Distance on a great circle

Percentage Accurate: 62.5% → 88.6%

Time: 28.9s

Alternatives: 32

Speedup: 1.0×

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: 62.5% 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: 88.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot \lambda_1\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \sin \left(0.5 \cdot \phi_1\right)\\ t_3 := \left(-\phi_2\right) \cdot 0.5\\ t_4 := \cos t\_3\\ t_5 := \cos \left(0.5 \cdot \phi_1\right) \cdot \sin t\_3\\ t_6 := \sin \left(0.5 \cdot \lambda_1\right)\\ t_7 := \left(-\lambda_2\right) \cdot 0.5\\ t_8 := t\_6 \cdot \cos t\_7 + t\_0 \cdot \sin t\_7\\ t_9 := \cos \phi_1 \cdot \cos \phi_2\\ t_10 := \left(t\_9 \cdot t\_1\right) \cdot t\_1\\ t_11 := {\left(\mathsf{fma}\left(t\_2, t\_4, t\_5\right)\right)}^{2} + t\_10\\ t_12 := {\left(t\_2 \cdot t\_4 + t\_5\right)}^{2} + t\_10\\ \mathbf{if}\;\phi_2 \leq -2.8 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_12}}{\sqrt{1 - t\_12}}\right)\\ \mathbf{elif}\;\phi_2 \leq 6.8 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(t\_9 \cdot t\_8\right) \cdot t\_8}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), t\_6, t\_0 \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_11}}{\sqrt{1 - t\_11}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 lambda1)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (sin (* 0.5 phi1)))
        (t_3 (* (- phi2) 0.5))
        (t_4 (cos t_3))
        (t_5 (* (cos (* 0.5 phi1)) (sin t_3)))
        (t_6 (sin (* 0.5 lambda1)))
        (t_7 (* (- lambda2) 0.5))
        (t_8 (+ (* t_6 (cos t_7)) (* t_0 (sin t_7))))
        (t_9 (* (cos phi1) (cos phi2)))
        (t_10 (* (* t_9 t_1) t_1))
        (t_11 (+ (pow (fma t_2 t_4 t_5) 2.0) t_10))
        (t_12 (+ (pow (+ (* t_2 t_4) t_5) 2.0) t_10)))
   (if (<= phi2 -2.8e-8)
     (* R (* 2.0 (atan2 (sqrt t_12) (sqrt (- 1.0 t_12)))))
     (if (<= phi2 6.8e-10)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* t_9 t_8) t_8)))
          (sqrt
           (-
            1.0
            (fma
             (cos phi1)
             (*
              (cos phi2)
              (pow
               (fma (cos (* -0.5 lambda2)) t_6 (* t_0 (sin (* -0.5 lambda2))))
               2.0))
             (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))))))))
       (* R (* 2.0 (atan2 (sqrt t_11) (sqrt (- 1.0 t_11)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((0.5 * lambda1));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = sin((0.5 * phi1));
	double t_3 = -phi2 * 0.5;
	double t_4 = cos(t_3);
	double t_5 = cos((0.5 * phi1)) * sin(t_3);
	double t_6 = sin((0.5 * lambda1));
	double t_7 = -lambda2 * 0.5;
	double t_8 = (t_6 * cos(t_7)) + (t_0 * sin(t_7));
	double t_9 = cos(phi1) * cos(phi2);
	double t_10 = (t_9 * t_1) * t_1;
	double t_11 = pow(fma(t_2, t_4, t_5), 2.0) + t_10;
	double t_12 = pow(((t_2 * t_4) + t_5), 2.0) + t_10;
	double tmp;
	if (phi2 <= -2.8e-8) {
		tmp = R * (2.0 * atan2(sqrt(t_12), sqrt((1.0 - t_12))));
	} else if (phi2 <= 6.8e-10) {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + ((t_9 * t_8) * t_8))), sqrt((1.0 - fma(cos(phi1), (cos(phi2) * pow(fma(cos((-0.5 * lambda2)), t_6, (t_0 * sin((-0.5 * lambda2)))), 2.0)), (0.5 - (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_11), sqrt((1.0 - t_11))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(0.5 * lambda1))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = sin(Float64(0.5 * phi1))
	t_3 = Float64(Float64(-phi2) * 0.5)
	t_4 = cos(t_3)
	t_5 = Float64(cos(Float64(0.5 * phi1)) * sin(t_3))
	t_6 = sin(Float64(0.5 * lambda1))
	t_7 = Float64(Float64(-lambda2) * 0.5)
	t_8 = Float64(Float64(t_6 * cos(t_7)) + Float64(t_0 * sin(t_7)))
	t_9 = Float64(cos(phi1) * cos(phi2))
	t_10 = Float64(Float64(t_9 * t_1) * t_1)
	t_11 = Float64((fma(t_2, t_4, t_5) ^ 2.0) + t_10)
	t_12 = Float64((Float64(Float64(t_2 * t_4) + t_5) ^ 2.0) + t_10)
	tmp = 0.0
	if (phi2 <= -2.8e-8)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_12), sqrt(Float64(1.0 - t_12)))));
	elseif (phi2 <= 6.8e-10)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(t_9 * t_8) * t_8))), sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * (fma(cos(Float64(-0.5 * lambda2)), t_6, Float64(t_0 * sin(Float64(-0.5 * lambda2)))) ^ 2.0)), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2))))))))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_11), sqrt(Float64(1.0 - t_11)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[((-phi2) * 0.5), $MachinePrecision]}, Block[{t$95$4 = N[Cos[t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[((-lambda2) * 0.5), $MachinePrecision]}, Block[{t$95$8 = N[(N[(t$95$6 * N[Cos[t$95$7], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Sin[t$95$7], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$9 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$10 = N[(N[(t$95$9 * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$11 = N[(N[Power[N[(t$95$2 * t$95$4 + t$95$5), $MachinePrecision], 2.0], $MachinePrecision] + t$95$10), $MachinePrecision]}, Block[{t$95$12 = N[(N[Power[N[(N[(t$95$2 * t$95$4), $MachinePrecision] + t$95$5), $MachinePrecision], 2.0], $MachinePrecision] + t$95$10), $MachinePrecision]}, If[LessEqual[phi2, -2.8e-8], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$12], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$12), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 6.8e-10], 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[(t$95$9 * t$95$8), $MachinePrecision] * t$95$8), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * t$95$6 + N[(t$95$0 * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$11], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$11), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -2.7999999999999999e-8

   1. Initial program 45.9%

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

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-*.f64 N/A

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

   3. Applied rewrites 47.8%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-*.f64 N/A

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

   5. Applied rewrites 78.2%

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

   if -2.7999999999999999e-8 < phi2 < 6.8000000000000003e-10

   1. Initial program 78.5%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   3. Applied rewrites 77.3%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   5. Applied rewrites 79.1%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   7. Applied rewrites 78.8%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   9. Applied rewrites 98.4%

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

   10. Taylor expanded in lambda1 around inf

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

       1. lower--.f64 N/A

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

       2. unpow2 N/A

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

       3. sqr-sin-a-rev N/A

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

   12. Applied rewrites 98.5%

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

   if 6.8000000000000003e-10 < phi2

   1. Initial program 48.3%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-sin.f64 N/A

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

   3. Applied rewrites 50.0%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-sin.f64 N/A

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

   5. Applied rewrites 79.7%

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

Alternative 2: 88.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sin \left(0.5 \cdot \phi_1\right)\\ t_2 := \left(-\phi_2\right) \cdot 0.5\\ t_3 := \cos \left(0.5 \cdot \phi_1\right) \cdot \sin t\_2\\ t_4 := \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0\\ t_5 := \cos t\_2\\ t_6 := {\left(\mathsf{fma}\left(t\_1, t\_5, t\_3\right)\right)}^{2} + t\_4\\ t_7 := {\left(t\_1 \cdot t\_5 + t\_3\right)}^{2} + t\_4\\ t_8 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ \mathbf{if}\;\phi_2 \leq -2.8 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_7}}{\sqrt{1 - t\_7}}\right)\\ \mathbf{elif}\;\phi_2 \leq 6.8 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot t\_8}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t\_8, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_6}}{\sqrt{1 - t\_6}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (sin (* 0.5 phi1)))
        (t_2 (* (- phi2) 0.5))
        (t_3 (* (cos (* 0.5 phi1)) (sin t_2)))
        (t_4 (* (* (* (cos phi1) (cos phi2)) t_0) t_0))
        (t_5 (cos t_2))
        (t_6 (+ (pow (fma t_1 t_5 t_3) 2.0) t_4))
        (t_7 (+ (pow (+ (* t_1 t_5) t_3) 2.0) t_4))
        (t_8
         (pow
          (fma
           (cos (* -0.5 lambda2))
           (sin (* 0.5 lambda1))
           (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
          2.0)))
   (if (<= phi2 -2.8e-8)
     (* R (* 2.0 (atan2 (sqrt t_7) (sqrt (- 1.0 t_7)))))
     (if (<= phi2 6.8e-10)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (cos phi1) t_8)))
          (sqrt
           (-
            1.0
            (fma
             (cos phi1)
             (* (cos phi2) t_8)
             (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))))))))
       (* 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 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = sin((0.5 * phi1));
	double t_2 = -phi2 * 0.5;
	double t_3 = cos((0.5 * phi1)) * sin(t_2);
	double t_4 = ((cos(phi1) * cos(phi2)) * t_0) * t_0;
	double t_5 = cos(t_2);
	double t_6 = pow(fma(t_1, t_5, t_3), 2.0) + t_4;
	double t_7 = pow(((t_1 * t_5) + t_3), 2.0) + t_4;
	double t_8 = pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0);
	double tmp;
	if (phi2 <= -2.8e-8) {
		tmp = R * (2.0 * atan2(sqrt(t_7), sqrt((1.0 - t_7))));
	} else if (phi2 <= 6.8e-10) {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi1) * t_8))), sqrt((1.0 - fma(cos(phi1), (cos(phi2) * t_8), (0.5 - (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))))))));
	} 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 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sin(Float64(0.5 * phi1))
	t_2 = Float64(Float64(-phi2) * 0.5)
	t_3 = Float64(cos(Float64(0.5 * phi1)) * sin(t_2))
	t_4 = Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0)
	t_5 = cos(t_2)
	t_6 = Float64((fma(t_1, t_5, t_3) ^ 2.0) + t_4)
	t_7 = Float64((Float64(Float64(t_1 * t_5) + t_3) ^ 2.0) + t_4)
	t_8 = fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0
	tmp = 0.0
	if (phi2 <= -2.8e-8)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_7), sqrt(Float64(1.0 - t_7)))));
	elseif (phi2 <= 6.8e-10)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi1) * t_8))), sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * t_8), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2))))))))))));
	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[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[((-phi2) * 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$5 = N[Cos[t$95$2], $MachinePrecision]}, Block[{t$95$6 = N[(N[Power[N[(t$95$1 * t$95$5 + t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + t$95$4), $MachinePrecision]}, Block[{t$95$7 = N[(N[Power[N[(N[(t$95$1 * t$95$5), $MachinePrecision] + t$95$3), $MachinePrecision], 2.0], $MachinePrecision] + t$95$4), $MachinePrecision]}, Block[{t$95$8 = N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[phi2, -2.8e-8], 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], If[LessEqual[phi2, 6.8e-10], 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[phi1], $MachinePrecision] * t$95$8), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$8), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 3 regimes

2. if phi2 < -2.7999999999999999e-8

   1. Initial program 45.9%

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

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-*.f64 N/A

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

   3. Applied rewrites 47.8%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. lower-*.f64 N/A

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

   5. Applied rewrites 78.2%

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

   if -2.7999999999999999e-8 < phi2 < 6.8000000000000003e-10

   1. Initial program 78.5%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   3. Applied rewrites 77.3%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   5. Applied rewrites 79.1%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   7. Applied rewrites 78.8%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   9. Applied rewrites 98.4%

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

   10. Taylor expanded in lambda1 around inf

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

       1. lower--.f64 N/A

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

       2. unpow2 N/A

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

       3. sqr-sin-a-rev N/A

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

   12. Applied rewrites 98.5%

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

   13. Taylor expanded in phi2 around 0

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

       1. lower-*.f64 N/A

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

       2. lift-cos.f64 N/A

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

   15. Applied rewrites 98.5%

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

   if 6.8000000000000003e-10 < phi2

   1. Initial program 48.3%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-sin.f64 N/A

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

   3. Applied rewrites 50.0%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-sin.f64 N/A

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

   5. Applied rewrites 79.7%

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

Alternative 3: 88.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_2 := \left(-\phi_2\right) \cdot 0.5\\ t_3 := {\left(\mathsf{fma}\left(\sin \left(0.5 \cdot \phi_1\right), \cos t\_2, \cos \left(0.5 \cdot \phi_1\right) \cdot \sin t\_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0\\ t_4 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{if}\;\phi_2 \leq -2.8 \cdot 10^{-8}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;\phi_2 \leq 6.8 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot t\_1}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t\_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (pow
          (fma
           (cos (* -0.5 lambda2))
           (sin (* 0.5 lambda1))
           (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
          2.0))
        (t_2 (* (- phi2) 0.5))
        (t_3
         (+
          (pow
           (fma (sin (* 0.5 phi1)) (cos t_2) (* (cos (* 0.5 phi1)) (sin t_2)))
           2.0)
          (* (* (* (cos phi1) (cos phi2)) t_0) t_0)))
        (t_4 (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))))
   (if (<= phi2 -2.8e-8)
     t_4
     (if (<= phi2 6.8e-10)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (cos phi1) t_1)))
          (sqrt
           (-
            1.0
            (fma
             (cos phi1)
             (* (cos phi2) t_1)
             (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))))))))
       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 = pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0);
	double t_2 = -phi2 * 0.5;
	double t_3 = pow(fma(sin((0.5 * phi1)), cos(t_2), (cos((0.5 * phi1)) * sin(t_2))), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	double t_4 = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	double tmp;
	if (phi2 <= -2.8e-8) {
		tmp = t_4;
	} else if (phi2 <= 6.8e-10) {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi1) * t_1))), sqrt((1.0 - fma(cos(phi1), (cos(phi2) * t_1), (0.5 - (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))))))));
	} 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 = fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0
	t_2 = Float64(Float64(-phi2) * 0.5)
	t_3 = Float64((fma(sin(Float64(0.5 * phi1)), cos(t_2), Float64(cos(Float64(0.5 * phi1)) * sin(t_2))) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))
	t_4 = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))))
	tmp = 0.0
	if (phi2 <= -2.8e-8)
		tmp = t_4;
	elseif (phi2 <= 6.8e-10)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi1) * t_1))), sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * t_1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2))))))))))));
	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[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[((-phi2) * 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision] + N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$2], $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]}, Block[{t$95$4 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -2.8e-8], t$95$4, If[LessEqual[phi2, 6.8e-10], 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[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.7999999999999999e-8 or 6.8000000000000003e-10 < phi2

   1. Initial program 47.1%

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

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-sin.f64 N/A

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

   3. Applied rewrites 48.9%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-sin.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-cos.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-*.f64 N/A

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

      18. lower-neg.f64 N/A

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

      19. lower-*.f64 N/A

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

      20. lower-cos.f64 N/A

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

      21. lower-*.f64 N/A

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

      22. lower-sin.f64 N/A

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

   5. Applied rewrites 79.0%

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

   if -2.7999999999999999e-8 < phi2 < 6.8000000000000003e-10

   1. Initial program 78.5%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   3. Applied rewrites 77.3%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   5. Applied rewrites 79.1%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   7. Applied rewrites 78.8%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   9. Applied rewrites 98.4%

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

   10. Taylor expanded in lambda1 around inf

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

       1. lower--.f64 N/A

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

       2. unpow2 N/A

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

       3. sqr-sin-a-rev N/A

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

   12. Applied rewrites 98.5%

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

   13. Taylor expanded in phi2 around 0

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

       1. lower-*.f64 N/A

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

       2. lift-cos.f64 N/A

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

   15. Applied rewrites 98.5%

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

Alternative 4: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_2 := \sqrt{t\_1}\\ t_3 := \sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t\_0, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}\\ \mathbf{if}\;\phi_2 \leq -3.1 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_2}{t\_3}\right)\\ \mathbf{elif}\;\phi_2 \leq 4.8 \cdot 10^{-65}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot t\_0}}{t\_3}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_2}{\sqrt{1 - t\_1}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (fma
           (cos (* -0.5 lambda2))
           (sin (* 0.5 lambda1))
           (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
          2.0))
        (t_1 (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0)))
        (t_2 (sqrt t_1))
        (t_3
         (sqrt
          (-
           1.0
           (fma
            (cos phi1)
            (* (cos phi2) t_0)
            (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2)))))))))))
   (if (<= phi2 -3.1e-5)
     (* R (* 2.0 (atan2 t_2 t_3)))
     (if (<= phi2 4.8e-65)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (cos phi1) t_0)))
          t_3)))
       (* R (* 2.0 (atan2 t_2 (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 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0);
	double t_1 = fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0));
	double t_2 = sqrt(t_1);
	double t_3 = sqrt((1.0 - fma(cos(phi1), (cos(phi2) * t_0), (0.5 - (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))))));
	double tmp;
	if (phi2 <= -3.1e-5) {
		tmp = R * (2.0 * atan2(t_2, t_3));
	} else if (phi2 <= 4.8e-65) {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi1) * t_0))), t_3));
	} else {
		tmp = R * (2.0 * atan2(t_2, sqrt((1.0 - t_1))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0
	t_1 = fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0))
	t_2 = sqrt(t_1)
	t_3 = sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * t_0), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))))))
	tmp = 0.0
	if (phi2 <= -3.1e-5)
		tmp = Float64(R * Float64(2.0 * atan(t_2, t_3)));
	elseif (phi2 <= 4.8e-65)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi1) * t_0))), t_3)));
	else
		tmp = Float64(R * Float64(2.0 * atan(t_2, 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 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -3.1e-5], N[(R * N[(2.0 * N[ArcTan[t$95$2 / t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 4.8e-65], 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[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[t$95$2 / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -3.10000000000000014e-5

   1. Initial program 45.7%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   3. Applied rewrites 45.4%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   5. Applied rewrites 46.0%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   7. Applied rewrites 45.7%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   9. Applied rewrites 56.4%

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

   10. Taylor expanded in lambda1 around inf

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

       1. lower--.f64 N/A

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

       2. unpow2 N/A

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

       3. sqr-sin-a-rev N/A

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

   12. Applied rewrites 56.5%

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

   13. Taylor expanded in phi1 around 0

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

       1. lower-fma.f64 N/A

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

   15. Applied rewrites 57.5%

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

   if -3.10000000000000014e-5 < phi2 < 4.8000000000000003e-65

   1. Initial program 78.4%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   3. Applied rewrites 77.1%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   5. Applied rewrites 79.0%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   7. Applied rewrites 78.7%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   9. Applied rewrites 98.3%

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

   10. Taylor expanded in lambda1 around inf

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

       1. lower--.f64 N/A

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

       2. unpow2 N/A

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

       3. sqr-sin-a-rev N/A

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

   12. Applied rewrites 98.4%

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

   13. Taylor expanded in phi2 around 0

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

       1. lower-*.f64 N/A

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

       2. lift-cos.f64 N/A

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

   15. Applied rewrites 98.4%

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

   if 4.8000000000000003e-65 < phi2

   1. Initial program 52.9%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   3. Applied rewrites 52.4%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   5. Applied rewrites 53.2%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   7. Applied rewrites 52.9%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   9. Applied rewrites 64.3%

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

   10. Taylor expanded in phi1 around 0

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

       1. lower-fma.f64 N/A

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

   12. Applied rewrites 59.0%

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

   13. Taylor expanded in phi1 around 0

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

       1. lower-fma.f64 N/A

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

   15. Applied rewrites 59.5%

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

Alternative 5: 76.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot t\_0\\ t_2 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_3 := \sqrt{t\_2}\\ \mathbf{if}\;\phi_2 \leq -1 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_3}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t\_0, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right)\\ \mathbf{elif}\;\phi_2 \leq 4.8 \cdot 10^{-65}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_3}{\sqrt{1 - t\_2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (fma
           (cos (* -0.5 lambda2))
           (sin (* 0.5 lambda1))
           (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
          2.0))
        (t_1 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (cos phi1) t_0)))
        (t_2 (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0)))
        (t_3 (sqrt t_2)))
   (if (<= phi2 -1e-5)
     (*
      R
      (*
       2.0
       (atan2
        t_3
        (sqrt
         (-
          1.0
          (fma
           (cos phi1)
           (* (cos phi2) t_0)
           (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))))))))
     (if (<= phi2 4.8e-65)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       (* R (* 2.0 (atan2 t_3 (sqrt (- 1.0 t_2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0);
	double t_1 = 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 = sqrt(t_2);
	double tmp;
	if (phi2 <= -1e-5) {
		tmp = R * (2.0 * atan2(t_3, sqrt((1.0 - fma(cos(phi1), (cos(phi2) * t_0), (0.5 - (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))))))));
	} else if (phi2 <= 4.8e-65) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = R * (2.0 * atan2(t_3, sqrt((1.0 - t_2))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0
	t_1 = 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 = sqrt(t_2)
	tmp = 0.0
	if (phi2 <= -1e-5)
		tmp = Float64(R * Float64(2.0 * atan(t_3, sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * t_0), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2))))))))))));
	elseif (phi2 <= 4.8e-65)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(t_3, sqrt(Float64(1.0 - t_2)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[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[Sqrt[t$95$2], $MachinePrecision]}, If[LessEqual[phi2, -1e-5], N[(R * N[(2.0 * N[ArcTan[t$95$3 / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 4.8e-65], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[t$95$3 / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.00000000000000008e-5

   1. Initial program 45.7%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   3. Applied rewrites 45.4%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. negate-sub N/A

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

      6. mul-1-neg N/A

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

      7. metadata-eval N/A

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

      8. *-commutative N/A

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

      9. distribute-rgt-in N/A

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

      10. *-commutative N/A

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

      11. sin-sum N/A

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

      12. lower-+.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-sin.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-cos.f64 N/A

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

      17. mul-1-neg N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

   5. Applied rewrites 46.0%

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

      1. lift-sin.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 45.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   9. Applied rewrites 56.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   11. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

       2. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + \sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}\right)}}\right) \]

       3. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right)\right)}}\right) \]

   12. Applied rewrites 56.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}}\right) \]

   13. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]

   15. Applied rewrites 57.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]

   if -1.00000000000000008e-5 < phi2 < 4.8000000000000003e-65

   1. Initial program 78.4%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 77.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 79.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   9. Applied rewrites 98.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       2. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 98.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   14. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]

       2. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

   15. Applied rewrites 98.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}\right)}}\right) \]

   if 4.8000000000000003e-65 < phi2

   1. Initial program 52.9%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 52.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 53.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 52.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   9. Applied rewrites 64.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 59.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 59.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 76.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot t\_0\\ t_2 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_2 \leq -1 \cdot 10^{-5}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_2 \leq 4.8 \cdot 10^{-65}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (fma
           (cos (* -0.5 lambda2))
           (sin (* 0.5 lambda1))
           (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
          2.0))
        (t_1 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (cos phi1) t_0)))
        (t_2 (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0)))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= phi2 -1e-5)
     t_3
     (if (<= phi2 4.8e-65)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0);
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi1) * t_0);
	double t_2 = fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi2 <= -1e-5) {
		tmp = t_3;
	} else if (phi2 <= 4.8e-65) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0
	t_1 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi1) * t_0))
	t_2 = fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi2 <= -1e-5)
		tmp = t_3;
	elseif (phi2 <= 4.8e-65)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1e-5], t$95$3, If[LessEqual[phi2, 4.8e-65], 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 < -1.00000000000000008e-5 or 4.8000000000000003e-65 < phi2

   1. Initial program 49.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 49.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 50.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 49.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   9. Applied rewrites 60.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 58.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 58.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   if -1.00000000000000008e-5 < phi2 < 4.8000000000000003e-65

   1. Initial program 78.4%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 77.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 79.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   9. Applied rewrites 98.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       2. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 98.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   14. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]

       2. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

   15. Applied rewrites 98.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 76.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 + -1 \cdot \phi_2\right)\right)}^{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos phi1)
          (*
           (cos phi2)
           (pow
            (fma
             (cos (* -0.5 lambda2))
             (sin (* 0.5 lambda1))
             (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
            2.0))
          (pow (sin (* 0.5 (+ phi1 (* -1.0 phi2)))) 2.0))))
   (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos(phi1), (cos(phi2) * pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0)), pow(sin((0.5 * (phi1 + (-1.0 * phi2)))), 2.0));
	return R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(phi1), Float64(cos(phi2) * (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0)), (sin(Float64(0.5 * Float64(phi1 + Float64(-1.0 * phi2)))) ^ 2.0))
	return Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * N[(phi1 + N[(-1.0 * 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 62.5%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. negate-sub N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. mul-1-neg N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. metadata-eval N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. *-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. distribute-rgt-in N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. *-commutative N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. sin-sum N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-+.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. mul-1-neg N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. lower-neg.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

3. Applied rewrites 61.7%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
4. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. negate-sub N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. mul-1-neg N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. metadata-eval N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. *-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. distribute-rgt-in N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. *-commutative N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. sin-sum N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-+.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. mul-1-neg N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. lower-neg.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Applied rewrites 62.9%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
6. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. negate-sub N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. mul-1-neg N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. metadata-eval N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. *-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. distribute-rgt-in N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. *-commutative N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. sin-sum N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-+.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. mul-1-neg N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. lower-neg.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

7. Applied rewrites 62.6%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
8. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

   2. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

   3. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

   4. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right)}}\right) \]

   5. negate-sub N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right)}}\right) \]

   6. mul-1-neg N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right)}}\right) \]

   7. metadata-eval N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)}}\right) \]

   8. *-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right)}}\right) \]

   9. distribute-rgt-in N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right)}}\right) \]

   10. *-commutative N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}\right) \]

   11. sin-sum N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}\right) \]

   12. lower-+.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}\right) \]

   13. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   14. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   15. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   16. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   17. mul-1-neg N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   19. lower-neg.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

9. Applied rewrites 77.5%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right)}}\right) \]

10. Taylor expanded in phi2 around -inf

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 + -1 \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
11. Step-by-step derivation

    1. lower-fma.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 + -1 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

12. Applied rewrites 77.6%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 + -1 \cdot \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right)}}\right) \]

13. Taylor expanded in phi2 around -inf

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 + -1 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 + -1 \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
14. Step-by-step derivation

    1. lower-fma.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 + -1 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 + -1 \cdot \phi_2\right)\right)}^{2}\right)}}\right) \]

15. Applied rewrites 77.5%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 + -1 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 + -1 \cdot \phi_2\right)\right)}^{2}\right)}}}\right) \]
16. Add Preprocessing

Alternative 8: 76.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_1, t\_0, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_1 \leq -9.8 \cdot 10^{-8}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_1 \leq 16000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (fma
           (cos (* -0.5 lambda2))
           (sin (* 0.5 lambda1))
           (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
          2.0))
        (t_1 (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0)))
        (t_2 (fma (cos phi1) t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= phi1 -9.8e-8)
     t_3
     (if (<= phi1 16000.0)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0);
	double t_1 = fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0));
	double t_2 = fma(cos(phi1), t_0, (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi1 <= -9.8e-8) {
		tmp = t_3;
	} else if (phi1 <= 16000.0) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0
	t_1 = fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0))
	t_2 = fma(cos(phi1), t_0, Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1))))))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi1 <= -9.8e-8)
		tmp = t_3;
	elseif (phi1 <= 16000.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(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, -9.8e-8], t$95$3, If[LessEqual[phi1, 16000.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -9.8000000000000004e-8 or 16000 < phi1

   1. Initial program 47.5%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 47.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 47.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 47.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   9. Applied rewrites 57.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       2. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 58.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left(\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}\right)}}\right) \]

       2. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left(\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)}\right)\right)}}\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

   15. Applied rewrites 59.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   if -9.8000000000000004e-8 < phi1 < 16000

   1. Initial program 77.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 76.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 78.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   9. Applied rewrites 97.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 95.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 95.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 69.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_1 := \mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := t\_0 + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot t\_2\right) \cdot t\_2\\ \mathbf{if}\;\phi_2 \leq -7200:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{elif}\;\phi_2 \leq 7.2 \cdot 10^{-31}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_2\right) \cdot t\_2}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot \phi_2\right)\right)\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_1
         (fma
          (cos phi1)
          (pow
           (fma
            (cos (* -0.5 lambda2))
            (sin (* 0.5 lambda1))
            (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
           2.0)
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3
         (+
          t_0
          (*
           (* (/ (+ (cos (+ phi2 phi1)) (cos (- phi2 phi1))) 2.0) t_2)
           t_2))))
   (if (<= phi2 -7200.0)
     (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
     (if (<= phi2 7.2e-31)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ t_0 (* (* (* (cos phi1) (cos phi2)) t_2) t_2)))
          (sqrt
           (-
            1.0
            (fma
             (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))
             (cos phi2)
             (- 0.5 (* 0.5 (cos (* 2.0 (* -0.5 phi2)))))))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_1 = fma(cos(phi1), pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))));
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = t_0 + ((((cos((phi2 + phi1)) + cos((phi2 - phi1))) / 2.0) * t_2) * t_2);
	double tmp;
	if (phi2 <= -7200.0) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else if (phi2 <= 7.2e-31) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((t_0 + (((cos(phi1) * cos(phi2)) * t_2) * t_2))), sqrt((1.0 - fma((0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))), cos(phi2), (0.5 - (0.5 * cos((2.0 * (-0.5 * phi2))))))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_1 = fma(cos(phi1), (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1))))))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64(t_0 + Float64(Float64(Float64(Float64(cos(Float64(phi2 + phi1)) + cos(Float64(phi2 - phi1))) / 2.0) * t_2) * t_2))
	tmp = 0.0
	if (phi2 <= -7200.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	elseif (phi2 <= 7.2e-31)
		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(Float64(t_0 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_2) * t_2))), sqrt(Float64(1.0 - fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))), cos(phi2), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(-0.5 * phi2)))))))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 + N[(N[(N[(N[(N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -7200.0], 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, 7.2e-31], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 + 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[(1.0 - N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(-0.5 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -7200

   1. Initial program 45.4%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\color{blue}{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\color{blue}{\cos \left(\phi_2 + \phi_1\right)} + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \color{blue}{\left(\phi_2 + \phi_1\right)} + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \color{blue}{\cos \left(\phi_2 - \phi_1\right)}}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower--.f64 46.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \color{blue}{\left(\phi_2 - \phi_1\right)}}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 46.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\color{blue}{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\color{blue}{\cos \left(\phi_2 + \phi_1\right)} + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \color{blue}{\left(\phi_2 + \phi_1\right)} + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \color{blue}{\cos \left(\phi_2 - \phi_1\right)}}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower--.f64 47.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \color{blue}{\left(\phi_2 - \phi_1\right)}}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 47.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   if -7200 < phi2 < 7.20000000000000007e-31

   1. Initial program 78.2%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 76.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 78.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 78.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right)}}\right) \]

      5. negate-sub N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \frac{1}{2}\right)\right)}}\right) \]

      6. mul-1-neg N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right) \cdot \frac{1}{2}\right)\right)}}\right) \]

      7. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\left(\lambda_1 + -1 \cdot \lambda_2\right) \cdot \color{blue}{\frac{1}{2}}\right)\right)}}\right) \]

      8. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}\right)}}\right) \]

      9. distribute-rgt-in N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)}\right)}}\right) \]

      10. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \sin \left(\color{blue}{\frac{1}{2} \cdot \lambda_1} + \left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}}\right) \]

      11. sin-sum N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}\right) \]

      12. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)}\right)}}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      14. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      15. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)} \cdot \cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      16. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \color{blue}{\cos \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      17. mul-1-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) \cdot \frac{1}{2}\right)} + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

      19. lower-neg.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\color{blue}{\left(-\lambda_2\right)} \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-1 \cdot \lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   9. Applied rewrites 98.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \color{blue}{\left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       2. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right) \cdot \left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 92.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \left(\sin \left(0.5 \cdot \lambda_1\right) \cdot \cos \left(\left(-\lambda_2\right) \cdot 0.5\right) + \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(\left(-\lambda_2\right) \cdot 0.5\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left(\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}\right)}}\right) \]

       2. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left(\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)}\right)\right)}}\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

   15. Applied rewrites 92.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   if 7.20000000000000007e-31 < phi2

   1. Initial program 50.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]

      2. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_2 + {\color{blue}{\sin \left(\frac{-1}{2} \cdot \phi_2\right)}}^{2}\right)}}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   4. Applied rewrites 49.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot \phi_2\right)\right)\right)}}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 62.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - 0.5 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot t\_1\right) \cdot t\_1\\ \mathbf{if}\;\lambda_1 \leq -1.4 \cdot 10^{+220}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (-
          0.5
          (*
           0.5
           (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (*
           (* (/ (+ (cos (+ phi2 phi1)) (cos (- phi2 phi1))) 2.0) t_1)
           t_1))))
   (if (<= lambda1 -1.4e+220)
     (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
     (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + ((((cos((phi2 + phi1)) + cos((phi2 - phi1))) / 2.0) * t_1) * t_1);
	double tmp;
	if (lambda1 <= -1.4e+220) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} 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 = Float64(0.5 - Float64(0.5 * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2)))))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(Float64(cos(Float64(phi2 + phi1)) + cos(Float64(phi2 - phi1))) / 2.0) * t_1) * t_1))
	tmp = 0.0
	if (lambda1 <= -1.4e+220)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[(N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -1.4e+220], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.4e220

   1. Initial program 48.9%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 41.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 42.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      4. lift--.f64 31.0

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   10. Applied rewrites 31.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - \color{blue}{0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       4. lift--.f64 30.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

   13. Applied rewrites 30.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(0.5 - \color{blue}{0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]
   14. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       2. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       3. cos-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \color{blue}{\sin \lambda_2}\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       4. cos-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \left(\cos \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       6. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       7. cos-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       10. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       11. lower-sin.f64 31.1

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

   15. Applied rewrites 31.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]
   16. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       2. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       3. cos-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \color{blue}{\sin \lambda_2}\right)\right)}}\right) \]

       4. cos-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \left(\cos \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}\right) \]

       6. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}\right) \]

       7. cos-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}\right) \]

       10. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}\right) \]

       11. lower-sin.f64 41.1

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}\right) \]

   17. Applied rewrites 41.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}\right) \]

   if -1.4e220 < lambda1

   1. Initial program 63.5%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\color{blue}{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\color{blue}{\cos \left(\phi_2 + \phi_1\right)} + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \color{blue}{\left(\phi_2 + \phi_1\right)} + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \color{blue}{\cos \left(\phi_2 - \phi_1\right)}}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower--.f64 64.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \color{blue}{\left(\phi_2 - \phi_1\right)}}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 64.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. cos-mult N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\color{blue}{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\color{blue}{\cos \left(\phi_2 + \phi_1\right)} + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \color{blue}{\left(\phi_2 + \phi_1\right)} + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \color{blue}{\cos \left(\phi_2 - \phi_1\right)}}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower--.f64 64.4

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \color{blue}{\left(\phi_2 - \phi_1\right)}}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 64.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\frac{\cos \left(\phi_2 + \phi_1\right) + \cos \left(\phi_2 - \phi_1\right)}{2}} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 62.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := 0.5 - 0.5 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\\ \mathbf{if}\;\lambda_1 \leq -1.4 \cdot 10^{+220}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0}}{\sqrt{\left(1 - \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (-
          0.5
          (*
           0.5
           (fma
            (cos lambda1)
            (cos lambda2)
            (* (sin lambda1) (sin lambda2)))))))
   (if (<= lambda1 -1.4e+220)
     (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_0) t_0)))
        (sqrt
         (-
          (-
           1.0
           (*
            (*
             (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))
             (cos phi2))
            (cos phi1)))
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = 0.5 - (0.5 * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2))));
	double tmp;
	if (lambda1 <= -1.4e+220) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(((1.0 - (((0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * cos(phi2)) * cos(phi1))) - (0.5 - (0.5 * cos((2.0 * (0.5 * (phi1 - phi2))))))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64(0.5 - Float64(0.5 * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2)))))
	tmp = 0.0
	if (lambda1 <= -1.4e+220)
		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(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(Float64(Float64(1.0 - Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))) * cos(phi2)) * cos(phi1))) - Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(0.5 * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -1.4e+220], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[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[(1.0 - N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.4e220

   1. Initial program 48.9%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 41.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 42.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      4. lift--.f64 31.0

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   10. Applied rewrites 31.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - \color{blue}{0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       4. lift--.f64 30.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

   13. Applied rewrites 30.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(0.5 - \color{blue}{0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]
   14. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       2. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       3. cos-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \color{blue}{\sin \lambda_2}\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       4. cos-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \left(\cos \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       6. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       7. cos-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       10. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       11. lower-sin.f64 31.1

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

   15. Applied rewrites 31.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]
   16. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       2. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       3. cos-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \color{blue}{\sin \lambda_2}\right)\right)}}\right) \]

       4. cos-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \left(\cos \lambda_1 \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right) + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}\right) \]

       6. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \left(\mathsf{neg}\left(\lambda_2\right)\right), \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}\right) \]

       7. cos-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}\right) \]

       8. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}\right) \]

       10. lower-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}\right) \]

       11. lower-sin.f64 41.1

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}\right) \]

   17. Applied rewrites 41.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}}\right) \]

   if -1.4e220 < lambda1

   1. Initial program 63.5%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 63.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\left(1 - \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 62.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := 0.5 \cdot \left(\lambda_1 - \lambda_2\right)\\ t_2 := \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot t\_1\right)\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_4 := t\_3 + \left(\sin t\_1 \cdot \cos \phi_1\right) \cdot t\_0\\ t_5 := t\_3 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0\\ \mathbf{if}\;2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}} \leq 0.28:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (* 0.5 (- lambda1 lambda2)))
        (t_2
         (+
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))
          (* (* (- 0.5 (* 0.5 (cos (* 2.0 t_1)))) (cos phi2)) (cos phi1))))
        (t_3 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_4 (+ t_3 (* (* (sin t_1) (cos phi1)) t_0)))
        (t_5 (+ t_3 (* (* (* (cos phi1) (cos phi2)) t_0) t_0))))
   (if (<= (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))) 0.28)
     (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
     (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = 0.5 * (lambda1 - lambda2);
	double t_2 = (0.5 - (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))) + (((0.5 - (0.5 * cos((2.0 * t_1)))) * cos(phi2)) * cos(phi1));
	double t_3 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_4 = t_3 + ((sin(t_1) * cos(phi1)) * t_0);
	double t_5 = t_3 + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	double tmp;
	if ((2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5)))) <= 0.28) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	}
	return tmp;
}

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
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    t_1 = 0.5d0 * (lambda1 - lambda2)
    t_2 = (0.5d0 - (0.5d0 * cos((2.0d0 * (0.5d0 * (phi1 - phi2)))))) + (((0.5d0 - (0.5d0 * cos((2.0d0 * t_1)))) * cos(phi2)) * cos(phi1))
    t_3 = sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0
    t_4 = t_3 + ((sin(t_1) * cos(phi1)) * t_0)
    t_5 = t_3 + (((cos(phi1) * cos(phi2)) * t_0) * t_0)
    if ((2.0d0 * atan2(sqrt(t_5), sqrt((1.0d0 - t_5)))) <= 0.28d0) then
        tmp = r * (2.0d0 * atan2(sqrt(t_4), sqrt((1.0d0 - t_4))))
    else
        tmp = r * (2.0d0 * atan2(sqrt(t_2), sqrt((1.0d0 - t_2))))
    end if
    code = tmp
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 = 0.5 * (lambda1 - lambda2);
	double t_2 = (0.5 - (0.5 * Math.cos((2.0 * (0.5 * (phi1 - phi2)))))) + (((0.5 - (0.5 * Math.cos((2.0 * t_1)))) * Math.cos(phi2)) * Math.cos(phi1));
	double t_3 = Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_4 = t_3 + ((Math.sin(t_1) * Math.cos(phi1)) * t_0);
	double t_5 = t_3 + (((Math.cos(phi1) * Math.cos(phi2)) * t_0) * t_0);
	double tmp;
	if ((2.0 * Math.atan2(Math.sqrt(t_5), Math.sqrt((1.0 - t_5)))) <= 0.28) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(t_4), Math.sqrt((1.0 - t_4))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(t_2), Math.sqrt((1.0 - t_2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	t_1 = 0.5 * (lambda1 - lambda2)
	t_2 = (0.5 - (0.5 * math.cos((2.0 * (0.5 * (phi1 - phi2)))))) + (((0.5 - (0.5 * math.cos((2.0 * t_1)))) * math.cos(phi2)) * math.cos(phi1))
	t_3 = math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0)
	t_4 = t_3 + ((math.sin(t_1) * math.cos(phi1)) * t_0)
	t_5 = t_3 + (((math.cos(phi1) * math.cos(phi2)) * t_0) * t_0)
	tmp = 0
	if (2.0 * math.atan2(math.sqrt(t_5), math.sqrt((1.0 - t_5)))) <= 0.28:
		tmp = R * (2.0 * math.atan2(math.sqrt(t_4), math.sqrt((1.0 - t_4))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt(t_2), math.sqrt((1.0 - t_2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64(0.5 * Float64(lambda1 - lambda2))
	t_2 = Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))) + Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_1)))) * cos(phi2)) * cos(phi1)))
	t_3 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_4 = Float64(t_3 + Float64(Float64(sin(t_1) * cos(phi1)) * t_0))
	t_5 = Float64(t_3 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))
	tmp = 0.0
	if (Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))) <= 0.28)
		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_2), sqrt(Float64(1.0 - t_2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = 0.5 * (lambda1 - lambda2);
	t_2 = (0.5 - (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))) + (((0.5 - (0.5 * cos((2.0 * t_1)))) * cos(phi2)) * cos(phi1));
	t_3 = sin(((phi1 - phi2) / 2.0)) ^ 2.0;
	t_4 = t_3 + ((sin(t_1) * cos(phi1)) * t_0);
	t_5 = t_3 + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	tmp = 0.0;
	if ((2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5)))) <= 0.28)
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	else
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + N[(N[(N[Sin[t$95$1], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 * N[ArcTan[N[Sqrt[t$95$5], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.28], 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$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $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))))))))) < 0.28000000000000003

   1. Initial program 83.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-cos.f64 81.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 81.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \cos \color{blue}{\phi_1}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. metadata-eval N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lift-cos.f64 81.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 81.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1\right)} \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   if 0.28000000000000003 < (*.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 60.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 60.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1}}{\sqrt{1 - \color{blue}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 62.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0\\ t_2 := \sqrt{t\_1}\\ t_3 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\\ t_4 := \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) + \left(t\_3 \cdot \cos \phi_2\right) \cdot \cos \phi_1\\ \mathbf{if}\;2 \cdot \tan^{-1}_* \frac{t\_2}{\sqrt{1 - t\_1}} \leq 0.05:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_2}{\sqrt{1 - \mathsf{fma}\left(t\_3, \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot \phi_2\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_0) t_0)))
        (t_2 (sqrt t_1))
        (t_3 (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2)))))))
        (t_4
         (+
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))
          (* (* t_3 (cos phi2)) (cos phi1)))))
   (if (<= (* 2.0 (atan2 t_2 (sqrt (- 1.0 t_1)))) 0.05)
     (*
      R
      (*
       2.0
       (atan2
        t_2
        (sqrt
         (-
          1.0
          (fma t_3 (cos phi2) (- 0.5 (* 0.5 (cos (* 2.0 (* -0.5 phi2)))))))))))
     (* 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 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	double t_2 = sqrt(t_1);
	double t_3 = 0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))));
	double t_4 = (0.5 - (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))) + ((t_3 * cos(phi2)) * cos(phi1));
	double tmp;
	if ((2.0 * atan2(t_2, sqrt((1.0 - t_1)))) <= 0.05) {
		tmp = R * (2.0 * atan2(t_2, sqrt((1.0 - fma(t_3, cos(phi2), (0.5 - (0.5 * cos((2.0 * (-0.5 * phi2))))))))));
	} 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 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))
	t_2 = sqrt(t_1)
	t_3 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2))))))
	t_4 = Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))) + Float64(Float64(t_3 * cos(phi2)) * cos(phi1)))
	tmp = 0.0
	if (Float64(2.0 * atan(t_2, sqrt(Float64(1.0 - t_1)))) <= 0.05)
		tmp = Float64(R * Float64(2.0 * atan(t_2, sqrt(Float64(1.0 - fma(t_3, cos(phi2), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(-0.5 * phi2)))))))))));
	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[(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]}, Block[{t$95$2 = N[Sqrt[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(2.0 * N[ArcTan[t$95$2 / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 0.05], N[(R * N[(2.0 * N[ArcTan[t$95$2 / N[Sqrt[N[(1.0 - N[(t$95$3 * N[Cos[phi2], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(-0.5 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 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))))))))) < 0.050000000000000003

   1. Initial program 94.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]

      2. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_2 + {\color{blue}{\sin \left(\frac{-1}{2} \cdot \phi_2\right)}}^{2}\right)}}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   4. Applied rewrites 92.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot \phi_2\right)\right)\right)}}}\right) \]

   if 0.050000000000000003 < (*.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 60.1%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 60.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1}}{\sqrt{1 - \color{blue}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 62.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\\ t_1 := \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) + \left(t\_0 \cdot \cos \phi_2\right) \cdot \cos \phi_1\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_2\right) \cdot t\_2\\ \mathbf{if}\;t\_3 \leq 5 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - \mathsf{fma}\left(t\_0, \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2)))))))
        (t_1
         (+
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))
          (* (* t_0 (cos phi2)) (cos phi1))))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_2) t_2))))
   (if (<= t_3 5e-7)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt t_3)
        (sqrt
         (-
          1.0
          (fma t_0 (cos phi1) (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))))))
     (* 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 = 0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))));
	double t_1 = (0.5 - (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))) + ((t_0 * cos(phi2)) * cos(phi1));
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_2) * t_2);
	double tmp;
	if (t_3 <= 5e-7) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - fma(t_0, cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))))))));
	} 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 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2))))))
	t_1 = Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))) + Float64(Float64(t_0 * cos(phi2)) * cos(phi1)))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_2) * t_2))
	tmp = 0.0
	if (t_3 <= 5e-7)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - fma(t_0, cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1)))))))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[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]}, If[LessEqual[t$95$3, 5e-7], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$0 * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))) < 4.99999999999999977e-7

   1. Initial program 69.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{1 - \left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]

      2. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   4. Applied rewrites 69.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   if 4.99999999999999977e-7 < (+.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 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 61.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1}}{\sqrt{1 - \color{blue}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 60.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\\ t_1 := \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{if}\;\phi_2 \leq -3.6 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 3.8 \cdot 10^{-55}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t\_0, {\sin \left(0.5 \cdot \left(\phi_1 + -1 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(t\_0, \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2)))))))
        (t_1
         (fma
          (cos phi2)
          (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))
          (pow (sin (* -0.5 phi2)) 2.0)))
        (t_2 (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))
   (if (<= phi2 -3.6e-6)
     t_2
     (if (<= phi2 3.8e-55)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (fma
            (cos phi1)
            (* (cos phi2) t_0)
            (pow (sin (* 0.5 (+ phi1 (* -1.0 phi2)))) 2.0)))
          (sqrt
           (-
            1.0
            (fma
             t_0
             (cos phi1)
             (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))));
	double t_1 = fma(cos(phi2), (0.5 - (0.5 * cos((lambda1 - lambda2)))), 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 <= -3.6e-6) {
		tmp = t_2;
	} else if (phi2 <= 3.8e-55) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), (cos(phi2) * t_0), pow(sin((0.5 * (phi1 + (-1.0 * phi2)))), 2.0))), sqrt((1.0 - fma(t_0, cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2))))))
	t_1 = fma(cos(phi2), Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))), (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 <= -3.6e-6)
		tmp = t_2;
	elseif (phi2 <= 3.8e-55)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * t_0), (sin(Float64(0.5 * Float64(phi1 + Float64(-1.0 * phi2)))) ^ 2.0))), sqrt(Float64(1.0 - fma(t_0, cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1)))))))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -3.6e-6], t$95$2, If[LessEqual[phi2, 3.8e-55], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * N[(phi1 + N[(-1.0 * phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$0 * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -3.59999999999999984e-6 or 3.7999999999999997e-55 < phi2

   1. Initial program 49.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 21.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 21.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\phi_1 \cdot \left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right) + \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \color{blue}{\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \color{blue}{\sin \left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \color{blue}{\left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\color{blue}{\frac{-1}{2}} \cdot \phi_2\right), \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      7. unpow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      8. sqr-sin-a-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

   10. Applied rewrites 18.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\phi_1, \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(-0.5 \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\right)}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \color{blue}{\left(\phi_1 \cdot \left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right) + \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}}\right) \]
   12. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \color{blue}{\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \color{blue}{\sin \left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \color{blue}{\left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\color{blue}{\frac{-1}{2}} \cdot \phi_2\right), \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       7. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       8. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       9. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

   13. Applied rewrites 39.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(-0.5 \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\phi_1, \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(-0.5 \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\right)}}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + \color{blue}{{\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       2. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \color{blue}{\frac{1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       3. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       4. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       5. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       6. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       7. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       9. lift-pow.f64 40.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(-0.5 \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

   16. Applied rewrites 40.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(-0.5 \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + \color{blue}{{\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}\right)}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \color{blue}{\frac{1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       3. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       4. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       5. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       6. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       7. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       9. lift-pow.f64 48.4

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 48.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   if -3.59999999999999984e-6 < phi2 < 3.7999999999999997e-55

   1. Initial program 78.4%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 68.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 68.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   8. Taylor expanded in phi2 around -inf

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 + -1 \cdot \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-sqrt.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 + -1 \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 + -1 \cdot \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

   10. Applied rewrites 73.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 + -1 \cdot \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 60.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t\_0 \cdot \cos \phi_2\\ t_2 := t\_0 \cdot \cos \phi_1 + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_1 \leq -1.4 \cdot 10^{-6}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_1 \leq 0.000155:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2)))))))
        (t_1 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* t_0 (cos phi2))))
        (t_2 (+ (* t_0 (cos phi1)) (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= phi1 -1.4e-6)
     t_3
     (if (<= phi1 0.000155)
       (* 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 = 0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))));
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (t_0 * cos(phi2));
	double t_2 = (t_0 * cos(phi1)) + (0.5 - (0.5 * cos((2.0 * (0.5 * phi1)))));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi1 <= -1.4e-6) {
		tmp = t_3;
	} else if (phi1 <= 0.000155) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

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
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = 0.5d0 - (0.5d0 * cos((2.0d0 * (0.5d0 * (lambda1 - lambda2)))))
    t_1 = (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (t_0 * cos(phi2))
    t_2 = (t_0 * cos(phi1)) + (0.5d0 - (0.5d0 * cos((2.0d0 * (0.5d0 * phi1)))))
    t_3 = r * (2.0d0 * atan2(sqrt(t_2), sqrt((1.0d0 - t_2))))
    if (phi1 <= (-1.4d-6)) then
        tmp = t_3
    else if (phi1 <= 0.000155d0) then
        tmp = r * (2.0d0 * atan2(sqrt(t_1), sqrt((1.0d0 - t_1))))
    else
        tmp = t_3
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * Math.cos((2.0 * (0.5 * (lambda1 - lambda2)))));
	double t_1 = Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + (t_0 * Math.cos(phi2));
	double t_2 = (t_0 * Math.cos(phi1)) + (0.5 - (0.5 * Math.cos((2.0 * (0.5 * phi1)))));
	double t_3 = R * (2.0 * Math.atan2(Math.sqrt(t_2), Math.sqrt((1.0 - t_2))));
	double tmp;
	if (phi1 <= -1.4e-6) {
		tmp = t_3;
	} else if (phi1 <= 0.000155) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(t_1), Math.sqrt((1.0 - t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = 0.5 - (0.5 * math.cos((2.0 * (0.5 * (lambda1 - lambda2)))))
	t_1 = math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + (t_0 * math.cos(phi2))
	t_2 = (t_0 * math.cos(phi1)) + (0.5 - (0.5 * math.cos((2.0 * (0.5 * phi1)))))
	t_3 = R * (2.0 * math.atan2(math.sqrt(t_2), math.sqrt((1.0 - t_2))))
	tmp = 0
	if phi1 <= -1.4e-6:
		tmp = t_3
	elif phi1 <= 0.000155:
		tmp = R * (2.0 * math.atan2(math.sqrt(t_1), math.sqrt((1.0 - t_1))))
	else:
		tmp = t_3
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2))))))
	t_1 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(t_0 * cos(phi2)))
	t_2 = Float64(Float64(t_0 * cos(phi1)) + Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1))))))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi1 <= -1.4e-6)
		tmp = t_3;
	elseif (phi1 <= 0.000155)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = 0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))));
	t_1 = (sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (t_0 * cos(phi2));
	t_2 = (t_0 * cos(phi1)) + (0.5 - (0.5 * cos((2.0 * (0.5 * phi1)))));
	t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	tmp = 0.0;
	if (phi1 <= -1.4e-6)
		tmp = t_3;
	elseif (phi1 <= 0.000155)
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.4e-6], t$95$3, If[LessEqual[phi1, 0.000155], 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 < -1.39999999999999994e-6 or 1.55e-4 < phi1

   1. Initial program 47.5%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 48.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 49.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   9. Applied rewrites 49.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_1 + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   11. Applied rewrites 49.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_1 + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_1 + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}\right)}}\right) \]

   if -1.39999999999999994e-6 < phi1 < 1.55e-4

   1. Initial program 78.1%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\cos \phi_2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\cos \phi_2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 73.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\cos \phi_2}\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\cos \phi_2}\right)}}\right) \]

   7. Applied rewrites 73.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 59.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_1\\ t_1 := \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{if}\;\phi_2 \leq -2.95 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 3.8 \cdot 10^{-55}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (*
           (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))
           (cos phi1))))
        (t_1
         (fma
          (cos phi2)
          (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))
          (pow (sin (* -0.5 phi2)) 2.0)))
        (t_2 (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))
   (if (<= phi2 -2.95e-5)
     t_2
     (if (<= phi2 3.8e-55)
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + ((0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * cos(phi1));
	double t_1 = fma(cos(phi2), (0.5 - (0.5 * cos((lambda1 - lambda2)))), 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 <= -2.95e-5) {
		tmp = t_2;
	} else if (phi2 <= 3.8e-55) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))) * cos(phi1)))
	t_1 = fma(cos(phi2), Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))), (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 <= -2.95e-5)
		tmp = t_2;
	elseif (phi2 <= 3.8e-55)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -2.95e-5], t$95$2, If[LessEqual[phi2, 3.8e-55], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.9499999999999999e-5 or 3.7999999999999997e-55 < phi2

   1. Initial program 49.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 21.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 21.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\phi_1 \cdot \left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right) + \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \color{blue}{\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \color{blue}{\sin \left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \color{blue}{\left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\color{blue}{\frac{-1}{2}} \cdot \phi_2\right), \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      7. unpow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      8. sqr-sin-a-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

   10. Applied rewrites 18.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\phi_1, \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(-0.5 \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\right)}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \color{blue}{\left(\phi_1 \cdot \left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right) + \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}}\right) \]
   12. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \color{blue}{\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \color{blue}{\sin \left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \color{blue}{\left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\color{blue}{\frac{-1}{2}} \cdot \phi_2\right), \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       7. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       8. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       9. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

   13. Applied rewrites 39.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(-0.5 \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\phi_1, \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(-0.5 \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\right)}}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + \color{blue}{{\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       2. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \color{blue}{\frac{1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       3. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       4. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       5. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       6. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       7. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       9. lift-pow.f64 40.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(-0.5 \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

   16. Applied rewrites 40.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(-0.5 \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + \color{blue}{{\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}\right)}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \color{blue}{\frac{1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       3. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       4. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       5. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       6. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       7. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       9. lift-pow.f64 48.4

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 48.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   if -2.9499999999999999e-5 < phi2 < 3.7999999999999997e-55

   1. Initial program 78.4%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\cos \phi_1}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\cos \phi_1}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 73.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_1}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_1}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_1}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\cos \phi_1}\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_1}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\cos \phi_1}\right)}}\right) \]

   7. Applied rewrites 73.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_1}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_1}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 59.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_1 := \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{if}\;\phi_2 \leq -6 \cdot 10^{-8}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 4.8 \cdot 10^{-65}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))
          (cos phi1)
          (pow (sin (* 0.5 phi1)) 2.0)))
        (t_1
         (fma
          (cos phi2)
          (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))
          (pow (sin (* -0.5 phi2)) 2.0)))
        (t_2 (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))
   (if (<= phi2 -6e-8)
     t_2
     (if (<= phi2 4.8e-65)
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))), cos(phi1), pow(sin((0.5 * phi1)), 2.0));
	double t_1 = fma(cos(phi2), (0.5 - (0.5 * cos((lambda1 - lambda2)))), 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 <= -6e-8) {
		tmp = t_2;
	} else if (phi2 <= 4.8e-65) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))), cos(phi1), (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_1 = fma(cos(phi2), Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))), (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 <= -6e-8)
		tmp = t_2;
	elseif (phi2 <= 4.8e-65)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -6e-8], t$95$2, If[LessEqual[phi2, 4.8e-65], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -5.99999999999999946e-8 or 4.8000000000000003e-65 < phi2

   1. Initial program 49.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 22.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 22.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\phi_1 \cdot \left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right) + \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \color{blue}{\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \color{blue}{\sin \left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \color{blue}{\left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\color{blue}{\frac{-1}{2}} \cdot \phi_2\right), \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      7. unpow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      8. sqr-sin-a-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

   10. Applied rewrites 18.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\phi_1, \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(-0.5 \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\right)}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \color{blue}{\left(\phi_1 \cdot \left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right) + \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}}\right) \]
   12. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \color{blue}{\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \color{blue}{\sin \left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \color{blue}{\left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\color{blue}{\frac{-1}{2}} \cdot \phi_2\right), \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       7. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       8. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       9. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

   13. Applied rewrites 39.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(-0.5 \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\phi_1, \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(-0.5 \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\right)}}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + \color{blue}{{\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       2. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \color{blue}{\frac{1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       3. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       4. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       5. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       6. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       7. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       9. lift-pow.f64 40.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(-0.5 \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

   16. Applied rewrites 40.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(-0.5 \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + \color{blue}{{\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}\right)}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \color{blue}{\frac{1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       3. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       4. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       5. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       6. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       7. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       9. lift-pow.f64 48.4

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 48.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   if -5.99999999999999946e-8 < phi2 < 4.8000000000000003e-65

   1. Initial program 78.5%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 68.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 68.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      3. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      6. sqr-sin-a-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      7. unpow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      9. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      10. lift-*.f64 70.8

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   9. Applied rewrites 70.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]
   10. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

       3. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

       5. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

       6. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)}}\right) \]

       7. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       9. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       10. lift-*.f64 70.7

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 70.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 59.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_3 := \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) + t\_0 \cdot \cos \phi_1\\ t_4 := t\_2 + t\_0\\ \mathbf{if}\;t\_2 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1 \leq 0.0004:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))
          (cos phi2)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_3
         (+
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))
          (* t_0 (cos phi1))))
        (t_4 (+ t_2 t_0)))
   (if (<= (+ t_2 (* (* (* (cos phi1) (cos phi2)) t_1) t_1)) 0.0004)
     (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
     (* 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 = (0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * cos(phi2);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_3 = (0.5 - (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))) + (t_0 * cos(phi1));
	double t_4 = t_2 + t_0;
	double tmp;
	if ((t_2 + (((cos(phi1) * cos(phi2)) * t_1) * t_1)) <= 0.0004) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	}
	return tmp;
}

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
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = (0.5d0 - (0.5d0 * cos((2.0d0 * (0.5d0 * (lambda1 - lambda2)))))) * cos(phi2)
    t_1 = sin(((lambda1 - lambda2) / 2.0d0))
    t_2 = sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0
    t_3 = (0.5d0 - (0.5d0 * cos((2.0d0 * (0.5d0 * (phi1 - phi2)))))) + (t_0 * cos(phi1))
    t_4 = t_2 + t_0
    if ((t_2 + (((cos(phi1) * cos(phi2)) * t_1) * t_1)) <= 0.0004d0) then
        tmp = r * (2.0d0 * atan2(sqrt(t_4), sqrt((1.0d0 - t_4))))
    else
        tmp = r * (2.0d0 * atan2(sqrt(t_3), sqrt((1.0d0 - t_3))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (0.5 - (0.5 * Math.cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * Math.cos(phi2);
	double t_1 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_2 = Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_3 = (0.5 - (0.5 * Math.cos((2.0 * (0.5 * (phi1 - phi2)))))) + (t_0 * Math.cos(phi1));
	double t_4 = t_2 + t_0;
	double tmp;
	if ((t_2 + (((Math.cos(phi1) * Math.cos(phi2)) * t_1) * t_1)) <= 0.0004) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(t_4), Math.sqrt((1.0 - t_4))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(t_3), Math.sqrt((1.0 - t_3))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (0.5 - (0.5 * math.cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * math.cos(phi2)
	t_1 = math.sin(((lambda1 - lambda2) / 2.0))
	t_2 = math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0)
	t_3 = (0.5 - (0.5 * math.cos((2.0 * (0.5 * (phi1 - phi2)))))) + (t_0 * math.cos(phi1))
	t_4 = t_2 + t_0
	tmp = 0
	if (t_2 + (((math.cos(phi1) * math.cos(phi2)) * t_1) * t_1)) <= 0.0004:
		tmp = R * (2.0 * math.atan2(math.sqrt(t_4), math.sqrt((1.0 - t_4))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt(t_3), math.sqrt((1.0 - t_3))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))) * cos(phi2))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_3 = Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))) + Float64(t_0 * cos(phi1)))
	t_4 = Float64(t_2 + t_0)
	tmp = 0.0
	if (Float64(t_2 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1)) <= 0.0004)
		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_3), sqrt(Float64(1.0 - t_3)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * cos(phi2);
	t_1 = sin(((lambda1 - lambda2) / 2.0));
	t_2 = sin(((phi1 - phi2) / 2.0)) ^ 2.0;
	t_3 = (0.5 - (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))) + (t_0 * cos(phi1));
	t_4 = t_2 + t_0;
	tmp = 0.0;
	if ((t_2 + (((cos(phi1) * cos(phi2)) * t_1) * t_1)) <= 0.0004)
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	else
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 + t$95$0), $MachinePrecision]}, If[LessEqual[N[(t$95$2 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 0.0004], 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$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $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))))) < 4.00000000000000019e-4

   1. Initial program 70.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\cos \phi_2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\cos \phi_2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\cos \phi_2}\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \color{blue}{\cos \phi_2}\right)}}\right) \]

   7. Applied rewrites 46.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2}\right)}}\right) \]

   if 4.00000000000000019e-4 < (+.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 61.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 61.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1}}{\sqrt{1 - \color{blue}{\left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) + \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_2\right) \cdot \cos \phi_1\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 58.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_1 := \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_1 + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{if}\;\phi_1 \leq -1.1 \cdot 10^{-16}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 16000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos phi2)
          (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))
          (pow (sin (* -0.5 phi2)) 2.0)))
        (t_1
         (+
          (*
           (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))
           (cos phi1))
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))
        (t_2 (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))
   (if (<= phi1 -1.1e-16)
     t_2
     (if (<= phi1 16000.0)
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos(phi2), (0.5 - (0.5 * cos((lambda1 - lambda2)))), pow(sin((-0.5 * phi2)), 2.0));
	double t_1 = ((0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * cos(phi1)) + (0.5 - (0.5 * cos((2.0 * (0.5 * phi1)))));
	double t_2 = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	double tmp;
	if (phi1 <= -1.1e-16) {
		tmp = t_2;
	} else if (phi1 <= 16000.0) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(phi2), Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))), (sin(Float64(-0.5 * phi2)) ^ 2.0))
	t_1 = Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))) * cos(phi1)) + Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1))))))
	t_2 = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
	tmp = 0.0
	if (phi1 <= -1.1e-16)
		tmp = t_2;
	elseif (phi1 <= 16000.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.1e-16], t$95$2, If[LessEqual[phi1, 16000.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.1e-16 or 16000 < phi1

   1. Initial program 47.9%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 48.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 48.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   9. Applied rewrites 48.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_1 + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   11. Applied rewrites 48.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_1 + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_1 + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}\right)}}\right) \]

   if -1.1e-16 < phi1 < 16000

   1. Initial program 77.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 37.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 36.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\phi_1 \cdot \left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right) + \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \color{blue}{\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \color{blue}{\sin \left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \color{blue}{\left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\color{blue}{\frac{-1}{2}} \cdot \phi_2\right), \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      5. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      7. unpow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      8. sqr-sin-a-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

   10. Applied rewrites 42.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\phi_1, \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(-0.5 \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\right)}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \color{blue}{\left(\phi_1 \cdot \left(\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)\right) + \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}}\right) \]
   12. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \color{blue}{\cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \color{blue}{\sin \left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \color{blue}{\left(\frac{-1}{2} \cdot \phi_2\right)}, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\color{blue}{\frac{-1}{2}} \cdot \phi_2\right), \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       5. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       7. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       8. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       9. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

   13. Applied rewrites 70.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1, \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(-0.5 \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\phi_1, \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(-0.5 \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\right)}}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + \color{blue}{{\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       2. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \color{blue}{\frac{1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       3. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       4. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       5. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       6. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       7. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\frac{-1}{2} \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

       9. lift-pow.f64 70.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(-0.5 \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

   16. Applied rewrites 70.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1, \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(-0.5 \cdot \phi_2\right), \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left(\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + \color{blue}{{\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}\right)}}\right) \]
   18. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       2. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \color{blue}{\frac{1}{2}} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       3. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       4. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       5. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       6. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       7. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

       9. lift-pow.f64 70.5

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 70.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 56.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\\ t_1 := t\_0 \cdot \cos \phi_1 + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)\\ t_2 := \mathsf{fma}\left(t\_0, \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot \phi_2\right)\right)\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_2 \leq -3.6 \cdot 10^{-6}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_2 \leq 4.8 \cdot 10^{-65}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2)))))))
        (t_1 (+ (* t_0 (cos phi1)) (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))
        (t_2 (fma t_0 (cos phi2) (- 0.5 (* 0.5 (cos (* 2.0 (* -0.5 phi2)))))))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= phi2 -3.6e-6)
     t_3
     (if (<= phi2 4.8e-65)
       (* 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 = 0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))));
	double t_1 = (t_0 * cos(phi1)) + (0.5 - (0.5 * cos((2.0 * (0.5 * phi1)))));
	double t_2 = fma(t_0, cos(phi2), (0.5 - (0.5 * cos((2.0 * (-0.5 * phi2))))));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi2 <= -3.6e-6) {
		tmp = t_3;
	} else if (phi2 <= 4.8e-65) {
		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 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2))))))
	t_1 = Float64(Float64(t_0 * cos(phi1)) + Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1))))))
	t_2 = fma(t_0, cos(phi2), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(-0.5 * phi2))))))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi2 <= -3.6e-6)
		tmp = t_3;
	elseif (phi2 <= 4.8e-65)
		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[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[Cos[phi2], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(-0.5 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -3.6e-6], t$95$3, If[LessEqual[phi2, 4.8e-65], 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 < -3.59999999999999984e-6 or 4.8000000000000003e-65 < phi2

   1. Initial program 49.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_2 + {\color{blue}{\sin \left(\frac{-1}{2} \cdot \phi_2\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot \phi_2\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_2 + {\color{blue}{\sin \left(\frac{-1}{2} \cdot \phi_2\right)}}^{2}\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 47.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot \phi_2\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot \phi_2\right)\right)\right)}}}\right) \]

   if -3.59999999999999984e-6 < phi2 < 4.8000000000000003e-65

   1. Initial program 78.4%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 68.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 68.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   9. Applied rewrites 68.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_1 + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   11. Applied rewrites 68.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_1 + \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left(\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \cos \phi_1 + \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 56.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\\ t_1 := \mathsf{fma}\left(t\_0, \cos \phi_1, 0.5 - 0.5 \cdot \cos \phi_1\right)\\ t_2 := \mathsf{fma}\left(t\_0, \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot \phi_2\right)\right)\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_2 \leq -3.6 \cdot 10^{-6}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_2 \leq 4.8 \cdot 10^{-65}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2)))))))
        (t_1 (fma t_0 (cos phi1) (- 0.5 (* 0.5 (cos phi1)))))
        (t_2 (fma t_0 (cos phi2) (- 0.5 (* 0.5 (cos (* 2.0 (* -0.5 phi2)))))))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= phi2 -3.6e-6)
     t_3
     (if (<= phi2 4.8e-65)
       (* 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 = 0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))));
	double t_1 = fma(t_0, cos(phi1), (0.5 - (0.5 * cos(phi1))));
	double t_2 = fma(t_0, cos(phi2), (0.5 - (0.5 * cos((2.0 * (-0.5 * phi2))))));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi2 <= -3.6e-6) {
		tmp = t_3;
	} else if (phi2 <= 4.8e-65) {
		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 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2))))))
	t_1 = fma(t_0, cos(phi1), Float64(0.5 - Float64(0.5 * cos(phi1))))
	t_2 = fma(t_0, cos(phi2), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(-0.5 * phi2))))))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi2 <= -3.6e-6)
		tmp = t_3;
	elseif (phi2 <= 4.8e-65)
		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[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[Cos[phi2], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(-0.5 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -3.6e-6], t$95$3, If[LessEqual[phi2, 4.8e-65], 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 < -3.59999999999999984e-6 or 4.8000000000000003e-65 < phi2

   1. Initial program 49.7%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_2 + {\color{blue}{\sin \left(\frac{-1}{2} \cdot \phi_2\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot \phi_2\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_2 + {\color{blue}{\sin \left(\frac{-1}{2} \cdot \phi_2\right)}}^{2}\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{-1}{2} \cdot \phi_2\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 47.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot \phi_2\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot \phi_2\right)\right)\right)}}}\right) \]

   if -3.59999999999999984e-6 < phi2 < 4.8000000000000003e-65

   1. Initial program 78.4%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 68.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 68.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 68.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \phi_1\right)}}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 68.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \phi_1\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 52.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \phi_1\right)\\ t_1 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ t_2 := \frac{\lambda_1 - \lambda_2}{2}\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)\\ \mathbf{if}\;t\_2 \leq -0.0005:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))
          (cos phi1)
          (- 0.5 (* 0.5 (cos phi1)))))
        (t_1 (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0))))))
        (t_2 (/ (- lambda1 lambda2) 2.0))
        (t_3
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (cos phi2) (* -0.5 lambda2)) (* -0.5 lambda2)))))
   (if (<= t_2 -0.0005)
     t_1
     (if (<= t_2 5e-6)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       t_1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))), cos(phi1), (0.5 - (0.5 * cos(phi1))));
	double t_1 = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	double t_2 = (lambda1 - lambda2) / 2.0;
	double t_3 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + ((cos(phi2) * (-0.5 * lambda2)) * (-0.5 * lambda2));
	double tmp;
	if (t_2 <= -0.0005) {
		tmp = t_1;
	} else if (t_2 <= 5e-6) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))), cos(phi1), Float64(0.5 - Float64(0.5 * cos(phi1))))
	t_1 = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))))
	t_2 = Float64(Float64(lambda1 - lambda2) / 2.0)
	t_3 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(cos(phi2) * Float64(-0.5 * lambda2)) * Float64(-0.5 * lambda2)))
	tmp = 0.0
	if (t_2 <= -0.0005)
		tmp = t_1;
	elseif (t_2 <= 5e-6)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[Cos[phi2], $MachinePrecision] * N[(-0.5 * lambda2), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.0005], t$95$1, If[LessEqual[t$95$2, 5e-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$1]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)) < -5.0000000000000001e-4 or 5.00000000000000041e-6 < (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))

   1. Initial program 57.4%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 46.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \phi_1\right)}}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 46.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \phi_1\right)}}\right) \]

   if -5.0000000000000001e-4 < (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)) < 5.00000000000000041e-6

   1. Initial program 78.1%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) + \frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) + \color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) \cdot \frac{-1}{2} + \sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right), \color{blue}{\frac{-1}{2}}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 78.1

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 78.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) + \frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) + \color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \left(\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) \cdot \frac{-1}{2} + \sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right), \color{blue}{\frac{-1}{2}}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 78.1

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 78.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) + \frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) + \color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) \cdot \frac{-1}{2} + \sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right), \color{blue}{\frac{-1}{2}}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 78.1

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. Applied rewrites 78.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) + \frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}\right)}}\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) + \color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}\right)\right)}}\right) \]

       2. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \left(\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) \cdot \frac{-1}{2} + \sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)}\right)\right)}}\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right), \color{blue}{\frac{-1}{2}}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]

       4. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]

       6. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]

       9. lower-*.f64 78.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   13. Applied rewrites 78.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}\right)}}\right) \]

   14. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \color{blue}{\lambda_2}\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lift-*.f64 72.4

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   16. Applied rewrites 72.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \color{blue}{\lambda_2}\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \color{blue}{\lambda_2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]
   18. Step-by-step derivation

       1. lift-*.f64 72.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   19. Applied rewrites 72.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \color{blue}{\lambda_2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   20. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \color{blue}{\lambda_2}\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]
   21. Step-by-step derivation

       1. lift-*.f64 72.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   22. Applied rewrites 72.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \color{blue}{\lambda_2}\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   23. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \color{blue}{\lambda_2}\right)\right)}}\right) \]
   24. Step-by-step derivation

       1. lift-*.f64 72.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right)}}\right) \]

   25. Applied rewrites 72.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \color{blue}{\lambda_2}\right)\right)}}\right) \]

   26. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_2} \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}\right) \]
   27. Step-by-step derivation

       1. lift-cos.f64 72.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right)}}\right) \]

   28. Applied rewrites 72.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_2} \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right)}}\right) \]

   29. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_2} \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}\right) \]
   30. Step-by-step derivation

       1. lift-cos.f64 72.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right)}}\right) \]

   31. Applied rewrites 72.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_2} \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 52.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(0.5 + \cos \phi_1 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - 0.5 \cdot \cos \phi_1\\ t_1 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)\\ \mathbf{if}\;t\_2 \leq -0.0005:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (-
          (+ 0.5 (* (cos phi1) (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))))
          (* 0.5 (cos phi1))))
        (t_1 (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0))))))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (cos phi2) (* -0.5 lambda2)) (* -0.5 lambda2)))))
   (if (<= t_2 -0.0005)
     t_1
     (if (<= t_2 5e-6)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       t_1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (0.5 + (cos(phi1) * (0.5 - (0.5 * cos((lambda1 - lambda2)))))) - (0.5 * cos(phi1));
	double t_1 = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + ((cos(phi2) * (-0.5 * lambda2)) * (-0.5 * lambda2));
	double tmp;
	if (t_2 <= -0.0005) {
		tmp = t_1;
	} else if (t_2 <= 5e-6) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

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
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (0.5d0 + (cos(phi1) * (0.5d0 - (0.5d0 * cos((lambda1 - lambda2)))))) - (0.5d0 * cos(phi1))
    t_1 = r * (2.0d0 * atan2(sqrt(t_0), sqrt((1.0d0 - t_0))))
    t_2 = sin(((lambda1 - lambda2) / 2.0d0))
    t_3 = (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + ((cos(phi2) * ((-0.5d0) * lambda2)) * ((-0.5d0) * lambda2))
    if (t_2 <= (-0.0005d0)) then
        tmp = t_1
    else if (t_2 <= 5d-6) then
        tmp = r * (2.0d0 * atan2(sqrt(t_3), sqrt((1.0d0 - t_3))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (0.5 + (Math.cos(phi1) * (0.5 - (0.5 * Math.cos((lambda1 - lambda2)))))) - (0.5 * Math.cos(phi1));
	double t_1 = R * (2.0 * Math.atan2(Math.sqrt(t_0), Math.sqrt((1.0 - t_0))));
	double t_2 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_3 = Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + ((Math.cos(phi2) * (-0.5 * lambda2)) * (-0.5 * lambda2));
	double tmp;
	if (t_2 <= -0.0005) {
		tmp = t_1;
	} else if (t_2 <= 5e-6) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(t_3), Math.sqrt((1.0 - t_3))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (0.5 + (math.cos(phi1) * (0.5 - (0.5 * math.cos((lambda1 - lambda2)))))) - (0.5 * math.cos(phi1))
	t_1 = R * (2.0 * math.atan2(math.sqrt(t_0), math.sqrt((1.0 - t_0))))
	t_2 = math.sin(((lambda1 - lambda2) / 2.0))
	t_3 = math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + ((math.cos(phi2) * (-0.5 * lambda2)) * (-0.5 * lambda2))
	tmp = 0
	if t_2 <= -0.0005:
		tmp = t_1
	elif t_2 <= 5e-6:
		tmp = R * (2.0 * math.atan2(math.sqrt(t_3), math.sqrt((1.0 - t_3))))
	else:
		tmp = t_1
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(0.5 + Float64(cos(phi1) * Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))))) - Float64(0.5 * cos(phi1)))
	t_1 = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(cos(phi2) * Float64(-0.5 * lambda2)) * Float64(-0.5 * lambda2)))
	tmp = 0.0
	if (t_2 <= -0.0005)
		tmp = t_1;
	elseif (t_2 <= 5e-6)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (0.5 + (cos(phi1) * (0.5 - (0.5 * cos((lambda1 - lambda2)))))) - (0.5 * cos(phi1));
	t_1 = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	t_2 = sin(((lambda1 - lambda2) / 2.0));
	t_3 = (sin(((phi1 - phi2) / 2.0)) ^ 2.0) + ((cos(phi2) * (-0.5 * lambda2)) * (-0.5 * lambda2));
	tmp = 0.0;
	if (t_2 <= -0.0005)
		tmp = t_1;
	elseif (t_2 <= 5e-6)
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(0.5 + N[(N[Cos[phi1], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[Cos[phi2], $MachinePrecision] * N[(-0.5 * lambda2), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.0005], t$95$1, If[LessEqual[t$95$2, 5e-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$1]]]]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -5.0000000000000001e-4 or 5.00000000000000041e-6 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 57.4%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   8. Taylor expanded in lambda1 around inf

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \color{blue}{\frac{1}{2} \cdot \cos \phi_1}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \color{blue}{\cos \phi_1}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \color{blue}{\phi_1}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_1}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      4. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_1}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      5. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_1}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_1}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      7. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_1}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      8. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_1}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_1}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      10. lift-cos.f64 46.3

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 + \cos \phi_1 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - 0.5 \cdot \cos \phi_1}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   10. Applied rewrites 46.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 + \cos \phi_1 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \color{blue}{0.5 \cdot \cos \phi_1}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   11. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_1}}{\sqrt{1 - \left(\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \color{blue}{\frac{1}{2} \cdot \cos \phi_1}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_1}}{\sqrt{1 - \left(\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \color{blue}{\cos \phi_1}\right)}}\right) \]

       2. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_1}}{\sqrt{1 - \left(\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \color{blue}{\phi_1}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_1}}{\sqrt{1 - \left(\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_1\right)}}\right) \]

       4. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_1}}{\sqrt{1 - \left(\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_1\right)}}\right) \]

       5. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_1}}{\sqrt{1 - \left(\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_1\right)}}\right) \]

       6. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_1}}{\sqrt{1 - \left(\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_1\right)}}\right) \]

       7. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_1}}{\sqrt{1 - \left(\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_1\right)}}\right) \]

       8. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_1}}{\sqrt{1 - \left(\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_1\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_1}}{\sqrt{1 - \left(\left(\frac{1}{2} + \cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \frac{1}{2} \cdot \cos \phi_1\right)}}\right) \]

       10. lift-cos.f64 46.2

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 + \cos \phi_1 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - 0.5 \cdot \cos \phi_1}}{\sqrt{1 - \left(\left(0.5 + \cos \phi_1 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - 0.5 \cdot \cos \phi_1\right)}}\right) \]

   13. Applied rewrites 46.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 + \cos \phi_1 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - 0.5 \cdot \cos \phi_1}}{\sqrt{1 - \left(\left(0.5 + \cos \phi_1 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) - \color{blue}{0.5 \cdot \cos \phi_1}\right)}}\right) \]

   if -5.0000000000000001e-4 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 5.00000000000000041e-6

   1. Initial program 78.1%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) + \frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) + \color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) \cdot \frac{-1}{2} + \sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right), \color{blue}{\frac{-1}{2}}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 78.1

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 78.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) + \frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) + \color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \left(\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) \cdot \frac{-1}{2} + \sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right), \color{blue}{\frac{-1}{2}}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 78.1

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 78.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) + \frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) + \color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) \cdot \frac{-1}{2} + \sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right), \color{blue}{\frac{-1}{2}}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 78.1

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. Applied rewrites 78.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) + \frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}\right)}}\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) + \color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}\right)\right)}}\right) \]

       2. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \left(\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) \cdot \frac{-1}{2} + \sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)}\right)\right)}}\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right), \color{blue}{\frac{-1}{2}}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]

       4. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]

       6. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]

       9. lower-*.f64 78.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   13. Applied rewrites 78.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}\right)}}\right) \]

   14. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \color{blue}{\lambda_2}\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lift-*.f64 72.4

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   16. Applied rewrites 72.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \color{blue}{\lambda_2}\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \color{blue}{\lambda_2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]
   18. Step-by-step derivation

       1. lift-*.f64 72.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   19. Applied rewrites 72.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \color{blue}{\lambda_2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   20. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \color{blue}{\lambda_2}\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]
   21. Step-by-step derivation

       1. lift-*.f64 72.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   22. Applied rewrites 72.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \color{blue}{\lambda_2}\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   23. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \color{blue}{\lambda_2}\right)\right)}}\right) \]
   24. Step-by-step derivation

       1. lift-*.f64 72.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right)}}\right) \]

   25. Applied rewrites 72.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \color{blue}{\lambda_2}\right)\right)}}\right) \]

   26. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_2} \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}\right) \]
   27. Step-by-step derivation

       1. lift-cos.f64 72.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right)}}\right) \]

   28. Applied rewrites 72.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_2} \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right)}}\right) \]

   29. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_2} \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}\right) \]
   30. Step-by-step derivation

       1. lift-cos.f64 72.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right)}}\right) \]

   31. Applied rewrites 72.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_2} \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 43.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)\\ t_1 := \frac{\lambda_1 - \lambda_2}{2}\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right)\\ \mathbf{if}\;t\_1 \leq -0.0005:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (cos phi2) (* -0.5 lambda2)) (* -0.5 lambda2))))
        (t_1 (/ (- lambda1 lambda2) 2.0))
        (t_2
         (*
          R
          (*
           2.0
           (atan2
            (sqrt
             (fma
              (cos phi1)
              (*
               (cos phi2)
               (pow (sin (* 0.5 (+ lambda1 (* -1.0 lambda2)))) 2.0))
              (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))))
            (sqrt (- 1.0 (- 0.5 (* 0.5 (cos (- lambda1 lambda2)))))))))))
   (if (<= t_1 -0.0005)
     t_2
     (if (<= t_1 5e-6)
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + ((cos(phi2) * (-0.5 * lambda2)) * (-0.5 * lambda2));
	double t_1 = (lambda1 - lambda2) / 2.0;
	double t_2 = R * (2.0 * atan2(sqrt(fma(cos(phi1), (cos(phi2) * pow(sin((0.5 * (lambda1 + (-1.0 * lambda2)))), 2.0)), (0.5 - (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))))), sqrt((1.0 - (0.5 - (0.5 * cos((lambda1 - lambda2))))))));
	double tmp;
	if (t_1 <= -0.0005) {
		tmp = t_2;
	} else if (t_1 <= 5e-6) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(cos(phi2) * Float64(-0.5 * lambda2)) * Float64(-0.5 * lambda2)))
	t_1 = Float64(Float64(lambda1 - lambda2) / 2.0)
	t_2 = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * (sin(Float64(0.5 * Float64(lambda1 + Float64(-1.0 * lambda2)))) ^ 2.0)), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))))), sqrt(Float64(1.0 - Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))))))))
	tmp = 0.0
	if (t_1 <= -0.0005)
		tmp = t_2;
	elseif (t_1 <= 5e-6)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[Cos[phi2], $MachinePrecision] * N[(-0.5 * lambda2), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 + N[(-1.0 * lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.0005], t$95$2, If[LessEqual[t$95$1, 5e-6], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)) < -5.0000000000000001e-4 or 5.00000000000000041e-6 < (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))

   1. Initial program 57.4%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      4. lift--.f64 33.3

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   10. Applied rewrites 33.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - \color{blue}{0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       4. lift--.f64 32.8

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

   13. Applied rewrites 32.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(0.5 - \color{blue}{0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]

   14. Taylor expanded in lambda2 around -inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       2. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}^{2}\right) + \sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       3. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}^{2}\right) + \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}^{2}, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

   16. Applied rewrites 33.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 + -1 \cdot \lambda_2\right)\right)}^{2}, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

   if -5.0000000000000001e-4 < (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)) < 5.00000000000000041e-6

   1. Initial program 78.1%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) + \frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) + \color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) \cdot \frac{-1}{2} + \sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right), \color{blue}{\frac{-1}{2}}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 78.1

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 78.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) + \frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) + \color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \left(\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) \cdot \frac{-1}{2} + \sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right), \color{blue}{\frac{-1}{2}}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 78.1

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 78.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) + \frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) + \color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) \cdot \frac{-1}{2} + \sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right), \color{blue}{\frac{-1}{2}}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 78.1

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. Applied rewrites 78.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) + \frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}\right)}}\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) + \color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}\right)\right)}}\right) \]

       2. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \left(\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) \cdot \frac{-1}{2} + \sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)}\right)\right)}}\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right), \color{blue}{\frac{-1}{2}}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]

       4. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]

       6. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]

       9. lower-*.f64 78.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   13. Applied rewrites 78.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}\right)}}\right) \]

   14. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \color{blue}{\lambda_2}\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lift-*.f64 72.4

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   16. Applied rewrites 72.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \color{blue}{\lambda_2}\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \color{blue}{\lambda_2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]
   18. Step-by-step derivation

       1. lift-*.f64 72.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   19. Applied rewrites 72.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \color{blue}{\lambda_2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   20. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \color{blue}{\lambda_2}\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]
   21. Step-by-step derivation

       1. lift-*.f64 72.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   22. Applied rewrites 72.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \color{blue}{\lambda_2}\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   23. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \color{blue}{\lambda_2}\right)\right)}}\right) \]
   24. Step-by-step derivation

       1. lift-*.f64 72.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right)}}\right) \]

   25. Applied rewrites 72.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \color{blue}{\lambda_2}\right)\right)}}\right) \]

   26. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_2} \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}\right) \]
   27. Step-by-step derivation

       1. lift-cos.f64 72.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right)}}\right) \]

   28. Applied rewrites 72.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_2} \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right)}}\right) \]

   29. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_2} \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}\right) \]
   30. Step-by-step derivation

       1. lift-cos.f64 72.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right)}}\right) \]

   31. Applied rewrites 72.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_2} \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 43.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\lambda_1 - \lambda_2}{2}\\ t_1 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)\\ \mathbf{if}\;t\_0 \leq -0.0005:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (/ (- lambda1 lambda2) 2.0))
        (t_1
         (*
          R
          (*
           2.0
           (atan2
            (sqrt
             (fma
              (cos phi1)
              (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))
              (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))
            (sqrt (- 1.0 (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))))))))
        (t_2
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (cos phi2) (* -0.5 lambda2)) (* -0.5 lambda2)))))
   (if (<= t_0 -0.0005)
     t_1
     (if (<= t_0 5e-6)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
       t_1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) / 2.0;
	double t_1 = R * (2.0 * atan2(sqrt(fma(cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))))), sqrt((1.0 - (0.5 - (0.5 * cos((lambda1 - lambda2))))))));
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + ((cos(phi2) * (-0.5 * lambda2)) * (-0.5 * lambda2));
	double tmp;
	if (t_0 <= -0.0005) {
		tmp = t_1;
	} else if (t_0 <= 5e-6) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) / 2.0)
	t_1 = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1))))))), sqrt(Float64(1.0 - Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))))))))
	t_2 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(cos(phi2) * Float64(-0.5 * lambda2)) * Float64(-0.5 * lambda2)))
	tmp = 0.0
	if (t_0 <= -0.0005)
		tmp = t_1;
	elseif (t_0 <= 5e-6)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$1 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[Cos[phi2], $MachinePrecision] * N[(-0.5 * lambda2), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.0005], t$95$1, If[LessEqual[t$95$0, 5e-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], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)) < -5.0000000000000001e-4 or 5.00000000000000041e-6 < (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))

   1. Initial program 57.4%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      4. lift--.f64 33.3

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   10. Applied rewrites 33.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - \color{blue}{0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       4. lift--.f64 32.8

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

   13. Applied rewrites 32.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(0.5 - \color{blue}{0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]

   14. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       2. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       3. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

   16. Applied rewrites 33.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

   if -5.0000000000000001e-4 < (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)) < 5.00000000000000041e-6

   1. Initial program 78.1%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) + \frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) + \color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) \cdot \frac{-1}{2} + \sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right), \color{blue}{\frac{-1}{2}}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 78.1

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 78.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) + \frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) + \color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \left(\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) \cdot \frac{-1}{2} + \sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right), \color{blue}{\frac{-1}{2}}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 78.1

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 78.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Taylor expanded in lambda2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) + \frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) + \color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) \cdot \frac{-1}{2} + \sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right), \color{blue}{\frac{-1}{2}}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 78.1

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. Applied rewrites 78.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \lambda_1\right) + \frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}\right)}}\right) \]
   12. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \left(\frac{-1}{2} \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) + \color{blue}{\sin \left(\frac{1}{2} \cdot \lambda_1\right)}\right)\right)}}\right) \]

       2. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \left(\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right)\right) \cdot \frac{-1}{2} + \sin \color{blue}{\left(\frac{1}{2} \cdot \lambda_1\right)}\right)\right)}}\right) \]

       3. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \lambda_1\right), \color{blue}{\frac{-1}{2}}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]

       4. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]

       5. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]

       6. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]

       9. lower-*.f64 78.1

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   13. Applied rewrites 78.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}\right)}}\right) \]

   14. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \color{blue}{\lambda_2}\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. lift-*.f64 72.4

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   16. Applied rewrites 72.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \color{blue}{\lambda_2}\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \color{blue}{\lambda_2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]
   18. Step-by-step derivation

       1. lift-*.f64 72.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   19. Applied rewrites 72.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \color{blue}{\lambda_2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   20. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \color{blue}{\lambda_2}\right)\right) \cdot \mathsf{fma}\left(\cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \lambda_2, \frac{-1}{2}, \sin \left(\frac{1}{2} \cdot \lambda_1\right)\right)\right)}}\right) \]
   21. Step-by-step derivation

       1. lift-*.f64 72.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   22. Applied rewrites 72.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \color{blue}{\lambda_2}\right)\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \lambda_1\right) \cdot \lambda_2, -0.5, \sin \left(0.5 \cdot \lambda_1\right)\right)\right)}}\right) \]

   23. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \color{blue}{\lambda_2}\right)\right)}}\right) \]
   24. Step-by-step derivation

       1. lift-*.f64 72.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right)}}\right) \]

   25. Applied rewrites 72.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \color{blue}{\lambda_2}\right)\right)}}\right) \]

   26. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_2} \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}\right) \]
   27. Step-by-step derivation

       1. lift-cos.f64 72.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right)}}\right) \]

   28. Applied rewrites 72.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_2} \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right)}}\right) \]

   29. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_2} \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right) \cdot \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}\right) \]
   30. Step-by-step derivation

       1. lift-cos.f64 72.6

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right)}}\right) \]

   31. Applied rewrites 72.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_2 \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\color{blue}{\cos \phi_2} \cdot \left(-0.5 \cdot \lambda_2\right)\right) \cdot \left(-0.5 \cdot \lambda_2\right)\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 31.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \lambda_1, 1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)\\ \mathbf{if}\;\phi_1 \leq 2.7 \cdot 10^{+137}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (- 0.5 (* 0.5 (cos lambda1)))
          1.0
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1))))))))
   (if (<= phi1 2.7e+137)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (fma
          (cos phi2)
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))
          (pow (sin (* -0.5 phi2)) 2.0)))
        (sqrt (- 1.0 (- 0.5 (* 0.5 (cos (- lambda1 lambda2)))))))))
     (* 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((0.5 - (0.5 * cos(lambda1))), 1.0, (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))));
	double tmp;
	if (phi1 <= 2.7e+137) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi2), (0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))), pow(sin((-0.5 * phi2)), 2.0))), sqrt((1.0 - (0.5 - (0.5 * cos((lambda1 - lambda2))))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(0.5 - Float64(0.5 * cos(lambda1))), 1.0, Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1))))))
	tmp = 0.0
	if (phi1 <= 2.7e+137)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi2), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))), (sin(Float64(-0.5 * phi2)) ^ 2.0))), sqrt(Float64(1.0 - Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(0.5 - N[(0.5 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.0 + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, 2.7e+137], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < 2.70000000000000017e137

   1. Initial program 64.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 41.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 41.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      4. lift--.f64 28.2

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   10. Applied rewrites 28.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - \color{blue}{0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       4. lift--.f64 27.9

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

   13. Applied rewrites 27.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(0.5 - \color{blue}{0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       2. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

   16. Applied rewrites 32.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

   if 2.70000000000000017e137 < phi1

   1. Initial program 48.1%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 49.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   8. Taylor expanded in lambda1 around inf

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 40.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \lambda_1, \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   11. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 40.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \lambda_1, \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \lambda_1, \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, 1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]
   15. Step-by-step derivation

   16. Applied rewrites 29.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \lambda_1, 1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \lambda_1, \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, 1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1, 1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]
   18. Step-by-step derivation

   19. Applied rewrites 24.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \lambda_1, 1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \lambda_1, 1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 28: 28.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (*
   2.0
   (atan2
    (sqrt
     (fma
      (cos phi1)
      (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))
      (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))
    (sqrt (- 1.0 (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * (2.0 * atan2(sqrt(fma(cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))))), sqrt((1.0 - (0.5 - (0.5 * cos((lambda1 - lambda2))))))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1))))))), sqrt(Float64(1.0 - Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))))))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.5%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

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. *-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 42.8%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

   2. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

7. Applied rewrites 42.9%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

8. Taylor expanded in phi1 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]
9. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

   3. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

   4. lift--.f64 26.7

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

10. Applied rewrites 26.7%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - \color{blue}{0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

11. Taylor expanded in phi1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]
12. Step-by-step derivation

    1. lower--.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]

    2. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

    3. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

    4. lift--.f64 26.3

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

13. Applied rewrites 26.3%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(0.5 - \color{blue}{0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]

14. Taylor expanded in phi2 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]
15. Step-by-step derivation

    1. lower-sqrt.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

    2. unpow2 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

    3. sqr-sin-a-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

    4. lower-fma.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

16. Applied rewrites 28.8%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]
17. Add Preprocessing

Alternative 29: 26.3% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))))
   (* 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 = 0.5 - (0.5 * cos((lambda1 - lambda2)));
	return R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = 0.5d0 - (0.5d0 * cos((lambda1 - lambda2)))
    code = r * (2.0d0 * atan2(sqrt(t_0), sqrt((1.0d0 - t_0))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * Math.cos((lambda1 - lambda2)));
	return R * (2.0 * Math.atan2(Math.sqrt(t_0), Math.sqrt((1.0 - t_0))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = 0.5 - (0.5 * math.cos((lambda1 - lambda2)))
	return R * (2.0 * math.atan2(math.sqrt(t_0), math.sqrt((1.0 - t_0))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2))))
	return Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = 0.5 - (0.5 * cos((lambda1 - lambda2)));
	tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.5%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

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. *-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 42.8%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

   2. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

7. Applied rewrites 42.9%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

8. Taylor expanded in phi1 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]
9. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

   3. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

   4. lift--.f64 26.7

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

10. Applied rewrites 26.7%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - \color{blue}{0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

11. Taylor expanded in phi1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]
12. Step-by-step derivation

    1. lower--.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]

    2. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

    3. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

    4. lift--.f64 26.3

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

13. Applied rewrites 26.3%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(0.5 - \color{blue}{0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]
14. Add Preprocessing

Alternative 30: 25.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \lambda_1\\ t_1 := 0.5 - 0.5 \cdot \cos \lambda_2\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{if}\;\lambda_2 \leq -1.7 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\lambda_2 \leq 4.6 \cdot 10^{+14}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos lambda1))))
        (t_1 (- 0.5 (* 0.5 (cos lambda2))))
        (t_2 (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))
   (if (<= lambda2 -1.7e-6)
     t_2
     (if (<= lambda2 4.6e+14)
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos(lambda1));
	double t_1 = 0.5 - (0.5 * cos(lambda2));
	double t_2 = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	double tmp;
	if (lambda2 <= -1.7e-6) {
		tmp = t_2;
	} else if (lambda2 <= 4.6e+14) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

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
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 0.5d0 - (0.5d0 * cos(lambda1))
    t_1 = 0.5d0 - (0.5d0 * cos(lambda2))
    t_2 = r * (2.0d0 * atan2(sqrt(t_1), sqrt((1.0d0 - t_1))))
    if (lambda2 <= (-1.7d-6)) then
        tmp = t_2
    else if (lambda2 <= 4.6d+14) then
        tmp = r * (2.0d0 * atan2(sqrt(t_0), sqrt((1.0d0 - t_0))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * Math.cos(lambda1));
	double t_1 = 0.5 - (0.5 * Math.cos(lambda2));
	double t_2 = R * (2.0 * Math.atan2(Math.sqrt(t_1), Math.sqrt((1.0 - t_1))));
	double tmp;
	if (lambda2 <= -1.7e-6) {
		tmp = t_2;
	} else if (lambda2 <= 4.6e+14) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(t_0), Math.sqrt((1.0 - t_0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = 0.5 - (0.5 * math.cos(lambda1))
	t_1 = 0.5 - (0.5 * math.cos(lambda2))
	t_2 = R * (2.0 * math.atan2(math.sqrt(t_1), math.sqrt((1.0 - t_1))))
	tmp = 0
	if lambda2 <= -1.7e-6:
		tmp = t_2
	elif lambda2 <= 4.6e+14:
		tmp = R * (2.0 * math.atan2(math.sqrt(t_0), math.sqrt((1.0 - t_0))))
	else:
		tmp = t_2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(lambda1)))
	t_1 = Float64(0.5 - Float64(0.5 * cos(lambda2)))
	t_2 = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
	tmp = 0.0
	if (lambda2 <= -1.7e-6)
		tmp = t_2;
	elseif (lambda2 <= 4.6e+14)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = 0.5 - (0.5 * cos(lambda1));
	t_1 = 0.5 - (0.5 * cos(lambda2));
	t_2 = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	tmp = 0.0;
	if (lambda2 <= -1.7e-6)
		tmp = t_2;
	elseif (lambda2 <= 4.6e+14)
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(0.5 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -1.7e-6], t$95$2, If[LessEqual[lambda2, 4.6e+14], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -1.70000000000000003e-6 or 4.6e14 < lambda2

   1. Initial program 46.9%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 38.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 39.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      4. lift--.f64 29.2

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   10. Applied rewrites 29.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - \color{blue}{0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       4. lift--.f64 28.8

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

   13. Applied rewrites 28.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(0.5 - \color{blue}{0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]

   14. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]
   15. Step-by-step derivation

       1. cos-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_2}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       2. lower-cos.f64 28.8

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \lambda_2}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

   16. Applied rewrites 28.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \lambda_2}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_2}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}}\right) \]
   18. Step-by-step derivation

       1. cos-neg N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_2}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_2\right)}}\right) \]

       2. lower-cos.f64 28.7

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \lambda_2}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \lambda_2\right)}}\right) \]

   19. Applied rewrites 28.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \lambda_2}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \lambda_2\right)}}\right) \]

   if -1.70000000000000003e-6 < lambda2 < 4.6e14

   1. Initial program 76.9%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   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. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

      2. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

      4. lift--.f64 24.3

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   10. Applied rewrites 24.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - \color{blue}{0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

       4. lift--.f64 24.0

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

   13. Applied rewrites 24.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(0.5 - \color{blue}{0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]

   14. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]
   15. Step-by-step derivation

   16. Applied rewrites 23.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \lambda_1}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

   17. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_1\right)}}\right) \]
   18. Step-by-step derivation

   19. Applied rewrites 23.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \lambda_1}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \lambda_1\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 31: 16.7% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \lambda_2\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos lambda2)))))
   (* 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 = 0.5 - (0.5 * cos(lambda2));
	return R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = 0.5d0 - (0.5d0 * cos(lambda2))
    code = r * (2.0d0 * atan2(sqrt(t_0), sqrt((1.0d0 - t_0))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * Math.cos(lambda2));
	return R * (2.0 * Math.atan2(Math.sqrt(t_0), Math.sqrt((1.0 - t_0))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = 0.5 - (0.5 * math.cos(lambda2))
	return R * (2.0 * math.atan2(math.sqrt(t_0), math.sqrt((1.0 - t_0))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(lambda2)))
	return Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = 0.5 - (0.5 * cos(lambda2));
	tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.5%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

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. *-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 42.8%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

   2. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

7. Applied rewrites 42.9%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

8. Taylor expanded in phi1 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]
9. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

   3. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

   4. lift--.f64 26.7

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

10. Applied rewrites 26.7%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - \color{blue}{0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

11. Taylor expanded in phi1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]
12. Step-by-step derivation

    1. lower--.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]

    2. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

    3. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

    4. lift--.f64 26.3

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

13. Applied rewrites 26.3%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(0.5 - \color{blue}{0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]

14. Taylor expanded in lambda1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]
15. Step-by-step derivation

    1. cos-neg N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_2}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

    2. lower-cos.f64 16.8

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \lambda_2}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

16. Applied rewrites 16.8%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \lambda_2}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

17. Taylor expanded in lambda1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_2}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}}\right) \]
18. Step-by-step derivation

    1. cos-neg N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_2}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_2\right)}}\right) \]

    2. lower-cos.f64 16.7

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \lambda_2}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \lambda_2\right)}}\right) \]

19. Applied rewrites 16.7%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \lambda_2}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \lambda_2\right)}}\right) \]
20. Add Preprocessing

Alternative 32: 4.4% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.5 - 0.5 \cdot 1\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 1.0))))
   (* 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 = 0.5 - (0.5 * 1.0);
	return R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = 0.5d0 - (0.5d0 * 1.0d0)
    code = r * (2.0d0 * atan2(sqrt(t_0), sqrt((1.0d0 - t_0))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * 1.0);
	return R * (2.0 * Math.atan2(Math.sqrt(t_0), Math.sqrt((1.0 - t_0))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = 0.5 - (0.5 * 1.0)
	return R * (2.0 * math.atan2(math.sqrt(t_0), math.sqrt((1.0 - t_0))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * 1.0))
	return Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = 0.5 - (0.5 * 1.0);
	tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * 1.0), $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 62.5%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

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. *-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied rewrites 42.8%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Taylor expanded in phi2 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

   2. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

7. Applied rewrites 42.9%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

8. Taylor expanded in phi1 around 0

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]
9. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

   2. lower-*.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

   3. lower-cos.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}}\right) \]

   4. lift--.f64 26.7

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

10. Applied rewrites 26.7%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - \color{blue}{0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right) \]

11. Taylor expanded in phi1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]
12. Step-by-step derivation

    1. lower--.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]

    2. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

    3. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

    4. lift--.f64 26.3

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

13. Applied rewrites 26.3%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}{\sqrt{1 - \left(0.5 - \color{blue}{0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)}}\right) \]

14. Taylor expanded in lambda1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]
15. Step-by-step derivation

    1. cos-neg N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_2}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

    2. lower-cos.f64 16.8

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \lambda_2}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

16. Applied rewrites 16.8%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \lambda_2}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right) \]

17. Taylor expanded in lambda1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_2}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}}\right) \]
18. Step-by-step derivation

    1. cos-neg N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_2}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_2\right)}}\right) \]

    2. lower-cos.f64 16.7

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \lambda_2}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \lambda_2\right)}}\right) \]

19. Applied rewrites 16.7%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot \cos \lambda_2}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \lambda_2\right)}}\right) \]

20. Taylor expanded in lambda2 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot 1}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \lambda_2\right)}}\right) \]
21. Step-by-step derivation

22. Applied rewrites 4.4%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot 1}}{\sqrt{1 - \left(0.5 - 0.5 \cdot \cos \lambda_2\right)}}\right) \]

23. Taylor expanded in lambda2 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\frac{1}{2} - \frac{1}{2} \cdot 1}}{\sqrt{1 - \left(\frac{1}{2} - \frac{1}{2} \cdot 1\right)}}\right) \]
24. Step-by-step derivation

25. Applied rewrites 4.4%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{0.5 - 0.5 \cdot 1}}{\sqrt{1 - \left(0.5 - 0.5 \cdot 1\right)}}\right) \]
26. Add Preprocessing

Reproduce

herbie shell --seed 2025120 
(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))))))))))