambda_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \lambda_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

   21. Applied rewrites 33.2%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \lambda_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{0.5 - 0.5 \cdot \cos \lambda_1}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

   if -3.0999999999999998e-34 < phi1 < 8.5999999999999998e-45

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   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 31.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   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 21.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 21.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       7. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       10. lower-*.f64 23.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   16. Applied rewrites 23.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 45: 33.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_2 := \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - t\_1}} \cdot \left(2 \cdot R\right)\\ \mathbf{if}\;\phi_1 \leq -1.7 \cdot 10^{-35}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 6.5 \cdot 10^{-50}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, t\_0, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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
         (*
          (atan2
           (sqrt
            (fma
             (* phi1 0.5)
             (* phi1 0.5)
             (*
              (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda2 lambda1) -0.5)))))
              (cos phi1))))
           (sqrt (- 1.0 t_1)))
          (* 2.0 R))))
   (if (<= phi1 -1.7e-35)
     t_2
     (if (<= phi1 6.5e-50)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt t_1)
          (sqrt (- 1.0 (fma (cos phi1) t_0 (pow (* 0.5 phi1) 2.0)))))))
       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(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0));
	double t_2 = atan2(sqrt(fma((phi1 * 0.5), (phi1 * 0.5), ((0.5 - (0.5 * cos((2.0 * ((lambda2 - lambda1) * -0.5))))) * cos(phi1)))), sqrt((1.0 - t_1))) * (2.0 * R);
	double tmp;
	if (phi1 <= -1.7e-35) {
		tmp = t_2;
	} else if (phi1 <= 6.5e-50) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - fma(cos(phi1), t_0, pow((0.5 * phi1), 2.0))))));
	} else {
		tmp = 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(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0))
	t_2 = 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(lambda2 - lambda1) * -0.5))))) * cos(phi1)))), sqrt(Float64(1.0 - t_1))) * Float64(2.0 * R))
	tmp = 0.0
	if (phi1 <= -1.7e-35)
		tmp = t_2;
	elseif (phi1 <= 6.5e-50)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - fma(cos(phi1), t_0, (Float64(0.5 * phi1) ^ 2.0)))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[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[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.7e-35], t$95$2, If[LessEqual[phi1, 6.5e-50], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[(0.5 * phi1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.7000000000000001e-35 or 6.49999999999999987e-50 < phi1

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   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 31.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   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 21.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 21.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 19.5%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right)} \]

   16. Taylor expanded in phi1 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}} \cdot \left(2 \cdot R\right) \]
   17. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}} \cdot \left(2 \cdot R\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       7. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       10. lower-*.f64 29.4

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

   18. Applied rewrites 29.4%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}} \cdot \left(2 \cdot R\right) \]

   if -1.7000000000000001e-35 < phi1 < 6.49999999999999987e-50

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   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 31.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   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 21.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 21.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       7. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       10. lower-*.f64 23.0

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   16. Applied rewrites 23.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 46: 33.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)\\ t_1 := \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_2 := \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_1}} \cdot \left(2 \cdot R\right)\\ \mathbf{if}\;\phi_1 \leq -1.7 \cdot 10^{-35}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 6.5 \cdot 10^{-50}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_0}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (* (- lambda2 lambda1) -0.5)))))
           (cos phi1))))
        (t_1
         (fma
          (cos phi2)
          (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
          (pow (sin (* -0.5 phi2)) 2.0)))
        (t_2 (* (atan2 (sqrt t_0) (sqrt (- 1.0 t_1))) (* 2.0 R))))
   (if (<= phi1 -1.7e-35)
     t_2
     (if (<= phi1 6.5e-50)
       (* (atan2 (sqrt t_1) (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 * ((lambda2 - lambda1) * -0.5))))) * cos(phi1)));
	double t_1 = fma(cos(phi2), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow(sin((-0.5 * phi2)), 2.0));
	double t_2 = atan2(sqrt(t_0), sqrt((1.0 - t_1))) * (2.0 * R);
	double tmp;
	if (phi1 <= -1.7e-35) {
		tmp = t_2;
	} else if (phi1 <= 6.5e-50) {
		tmp = atan2(sqrt(t_1), 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(lambda2 - lambda1) * -0.5))))) * cos(phi1)))
	t_1 = fma(cos(phi2), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (sin(Float64(-0.5 * phi2)) ^ 2.0))
	t_2 = Float64(atan(sqrt(t_0), sqrt(Float64(1.0 - t_1))) * Float64(2.0 * R))
	tmp = 0.0
	if (phi1 <= -1.7e-35)
		tmp = t_2;
	elseif (phi1 <= 6.5e-50)
		tmp = Float64(atan(sqrt(t_1), sqrt(Float64(1.0 - t_0))) * Float64(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[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, Block[{t$95$2 = N[(N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.7e-35], t$95$2, If[LessEqual[phi1, 6.5e-50], N[(N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.7000000000000001e-35 or 6.49999999999999987e-50 < phi1

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   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 31.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   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 21.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 21.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 19.5%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right)} \]

   16. Taylor expanded in phi1 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}} \cdot \left(2 \cdot R\right) \]
   17. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}} \cdot \left(2 \cdot R\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       7. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       10. lower-*.f64 29.4

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

   18. Applied rewrites 29.4%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}} \cdot \left(2 \cdot R\right) \]

   if -1.7000000000000001e-35 < phi1 < 6.49999999999999987e-50

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   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 31.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   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 21.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 21.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 19.5%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right)} \]

   16. Taylor expanded in phi1 around 0

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]
   17. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       7. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       10. lower-*.f64 23.0

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

   18. Applied rewrites 23.0%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 47: 33.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, t\_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right)\\ \mathbf{if}\;\phi_1 \leq -1.7 \cdot 10^{-35}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 6.5 \cdot 10^{-50}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, t\_1, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - t\_0}} \cdot \left(2 \cdot R\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (* (- lambda2 lambda1) -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 -1.7e-35)
     t_2
     (if (<= phi1 6.5e-50)
       (*
        (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 * ((lambda2 - lambda1) * -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 <= -1.7e-35) {
		tmp = t_2;
	} else if (phi1 <= 6.5e-50) {
		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(lambda2 - lambda1) * -0.5))))) * cos(phi1)))
	t_1 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_2 = Float64(atan(sqrt(t_0), sqrt(Float64(1.0 - fma(cos(phi1), t_1, (sin(Float64(0.5 * phi1)) ^ 2.0))))) * Float64(2.0 * R))
	tmp = 0.0
	if (phi1 <= -1.7e-35)
		tmp = t_2;
	elseif (phi1 <= 6.5e-50)
		tmp = Float64(atan(sqrt(fma(cos(phi2), t_1, (sin(Float64(-0.5 * phi2)) ^ 2.0))), sqrt(Float64(1.0 - t_0))) * Float64(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[(lambda2 - lambda1), $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[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] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.7e-35], t$95$2, If[LessEqual[phi1, 6.5e-50], 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] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.7000000000000001e-35 or 6.49999999999999987e-50 < phi1

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   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 31.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   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 21.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 21.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 19.5%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right)} \]

   16. Taylor expanded in phi2 around 0

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}} \cdot \left(2 \cdot R\right) \]
   17. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}} \cdot \left(2 \cdot R\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       7. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

       10. lower-*.f64 29.1

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}} \cdot \left(2 \cdot R\right) \]

   18. Applied rewrites 29.1%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}} \cdot \left(2 \cdot R\right) \]

   if -1.7000000000000001e-35 < phi1 < 6.49999999999999987e-50

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   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 31.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   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 21.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 21.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 19.5%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right)} \]

   16. Taylor expanded in phi1 around 0

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]
   17. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       4. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       7. lower--.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

       10. lower-*.f64 23.0

           \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

   18. Applied rewrites 23.0%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 48: 31.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, t\_0, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt (fma (cos phi1) t_0 (pow (* 0.5 phi1) 2.0)))
      (sqrt (- 1.0 (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	return R * (2.0 * atan2(sqrt(fma(cos(phi1), t_0, pow((0.5 * phi1), 2.0))), sqrt((1.0 - fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0))))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	return Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), t_0, (Float64(0.5 * phi1) ^ 2.0))), sqrt(Float64(1.0 - fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0)))))))
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]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[(0.5 * phi1), $MachinePrecision], 2.0], $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]]

Derivation

1. Initial program 62.6%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

2. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. lower-*.f64 46.7

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 46.7%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

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 46.9

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

7. Applied rewrites 46.9%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

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 31.6

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

10. Applied rewrites 31.6%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

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 21.9

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

13. Applied rewrites 21.9%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

14. Taylor expanded in phi1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
15. Step-by-step derivation

    1. lower--.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]

    2. lower-fma.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

    3. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

    4. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

    5. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

    6. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

    7. lower--.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

    8. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

    9. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

    10. lower-*.f64 31.8

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

16. Applied rewrites 31.8%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]
17. Add Preprocessing

Alternative 49: 31.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, t\_0, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt (fma (cos phi1) t_0 (pow (* 0.5 phi1) 2.0)))
      (sqrt (- 1.0 (fma (cos phi1) t_0 (pow (sin (* 0.5 phi1)) 2.0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	return R * (2.0 * atan2(sqrt(fma(cos(phi1), t_0, pow((0.5 * phi1), 2.0))), sqrt((1.0 - fma(cos(phi1), t_0, pow(sin((0.5 * phi1)), 2.0))))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	return Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), t_0, (Float64(0.5 * phi1) ^ 2.0))), sqrt(Float64(1.0 - fma(cos(phi1), t_0, (sin(Float64(0.5 * phi1)) ^ 2.0)))))))
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]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[(0.5 * phi1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 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]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.6%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

2. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. lower-*.f64 46.7

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 46.7%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

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 46.9

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

7. Applied rewrites 46.9%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

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 31.6

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

10. Applied rewrites 31.6%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

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 21.9

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

13. Applied rewrites 21.9%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

14. Taylor expanded in phi2 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
15. Step-by-step derivation

    1. lower--.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]

    2. lower-fma.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

    3. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

    4. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

    5. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

    6. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

    7. lower--.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

    8. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

    9. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

    10. lower-*.f64 31.6

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

16. Applied rewrites 31.6%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
17. Add Preprocessing

Alternative 50: 27.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{{\left(1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)\right)}^{0.5}}\right) \end{array} \]

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 (* (- lambda2 lambda1) -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 * ((lambda2 - lambda1) * -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(lambda2 - lambda1) * -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[(lambda2 - lambda1), $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 62.6%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

2. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. lower-*.f64 46.7

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 46.7%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

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 46.9

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

7. Applied rewrites 46.9%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

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 31.6

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

10. Applied rewrites 31.6%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

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 21.9

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

13. Applied rewrites 21.9%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

15. Applied rewrites 27.2%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)\right)}^{0.5}}}\right) \]
16. Add Preprocessing

Alternative 51: 27.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(1 + -0.5 \cdot {\phi_1}^{2}, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1\\ \mathbf{if}\;2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}} \leq 0.072:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{{\left(1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_1\right)\right)}^{0.5}} \cdot \left(2 \cdot R\right)\\ \end{array} \end{array} \]

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)))) 0.072)
     (* 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 (* (- lambda2 lambda1) -0.5)))))
          (cos phi1))))
       (pow
        (-
         1.0
         (fma
          (* phi1 0.5)
          (* phi1 0.5)
          (* (fma (cos (* -1.0 (- lambda2 lambda1))) -0.5 0.5) (cos 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)))) <= 0.072) {
		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 * ((lambda2 - lambda1) * -0.5))))) * cos(phi1)))), pow((1.0 - fma((phi1 * 0.5), (phi1 * 0.5), (fma(cos((-1.0 * (lambda2 - lambda1))), -0.5, 0.5) * cos(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)))) <= 0.072)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = 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(lambda2 - lambda1) * -0.5))))) * cos(phi1)))), (Float64(1.0 - fma(Float64(phi1 * 0.5), Float64(phi1 * 0.5), Float64(fma(cos(Float64(-1.0 * Float64(lambda2 - lambda1))), -0.5, 0.5) * cos(phi1)))) ^ 0.5)) * Float64(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], 0.072], 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[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[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Power[N[(1.0 - N[(N[(phi1 * 0.5), $MachinePrecision] * N[(phi1 * 0.5), $MachinePrecision] + N[(N[(N[Cos[N[(-1.0 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 #s(literal 2 binary64) (atan2.f64 (sqrt.f64 (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))) (sqrt.f64 (-.f64 #s(literal 1 binary64) (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))))))) < 0.0719999999999999946

   1. Initial program 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   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 31.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   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 21.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 21.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   14. 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 21.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(1 + -0.5 \cdot {\phi_1}^{2}, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   16. Applied rewrites 21.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(1 + -0.5 \cdot {\phi_1}^{2}, {\color{blue}{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   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 21.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(1 + -0.5 \cdot {\phi_1}^{2}, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(1 + -0.5 \cdot {\phi_1}^{2}, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 21.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(1 + -0.5 \cdot {\phi_1}^{2}, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(1 + -0.5 \cdot {\phi_1}^{2}, {\color{blue}{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   if 0.0719999999999999946 < (*.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 62.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.7

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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 46.9

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   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 31.6

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   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 21.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 21.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 19.5%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right)} \]
   16. Step-by-step derivation

       1. lift-sqrt.f64 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\color{blue}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}} \cdot \left(2 \cdot R\right) \]

       2. pow1/2 N/A

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)\right)}^{\frac{1}{2}}}} \cdot \left(2 \cdot R\right) \]

       3. lower-pow.f64 24.7

          \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)\right)}^{0.5}}} \cdot \left(2 \cdot R\right) \]

   17. Applied rewrites 24.7%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_1\right)\right)}^{0.5}}} \cdot \left(2 \cdot R\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 52: 24.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{{\left(1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_1\right)\right)}^{0.5}} \cdot \left(2 \cdot R\right) \end{array} \]

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 (* (- lambda2 lambda1) -0.5)))))
      (cos phi1))))
   (pow
    (-
     1.0
     (fma
      (* phi1 0.5)
      (* phi1 0.5)
      (* (fma (cos (* -1.0 (- lambda2 lambda1))) -0.5 0.5) (cos 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 * ((lambda2 - lambda1) * -0.5))))) * cos(phi1)))), pow((1.0 - fma((phi1 * 0.5), (phi1 * 0.5), (fma(cos((-1.0 * (lambda2 - lambda1))), -0.5, 0.5) * cos(phi1)))), 0.5)) * (2.0 * R);
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return 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(lambda2 - lambda1) * -0.5))))) * cos(phi1)))), (Float64(1.0 - fma(Float64(phi1 * 0.5), Float64(phi1 * 0.5), Float64(fma(cos(Float64(-1.0 * Float64(lambda2 - lambda1))), -0.5, 0.5) * cos(phi1)))) ^ 0.5)) * Float64(2.0 * R))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := 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[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Power[N[(1.0 - N[(N[(phi1 * 0.5), $MachinePrecision] * N[(phi1 * 0.5), $MachinePrecision] + N[(N[(N[Cos[N[(-1.0 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5 + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.6%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

2. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. lower-*.f64 46.7

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 46.7%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

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 46.9

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

7. Applied rewrites 46.9%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

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 31.6

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

10. Applied rewrites 31.6%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

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 21.9

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

13. Applied rewrites 21.9%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

15. Applied rewrites 19.5%

    \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right)} \]
16. Step-by-step derivation

    1. lift-sqrt.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\color{blue}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}} \cdot \left(2 \cdot R\right) \]

    2. pow1/2 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)\right)}^{\frac{1}{2}}}} \cdot \left(2 \cdot R\right) \]

    3. lower-pow.f64 24.7

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)\right)}^{0.5}}} \cdot \left(2 \cdot R\right) \]

17. Applied rewrites 24.7%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \mathsf{fma}\left(\cos \left(-1 \cdot \left(\lambda_2 - \lambda_1\right)\right), -0.5, 0.5\right) \cdot \cos \phi_1\right)\right)}^{0.5}}} \cdot \left(2 \cdot R\right) \]
18. Add Preprocessing

Alternative 53: 19.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \left(1 + -0.5 \cdot {\phi_1}^{2}\right)\right)\\ \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}} \cdot \left(2 \cdot R\right) \end{array} \end{array} \]

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 (* (- lambda2 lambda1) -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 * ((lambda2 - lambda1) * -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(lambda2 - lambda1) * -0.5))))) * Float64(1.0 + Float64(-0.5 * (phi1 ^ 2.0)))))
	return Float64(atan(sqrt(t_0), sqrt(Float64(1.0 - t_0))) * Float64(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[(lambda2 - lambda1), $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[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.6%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

2. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. lower-*.f64 46.7

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 46.7%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

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 46.9

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

7. Applied rewrites 46.9%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

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 31.6

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

10. Applied rewrites 31.6%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

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 21.9

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

13. Applied rewrites 21.9%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

15. Applied rewrites 19.5%

    \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right)} \]

16. Taylor expanded in phi1 around 0

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]
17. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

    2. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

    3. lower-pow.f64 19.5

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \left(1 + -0.5 \cdot {\phi_1}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

18. Applied rewrites 19.5%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \left(1 + -0.5 \cdot {\phi_1}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right) \]

19. Taylor expanded in phi1 around 0

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right)\right)}} \cdot \left(2 \cdot R\right) \]
20. Step-by-step derivation

    1. lower-+.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right)\right)}} \cdot \left(2 \cdot R\right) \]

    2. lower-*.f64 N/A

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot \frac{1}{2}, \phi_1 \cdot \frac{1}{2}, \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot \frac{-1}{2}\right)\right)\right) \cdot \left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right)\right)}} \cdot \left(2 \cdot R\right) \]

    3. lower-pow.f64 19.5

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \left(1 + -0.5 \cdot {\phi_1}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \left(1 + -0.5 \cdot {\phi_1}^{2}\right)\right)}} \cdot \left(2 \cdot R\right) \]

21. Applied rewrites 19.5%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \left(1 + -0.5 \cdot {\phi_1}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)\right)\right) \cdot \left(1 + -0.5 \cdot {\phi_1}^{2}\right)\right)}} \cdot \left(2 \cdot R\right) \]
22. Add Preprocessing

Reproduce

herbie shell --seed 2025150 
(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))))))))))