Distance on a great circle

Percentage Accurate: 61.7% → 78.4%

Time: 1.2min

Alternatives: 29

Speedup: 1.1×

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

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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: 61.7% 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

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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: 78.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\phi_2 \cdot 0.5\right)\\ t_1 := \sin \left(0.5 \cdot \phi_1\right)\\ t_2 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_3 := \cos \phi_1 \cdot \cos \phi_2\\ t_4 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_5 := \sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\\ t_6 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t\_4 \cdot \left(t\_3 \cdot t\_4\right)\\ t_7 := \cos \left(0.5 \cdot \phi_1\right)\\ \mathbf{if}\;\tan^{-1}_* \frac{\sqrt{t\_6}}{\sqrt{1 - t\_6}} \leq 0.05:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{hypot}\left(t\_5, \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right) \cdot \sqrt{t\_3}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot t\_2\right), {t\_5}^{2}\right)}} \cdot \left(R \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(t\_0, t\_1, \sin \left(\phi_2 \cdot -0.5\right) \cdot t\_7\right)\right)}^{2} + t\_3 \cdot \left(0.5 - \frac{t\_2}{2}\right)}}{\sqrt{\left(1 - {\left(t\_0 \cdot t\_1 - t\_7 \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - t\_3 \cdot \left(t\_4 \cdot t\_4\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* phi2 0.5)))
        (t_1 (sin (* 0.5 phi1)))
        (t_2 (cos (- lambda1 lambda2)))
        (t_3 (* (cos phi1) (cos phi2)))
        (t_4 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_5 (sin (* 0.5 (- phi1 phi2))))
        (t_6 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* t_4 (* t_3 t_4))))
        (t_7 (cos (* 0.5 phi1))))
   (if (<= (atan2 (sqrt t_6) (sqrt (- 1.0 t_6))) 0.05)
     (*
      (atan2
       (hypot
        t_5
        (*
         (-
          (* (sin (/ lambda1 2.0)) (cos (/ lambda2 2.0)))
          (* (cos (/ lambda1 2.0)) (sin (/ lambda2 2.0))))
         (sqrt t_3)))
       (sqrt
        (-
         1.0
         (fma (cos phi1) (* (cos phi2) (- 0.5 (* 0.5 t_2))) (pow t_5 2.0)))))
      (* R 2.0))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (pow (fma t_0 t_1 (* (sin (* phi2 -0.5)) t_7)) 2.0)
          (* t_3 (- 0.5 (/ t_2 2.0)))))
        (sqrt
         (-
          (- 1.0 (pow (- (* t_0 t_1) (* t_7 (sin (* phi2 0.5)))) 2.0))
          (* t_3 (* t_4 t_4))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((phi2 * 0.5));
	double t_1 = sin((0.5 * phi1));
	double t_2 = cos((lambda1 - lambda2));
	double t_3 = cos(phi1) * cos(phi2);
	double t_4 = sin(((lambda1 - lambda2) / 2.0));
	double t_5 = sin((0.5 * (phi1 - phi2)));
	double t_6 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (t_4 * (t_3 * t_4));
	double t_7 = cos((0.5 * phi1));
	double tmp;
	if (atan2(sqrt(t_6), sqrt((1.0 - t_6))) <= 0.05) {
		tmp = atan2(hypot(t_5, (((sin((lambda1 / 2.0)) * cos((lambda2 / 2.0))) - (cos((lambda1 / 2.0)) * sin((lambda2 / 2.0)))) * sqrt(t_3))), sqrt((1.0 - fma(cos(phi1), (cos(phi2) * (0.5 - (0.5 * t_2))), pow(t_5, 2.0))))) * (R * 2.0);
	} else {
		tmp = R * (2.0 * atan2(sqrt((pow(fma(t_0, t_1, (sin((phi2 * -0.5)) * t_7)), 2.0) + (t_3 * (0.5 - (t_2 / 2.0))))), sqrt(((1.0 - pow(((t_0 * t_1) - (t_7 * sin((phi2 * 0.5)))), 2.0)) - (t_3 * (t_4 * t_4))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(phi2 * 0.5))
	t_1 = sin(Float64(0.5 * phi1))
	t_2 = cos(Float64(lambda1 - lambda2))
	t_3 = Float64(cos(phi1) * cos(phi2))
	t_4 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_5 = sin(Float64(0.5 * Float64(phi1 - phi2)))
	t_6 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(t_4 * Float64(t_3 * t_4)))
	t_7 = cos(Float64(0.5 * phi1))
	tmp = 0.0
	if (atan(sqrt(t_6), sqrt(Float64(1.0 - t_6))) <= 0.05)
		tmp = Float64(atan(hypot(t_5, Float64(Float64(Float64(sin(Float64(lambda1 / 2.0)) * cos(Float64(lambda2 / 2.0))) - Float64(cos(Float64(lambda1 / 2.0)) * sin(Float64(lambda2 / 2.0)))) * sqrt(t_3))), sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * t_2))), (t_5 ^ 2.0))))) * Float64(R * 2.0));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((fma(t_0, t_1, Float64(sin(Float64(phi2 * -0.5)) * t_7)) ^ 2.0) + Float64(t_3 * Float64(0.5 - Float64(t_2 / 2.0))))), sqrt(Float64(Float64(1.0 - (Float64(Float64(t_0 * t_1) - Float64(t_7 * sin(Float64(phi2 * 0.5)))) ^ 2.0)) - Float64(t_3 * Float64(t_4 * t_4)))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$4 * N[(t$95$3 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$7 = N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[ArcTan[N[Sqrt[t$95$6], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$6), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.05], N[(N[ArcTan[N[Sqrt[t$95$5 ^ 2 + N[(N[(N[(N[Sin[N[(lambda1 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(lambda2 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(lambda1 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(lambda2 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[t$95$3], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[t$95$5, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(R * 2.0), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[(t$95$0 * t$95$1 + N[(N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] * t$95$7), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$3 * N[(0.5 - N[(t$95$2 / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Power[N[(N[(t$95$0 * t$95$1), $MachinePrecision] - N[(t$95$7 * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - N[(t$95$3 * N[(t$95$4 * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (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 91.5%

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

      1. associate-*r* 91.5%

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

      2. *-commutative 91.5%

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

   3. Simplified 91.5%

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

   6. Applied egg-rr 95.9%

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

      1. *-lft-identity 95.9%

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

      2. *-commutative 95.9%

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

      3. *-commutative 95.9%

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

      4. metadata-eval 95.9%

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

      5. cancel-sign-sub-inv 95.9%

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

   8. Simplified 95.9%

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

      1. *-commutative 95.9%

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

      2. metadata-eval 95.9%

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

      3. div-inv 95.9%

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

      4. div-sub 95.9%

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

      5. sin-diff 96.1%

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

   10. Applied egg-rr 96.1%

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

   if 0.050000000000000003 < (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 60.3%

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

      1. associate-*l* 60.3%

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

   3. Simplified 60.3%

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

      1. div-sub 60.3%

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

      2. sin-diff 61.3%

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

      3. div-inv 61.3%

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

      4. metadata-eval 61.3%

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

      5. div-inv 61.3%

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

      6. metadata-eval 61.3%

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

      7. div-inv 61.3%

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

      8. metadata-eval 61.3%

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

      9. div-inv 61.3%

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

      10. metadata-eval 61.3%

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

   6. Applied egg-rr 61.3%

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

      1. div-sub 60.3%

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

      2. sin-diff 61.3%

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

      3. div-inv 61.3%

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

      4. metadata-eval 61.3%

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

      5. div-inv 61.3%

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

      6. metadata-eval 61.3%

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

      7. div-inv 61.3%

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

      8. metadata-eval 61.3%

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

      9. div-inv 61.3%

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

      10. metadata-eval 61.3%

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

   8. Applied egg-rr 75.6%

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

      1. *-commutative 75.6%

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

      2. *-commutative 75.6%

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

      3. fmm-def 75.6%

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

      4. *-commutative 75.6%

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

      5. *-commutative 75.6%

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

      6. *-commutative 75.6%

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

      7. distribute-lft-neg-in 75.6%

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

      8. sin-neg 75.6%

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

      9. distribute-rgt-neg-in 75.6%

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

      10. metadata-eval 75.6%

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

      11. *-commutative 75.6%

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

      12. *-commutative 75.6%

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

   10. Simplified 75.6%

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

       1. sin-mult 75.6%

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

       2. cos-sum 75.6%

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

       3. cos-2 75.6%

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

       4. div-sub 75.6%

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

       5. +-inverses 75.6%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \sin \left(\phi_1 \cdot 0.5\right), \sin \left(-0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right)}{2}\right)}}{\sqrt{\left(1 - {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. Applied egg-rr 75.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \sin \left(\phi_1 \cdot 0.5\right), \sin \left(-0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{\cos 0}{2} - \frac{\cos \left(\lambda_1 - \lambda_2\right)}{2}\right)}}}{\sqrt{\left(1 - {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   13. Step-by-step derivation

       1. cos-0 75.6%

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

       2. metadata-eval 75.6%

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

   14. Simplified 75.6%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}} \leq 0.05:\\ \;\;\;\;\tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{2}\right) - \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{2}\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \sin \left(0.5 \cdot \phi_1\right), \sin \left(\phi_2 \cdot -0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(0.5 - \frac{\cos \left(\lambda_1 - \lambda_2\right)}{2}\right)}}{\sqrt{\left(1 - {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_1\right)\\ t_1 := \cos \left(0.5 \cdot \phi_1\right)\\ t_2 := \cos \left(\phi_2 \cdot 0.5\right)\\ t_3 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_4 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t\_3 \cdot t\_3\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(t\_2, t\_0, \sin \left(\phi_2 \cdot -0.5\right) \cdot t\_1\right)\right)}^{2} + t\_4}}{\sqrt{\left(1 - {\left(t\_0 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(t\_2\right)\right) - t\_1 \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - t\_4}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 phi1)))
        (t_1 (cos (* 0.5 phi1)))
        (t_2 (cos (* phi2 0.5)))
        (t_3 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_4 (* (* (cos phi1) (cos phi2)) (* t_3 t_3))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt (+ (pow (fma t_2 t_0 (* (sin (* phi2 -0.5)) t_1)) 2.0) t_4))
      (sqrt
       (-
        (-
         1.0
         (pow (- (* t_0 (expm1 (log1p t_2))) (* t_1 (sin (* phi2 0.5)))) 2.0))
        t_4)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * phi1));
	double t_1 = cos((0.5 * phi1));
	double t_2 = cos((phi2 * 0.5));
	double t_3 = sin(((lambda1 - lambda2) / 2.0));
	double t_4 = (cos(phi1) * cos(phi2)) * (t_3 * t_3);
	return R * (2.0 * atan2(sqrt((pow(fma(t_2, t_0, (sin((phi2 * -0.5)) * t_1)), 2.0) + t_4)), sqrt(((1.0 - pow(((t_0 * expm1(log1p(t_2))) - (t_1 * sin((phi2 * 0.5)))), 2.0)) - t_4))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * phi1))
	t_1 = cos(Float64(0.5 * phi1))
	t_2 = cos(Float64(phi2 * 0.5))
	t_3 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_4 = Float64(Float64(cos(phi1) * cos(phi2)) * Float64(t_3 * t_3))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64((fma(t_2, t_0, Float64(sin(Float64(phi2 * -0.5)) * t_1)) ^ 2.0) + t_4)), sqrt(Float64(Float64(1.0 - (Float64(Float64(t_0 * expm1(log1p(t_2))) - Float64(t_1 * sin(Float64(phi2 * 0.5)))) ^ 2.0)) - t_4)))))
end

Wolfram

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

Derivation

1. Initial program 62.9%

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

   1. associate-*l* 62.9%

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

3. Simplified 63.0%

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

   1. div-sub 63.0%

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

   2. sin-diff 63.9%

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

   3. div-inv 63.9%

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

   4. metadata-eval 63.9%

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

   5. div-inv 63.9%

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

   6. metadata-eval 63.9%

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

   7. div-inv 63.9%

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

   8. metadata-eval 63.9%

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

   9. div-inv 63.9%

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

   10. metadata-eval 63.9%

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

6. Applied egg-rr 63.9%

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

   1. div-sub 63.0%

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

   2. sin-diff 63.9%

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

   3. div-inv 63.9%

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

   4. metadata-eval 63.9%

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

   5. div-inv 63.9%

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

   6. metadata-eval 63.9%

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

   7. div-inv 63.9%

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

   8. metadata-eval 63.9%

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

   9. div-inv 63.9%

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

   10. metadata-eval 63.9%

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

8. Applied egg-rr 77.0%

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

   1. *-commutative 77.0%

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

   2. *-commutative 77.0%

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

   3. fmm-def 77.0%

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

   4. *-commutative 77.0%

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

   5. *-commutative 77.0%

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

   6. *-commutative 77.0%

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

   7. distribute-lft-neg-in 77.0%

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

   8. sin-neg 77.0%

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

   9. distribute-rgt-neg-in 77.0%

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

   10. metadata-eval 77.0%

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

   11. *-commutative 77.0%

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

   12. *-commutative 77.0%

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

10. Simplified 77.0%

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

    1. expm1-log1p-u 77.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \sin \left(\phi_1 \cdot 0.5\right), \sin \left(-0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\phi_2 \cdot 0.5\right)\right)\right)} - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

12. Applied egg-rr 77.0%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \sin \left(\phi_1 \cdot 0.5\right), \sin \left(-0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\phi_2 \cdot 0.5\right)\right)\right)} - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

13. Final simplification 77.0%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \sin \left(0.5 \cdot \phi_1\right), \sin \left(\phi_2 \cdot -0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\phi_2 \cdot 0.5\right)\right)\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
14. Add Preprocessing

Alternative 3: 78.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

   1. associate-*l* 62.9%

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

3. Simplified 63.0%

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

   1. div-sub 63.0%

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

   2. sin-diff 63.9%

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

   3. div-inv 63.9%

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

   4. metadata-eval 63.9%

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

   5. div-inv 63.9%

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

   6. metadata-eval 63.9%

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

   7. div-inv 63.9%

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

   8. metadata-eval 63.9%

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

   9. div-inv 63.9%

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

   10. metadata-eval 63.9%

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

6. Applied egg-rr 63.9%

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

   1. div-sub 63.0%

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

   2. sin-diff 63.9%

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

   3. div-inv 63.9%

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

   4. metadata-eval 63.9%

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

   5. div-inv 63.9%

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

   6. metadata-eval 63.9%

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

   7. div-inv 63.9%

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

   8. metadata-eval 63.9%

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

   9. div-inv 63.9%

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

   10. metadata-eval 63.9%

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

8. Applied egg-rr 77.0%

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

   1. *-commutative 77.0%

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

   2. *-commutative 77.0%

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

   3. fmm-def 77.0%

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

   4. *-commutative 77.0%

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

   5. *-commutative 77.0%

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

   6. *-commutative 77.0%

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

   7. distribute-lft-neg-in 77.0%

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

   8. sin-neg 77.0%

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

   9. distribute-rgt-neg-in 77.0%

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

   10. metadata-eval 77.0%

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

   11. *-commutative 77.0%

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

   12. *-commutative 77.0%

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

10. Simplified 77.0%

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

    1. expm1-log1p-u 77.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \sin \left(\phi_1 \cdot 0.5\right), \sin \left(-0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\phi_2 \cdot 0.5\right)\right)\right)} - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

12. Applied egg-rr 77.0%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \sin \left(\phi_1 \cdot 0.5\right), \sin \left(-0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(\phi_2 \cdot 0.5\right)\right)\right)} - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
13. Step-by-step derivation

    1. expm1-undefine 77.0%

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

    2. log1p-undefine 77.0%

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

    3. rem-exp-log 77.0%

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

    4. *-commutative 77.0%

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

14. Applied egg-rr 77.0%

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

15. Final simplification 77.0%

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

Alternative 4: 78.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

   1. associate-*l* 62.9%

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

3. Simplified 63.0%

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

   1. div-sub 63.0%

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

   2. sin-diff 63.9%

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

   3. div-inv 63.9%

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

   4. metadata-eval 63.9%

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

   5. div-inv 63.9%

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

   6. metadata-eval 63.9%

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

   7. div-inv 63.9%

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

   8. metadata-eval 63.9%

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

   9. div-inv 63.9%

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

   10. metadata-eval 63.9%

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

6. Applied egg-rr 63.9%

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

   1. div-sub 63.0%

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

   2. sin-diff 63.9%

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

   3. div-inv 63.9%

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

   4. metadata-eval 63.9%

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

   5. div-inv 63.9%

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

   6. metadata-eval 63.9%

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

   7. div-inv 63.9%

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

   8. metadata-eval 63.9%

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

   9. div-inv 63.9%

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

   10. metadata-eval 63.9%

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

8. Applied egg-rr 77.0%

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

   1. *-commutative 77.0%

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

   2. *-commutative 77.0%

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

   3. fmm-def 77.0%

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

   4. *-commutative 77.0%

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

   5. *-commutative 77.0%

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

   6. *-commutative 77.0%

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

   7. distribute-lft-neg-in 77.0%

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

   8. sin-neg 77.0%

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

   9. distribute-rgt-neg-in 77.0%

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

   10. metadata-eval 77.0%

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

   11. *-commutative 77.0%

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

   12. *-commutative 77.0%

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

10. Simplified 77.0%

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

11. Final simplification 77.0%

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

Alternative 5: 77.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \cos \left(0.5 \cdot \phi_1\right)\\ t_3 := \sqrt{\left(1 - {\left(t\_0 - t\_2 \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t\_1 \cdot t\_1\right)}\\ t_4 := {\left(t\_0 + \sin \left(\phi_2 \cdot -0.5\right) \cdot t\_2\right)}^{2}\\ \mathbf{if}\;\lambda_2 \leq -2.2 \cdot 10^{-13} \lor \neg \left(\lambda_2 \leq 1.95 \cdot 10^{-8}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}\right) + t\_4}}{t\_3}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(0.5 \cdot \lambda_1\right)}^{2}\right)}}{t\_3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos (* phi2 0.5)) (sin (* 0.5 phi1))))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (cos (* 0.5 phi1)))
        (t_3
         (sqrt
          (-
           (- 1.0 (pow (- t_0 (* t_2 (sin (* phi2 0.5)))) 2.0))
           (* (* (cos phi1) (cos phi2)) (* t_1 t_1)))))
        (t_4 (pow (+ t_0 (* (sin (* phi2 -0.5)) t_2)) 2.0)))
   (if (or (<= lambda2 -2.2e-13) (not (<= lambda2 1.95e-8)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (* (cos phi1) (* (cos phi2) (pow (sin (* -0.5 lambda2)) 2.0)))
          t_4))
        t_3)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+ t_4 (* (cos phi1) (* (cos phi2) (pow (sin (* 0.5 lambda1)) 2.0)))))
        t_3))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((phi2 * 0.5)) * sin((0.5 * phi1));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = cos((0.5 * phi1));
	double t_3 = sqrt(((1.0 - pow((t_0 - (t_2 * sin((phi2 * 0.5)))), 2.0)) - ((cos(phi1) * cos(phi2)) * (t_1 * t_1))));
	double t_4 = pow((t_0 + (sin((phi2 * -0.5)) * t_2)), 2.0);
	double tmp;
	if ((lambda2 <= -2.2e-13) || !(lambda2 <= 1.95e-8)) {
		tmp = R * (2.0 * atan2(sqrt(((cos(phi1) * (cos(phi2) * pow(sin((-0.5 * lambda2)), 2.0))) + t_4)), t_3));
	} else {
		tmp = R * (2.0 * atan2(sqrt((t_4 + (cos(phi1) * (cos(phi2) * pow(sin((0.5 * lambda1)), 2.0))))), t_3));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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 = cos((phi2 * 0.5d0)) * sin((0.5d0 * phi1))
    t_1 = sin(((lambda1 - lambda2) / 2.0d0))
    t_2 = cos((0.5d0 * phi1))
    t_3 = sqrt(((1.0d0 - ((t_0 - (t_2 * sin((phi2 * 0.5d0)))) ** 2.0d0)) - ((cos(phi1) * cos(phi2)) * (t_1 * t_1))))
    t_4 = (t_0 + (sin((phi2 * (-0.5d0))) * t_2)) ** 2.0d0
    if ((lambda2 <= (-2.2d-13)) .or. (.not. (lambda2 <= 1.95d-8))) then
        tmp = r * (2.0d0 * atan2(sqrt(((cos(phi1) * (cos(phi2) * (sin(((-0.5d0) * lambda2)) ** 2.0d0))) + t_4)), t_3))
    else
        tmp = r * (2.0d0 * atan2(sqrt((t_4 + (cos(phi1) * (cos(phi2) * (sin((0.5d0 * lambda1)) ** 2.0d0))))), 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 = Math.cos((phi2 * 0.5)) * Math.sin((0.5 * phi1));
	double t_1 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_2 = Math.cos((0.5 * phi1));
	double t_3 = Math.sqrt(((1.0 - Math.pow((t_0 - (t_2 * Math.sin((phi2 * 0.5)))), 2.0)) - ((Math.cos(phi1) * Math.cos(phi2)) * (t_1 * t_1))));
	double t_4 = Math.pow((t_0 + (Math.sin((phi2 * -0.5)) * t_2)), 2.0);
	double tmp;
	if ((lambda2 <= -2.2e-13) || !(lambda2 <= 1.95e-8)) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(((Math.cos(phi1) * (Math.cos(phi2) * Math.pow(Math.sin((-0.5 * lambda2)), 2.0))) + t_4)), t_3));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_4 + (Math.cos(phi1) * (Math.cos(phi2) * Math.pow(Math.sin((0.5 * lambda1)), 2.0))))), t_3));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((phi2 * 0.5)) * math.sin((0.5 * phi1))
	t_1 = math.sin(((lambda1 - lambda2) / 2.0))
	t_2 = math.cos((0.5 * phi1))
	t_3 = math.sqrt(((1.0 - math.pow((t_0 - (t_2 * math.sin((phi2 * 0.5)))), 2.0)) - ((math.cos(phi1) * math.cos(phi2)) * (t_1 * t_1))))
	t_4 = math.pow((t_0 + (math.sin((phi2 * -0.5)) * t_2)), 2.0)
	tmp = 0
	if (lambda2 <= -2.2e-13) or not (lambda2 <= 1.95e-8):
		tmp = R * (2.0 * math.atan2(math.sqrt(((math.cos(phi1) * (math.cos(phi2) * math.pow(math.sin((-0.5 * lambda2)), 2.0))) + t_4)), t_3))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt((t_4 + (math.cos(phi1) * (math.cos(phi2) * math.pow(math.sin((0.5 * lambda1)), 2.0))))), t_3))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1)))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = cos(Float64(0.5 * phi1))
	t_3 = sqrt(Float64(Float64(1.0 - (Float64(t_0 - Float64(t_2 * sin(Float64(phi2 * 0.5)))) ^ 2.0)) - Float64(Float64(cos(phi1) * cos(phi2)) * Float64(t_1 * t_1))))
	t_4 = Float64(t_0 + Float64(sin(Float64(phi2 * -0.5)) * t_2)) ^ 2.0
	tmp = 0.0
	if ((lambda2 <= -2.2e-13) || !(lambda2 <= 1.95e-8))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(cos(phi1) * Float64(cos(phi2) * (sin(Float64(-0.5 * lambda2)) ^ 2.0))) + t_4)), t_3)));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_4 + Float64(cos(phi1) * Float64(cos(phi2) * (sin(Float64(0.5 * lambda1)) ^ 2.0))))), t_3)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((phi2 * 0.5)) * sin((0.5 * phi1));
	t_1 = sin(((lambda1 - lambda2) / 2.0));
	t_2 = cos((0.5 * phi1));
	t_3 = sqrt(((1.0 - ((t_0 - (t_2 * sin((phi2 * 0.5)))) ^ 2.0)) - ((cos(phi1) * cos(phi2)) * (t_1 * t_1))));
	t_4 = (t_0 + (sin((phi2 * -0.5)) * t_2)) ^ 2.0;
	tmp = 0.0;
	if ((lambda2 <= -2.2e-13) || ~((lambda2 <= 1.95e-8)))
		tmp = R * (2.0 * atan2(sqrt(((cos(phi1) * (cos(phi2) * (sin((-0.5 * lambda2)) ^ 2.0))) + t_4)), t_3));
	else
		tmp = R * (2.0 * atan2(sqrt((t_4 + (cos(phi1) * (cos(phi2) * (sin((0.5 * lambda1)) ^ 2.0))))), t_3));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[(1.0 - N[Power[N[(t$95$0 - N[(t$95$2 * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Power[N[(t$95$0 + N[(N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[Or[LessEqual[lambda2, -2.2e-13], N[Not[LessEqual[lambda2, 1.95e-8]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]], $MachinePrecision] / t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$4 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -2.19999999999999997e-13 or 1.94999999999999992e-8 < lambda2

   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. Step-by-step derivation

      1. associate-*l* 48.9%

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

   3. Simplified 48.9%

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

      1. div-sub 48.9%

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

      2. sin-diff 49.7%

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

      3. div-inv 49.7%

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

      4. metadata-eval 49.7%

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

      5. div-inv 49.7%

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

      6. metadata-eval 49.7%

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

      7. div-inv 49.7%

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

      8. metadata-eval 49.7%

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

      9. div-inv 49.7%

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

      10. metadata-eval 49.7%

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

   6. Applied egg-rr 49.7%

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

      1. div-sub 48.9%

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

      2. sin-diff 49.7%

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

      3. div-inv 49.7%

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

      4. metadata-eval 49.7%

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

      5. div-inv 49.7%

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

      6. metadata-eval 49.7%

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

      7. div-inv 49.7%

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

      8. metadata-eval 49.7%

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

      9. div-inv 49.7%

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

      10. metadata-eval 49.7%

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

   8. Applied egg-rr 58.6%

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

      1. *-commutative 58.6%

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

      2. *-commutative 58.6%

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

      3. fmm-def 58.7%

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

      4. *-commutative 58.7%

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

      5. *-commutative 58.7%

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

      6. *-commutative 58.7%

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

      7. distribute-lft-neg-in 58.7%

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

      8. sin-neg 58.7%

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

      9. distribute-rgt-neg-in 58.7%

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

      10. metadata-eval 58.7%

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

      11. *-commutative 58.7%

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

      12. *-commutative 58.7%

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

   10. Simplified 58.7%

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

   11. Taylor expanded in lambda1 around 0 58.6%

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

   if -2.19999999999999997e-13 < lambda2 < 1.94999999999999992e-8

   1. Initial program 79.1%

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

      1. associate-*l* 79.1%

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

   3. Simplified 79.1%

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

      1. div-sub 79.1%

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

      2. sin-diff 80.1%

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

      3. div-inv 80.1%

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

      4. metadata-eval 80.1%

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

      5. div-inv 80.1%

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

      6. metadata-eval 80.1%

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

      7. div-inv 80.1%

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

      8. metadata-eval 80.1%

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

      9. div-inv 80.1%

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

      10. metadata-eval 80.1%

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

   6. Applied egg-rr 80.1%

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

      1. div-sub 79.1%

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

      2. sin-diff 80.1%

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

      3. div-inv 80.1%

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

      4. metadata-eval 80.1%

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

      5. div-inv 80.1%

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

      6. metadata-eval 80.1%

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

      7. div-inv 80.1%

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

      8. metadata-eval 80.1%

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

      9. div-inv 80.1%

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

      10. metadata-eval 80.1%

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

   8. Applied egg-rr 98.1%

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

      1. *-commutative 98.1%

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

      2. *-commutative 98.1%

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

      3. fmm-def 98.1%

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

      4. *-commutative 98.1%

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

      5. *-commutative 98.1%

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

      6. *-commutative 98.1%

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

      7. distribute-lft-neg-in 98.1%

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

      8. sin-neg 98.1%

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

      9. distribute-rgt-neg-in 98.1%

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

      10. metadata-eval 98.1%

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

      11. *-commutative 98.1%

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

      12. *-commutative 98.1%

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

   10. Simplified 98.1%

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

   11. Taylor expanded in lambda2 around 0 97.3%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -2.2 \cdot 10^{-13} \lor \neg \left(\lambda_2 \leq 1.95 \cdot 10^{-8}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) + \sin \left(\phi_2 \cdot -0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}}}{\sqrt{\left(1 - {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) + \sin \left(\phi_2 \cdot -0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(0.5 \cdot \lambda_1\right)}^{2}\right)}}{\sqrt{\left(1 - {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \cos \left(0.5 \cdot \phi_1\right)\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := t\_0 \cdot \left(t\_2 \cdot t\_2\right)\\ t_4 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_5 := \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\\ t_6 := {\left(t\_5 - t\_1 \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\\ t_7 := 1 - t\_6\\ \mathbf{if}\;\lambda_1 \leq -6.2 \cdot 10^{-56}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3 + t\_6}}{\sqrt{\left(1 - t\_4\right) - t\_3}}\right)\\ \mathbf{elif}\;\lambda_1 \leq 0.00145:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}\right) + {\left(t\_5 + \sin \left(\phi_2 \cdot -0.5\right) \cdot t\_1\right)}^{2}}}{\sqrt{t\_7 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3 + t\_4}}{\sqrt{t\_7 - t\_0 \cdot {\sin \left(0.5 \cdot \lambda_1\right)}^{2}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2)))
        (t_1 (cos (* 0.5 phi1)))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3 (* t_0 (* t_2 t_2)))
        (t_4 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_5 (* (cos (* phi2 0.5)) (sin (* 0.5 phi1))))
        (t_6 (pow (- t_5 (* t_1 (sin (* phi2 0.5)))) 2.0))
        (t_7 (- 1.0 t_6)))
   (if (<= lambda1 -6.2e-56)
     (* R (* 2.0 (atan2 (sqrt (+ t_3 t_6)) (sqrt (- (- 1.0 t_4) t_3)))))
     (if (<= lambda1 0.00145)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (+
            (* (cos phi1) (* (cos phi2) (pow (sin (* -0.5 lambda2)) 2.0)))
            (pow (+ t_5 (* (sin (* phi2 -0.5)) t_1)) 2.0)))
          (sqrt (- t_7 t_3)))))
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ t_3 t_4))
          (sqrt (- t_7 (* t_0 (pow (sin (* 0.5 lambda1)) 2.0)))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * cos(phi2);
	double t_1 = cos((0.5 * phi1));
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = t_0 * (t_2 * t_2);
	double t_4 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_5 = cos((phi2 * 0.5)) * sin((0.5 * phi1));
	double t_6 = pow((t_5 - (t_1 * sin((phi2 * 0.5)))), 2.0);
	double t_7 = 1.0 - t_6;
	double tmp;
	if (lambda1 <= -6.2e-56) {
		tmp = R * (2.0 * atan2(sqrt((t_3 + t_6)), sqrt(((1.0 - t_4) - t_3))));
	} else if (lambda1 <= 0.00145) {
		tmp = R * (2.0 * atan2(sqrt(((cos(phi1) * (cos(phi2) * pow(sin((-0.5 * lambda2)), 2.0))) + pow((t_5 + (sin((phi2 * -0.5)) * t_1)), 2.0))), sqrt((t_7 - t_3))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((t_3 + t_4)), sqrt((t_7 - (t_0 * pow(sin((0.5 * lambda1)), 2.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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) :: t_6
    real(8) :: t_7
    real(8) :: tmp
    t_0 = cos(phi1) * cos(phi2)
    t_1 = cos((0.5d0 * phi1))
    t_2 = sin(((lambda1 - lambda2) / 2.0d0))
    t_3 = t_0 * (t_2 * t_2)
    t_4 = sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0
    t_5 = cos((phi2 * 0.5d0)) * sin((0.5d0 * phi1))
    t_6 = (t_5 - (t_1 * sin((phi2 * 0.5d0)))) ** 2.0d0
    t_7 = 1.0d0 - t_6
    if (lambda1 <= (-6.2d-56)) then
        tmp = r * (2.0d0 * atan2(sqrt((t_3 + t_6)), sqrt(((1.0d0 - t_4) - t_3))))
    else if (lambda1 <= 0.00145d0) then
        tmp = r * (2.0d0 * atan2(sqrt(((cos(phi1) * (cos(phi2) * (sin(((-0.5d0) * lambda2)) ** 2.0d0))) + ((t_5 + (sin((phi2 * (-0.5d0))) * t_1)) ** 2.0d0))), sqrt((t_7 - t_3))))
    else
        tmp = r * (2.0d0 * atan2(sqrt((t_3 + t_4)), sqrt((t_7 - (t_0 * (sin((0.5d0 * lambda1)) ** 2.0d0))))))
    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.cos(phi1) * Math.cos(phi2);
	double t_1 = Math.cos((0.5 * phi1));
	double t_2 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_3 = t_0 * (t_2 * t_2);
	double t_4 = Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_5 = Math.cos((phi2 * 0.5)) * Math.sin((0.5 * phi1));
	double t_6 = Math.pow((t_5 - (t_1 * Math.sin((phi2 * 0.5)))), 2.0);
	double t_7 = 1.0 - t_6;
	double tmp;
	if (lambda1 <= -6.2e-56) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_3 + t_6)), Math.sqrt(((1.0 - t_4) - t_3))));
	} else if (lambda1 <= 0.00145) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(((Math.cos(phi1) * (Math.cos(phi2) * Math.pow(Math.sin((-0.5 * lambda2)), 2.0))) + Math.pow((t_5 + (Math.sin((phi2 * -0.5)) * t_1)), 2.0))), Math.sqrt((t_7 - t_3))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_3 + t_4)), Math.sqrt((t_7 - (t_0 * Math.pow(Math.sin((0.5 * lambda1)), 2.0))))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.cos(phi2)
	t_1 = math.cos((0.5 * phi1))
	t_2 = math.sin(((lambda1 - lambda2) / 2.0))
	t_3 = t_0 * (t_2 * t_2)
	t_4 = math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0)
	t_5 = math.cos((phi2 * 0.5)) * math.sin((0.5 * phi1))
	t_6 = math.pow((t_5 - (t_1 * math.sin((phi2 * 0.5)))), 2.0)
	t_7 = 1.0 - t_6
	tmp = 0
	if lambda1 <= -6.2e-56:
		tmp = R * (2.0 * math.atan2(math.sqrt((t_3 + t_6)), math.sqrt(((1.0 - t_4) - t_3))))
	elif lambda1 <= 0.00145:
		tmp = R * (2.0 * math.atan2(math.sqrt(((math.cos(phi1) * (math.cos(phi2) * math.pow(math.sin((-0.5 * lambda2)), 2.0))) + math.pow((t_5 + (math.sin((phi2 * -0.5)) * t_1)), 2.0))), math.sqrt((t_7 - t_3))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt((t_3 + t_4)), math.sqrt((t_7 - (t_0 * math.pow(math.sin((0.5 * lambda1)), 2.0))))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	t_1 = cos(Float64(0.5 * phi1))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64(t_0 * Float64(t_2 * t_2))
	t_4 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_5 = Float64(cos(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1)))
	t_6 = Float64(t_5 - Float64(t_1 * sin(Float64(phi2 * 0.5)))) ^ 2.0
	t_7 = Float64(1.0 - t_6)
	tmp = 0.0
	if (lambda1 <= -6.2e-56)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_3 + t_6)), sqrt(Float64(Float64(1.0 - t_4) - t_3)))));
	elseif (lambda1 <= 0.00145)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(cos(phi1) * Float64(cos(phi2) * (sin(Float64(-0.5 * lambda2)) ^ 2.0))) + (Float64(t_5 + Float64(sin(Float64(phi2 * -0.5)) * t_1)) ^ 2.0))), sqrt(Float64(t_7 - t_3)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_3 + t_4)), sqrt(Float64(t_7 - Float64(t_0 * (sin(Float64(0.5 * lambda1)) ^ 2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * cos(phi2);
	t_1 = cos((0.5 * phi1));
	t_2 = sin(((lambda1 - lambda2) / 2.0));
	t_3 = t_0 * (t_2 * t_2);
	t_4 = sin(((phi1 - phi2) / 2.0)) ^ 2.0;
	t_5 = cos((phi2 * 0.5)) * sin((0.5 * phi1));
	t_6 = (t_5 - (t_1 * sin((phi2 * 0.5)))) ^ 2.0;
	t_7 = 1.0 - t_6;
	tmp = 0.0;
	if (lambda1 <= -6.2e-56)
		tmp = R * (2.0 * atan2(sqrt((t_3 + t_6)), sqrt(((1.0 - t_4) - t_3))));
	elseif (lambda1 <= 0.00145)
		tmp = R * (2.0 * atan2(sqrt(((cos(phi1) * (cos(phi2) * (sin((-0.5 * lambda2)) ^ 2.0))) + ((t_5 + (sin((phi2 * -0.5)) * t_1)) ^ 2.0))), sqrt((t_7 - t_3))));
	else
		tmp = R * (2.0 * atan2(sqrt((t_3 + t_4)), sqrt((t_7 - (t_0 * (sin((0.5 * lambda1)) ^ 2.0))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * phi1), $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[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Power[N[(t$95$5 - N[(t$95$1 * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$7 = N[(1.0 - t$95$6), $MachinePrecision]}, If[LessEqual[lambda1, -6.2e-56], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$3 + t$95$6), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - t$95$4), $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, 0.00145], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(t$95$5 + N[(N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(t$95$7 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$3 + t$95$4), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(t$95$7 - N[(t$95$0 * N[Power[N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -6.19999999999999975e-56

   1. Initial program 52.7%

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

      1. associate-*l* 52.7%

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

   3. Simplified 52.7%

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

      1. div-sub 52.7%

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

      2. sin-diff 53.1%

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

      3. div-inv 53.1%

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

      4. metadata-eval 53.1%

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

      5. div-inv 53.1%

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

      6. metadata-eval 53.1%

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

      7. div-inv 53.1%

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

      8. metadata-eval 53.1%

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

      9. div-inv 53.1%

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

      10. metadata-eval 53.1%

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

   6. Applied egg-rr 53.3%

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

   if -6.19999999999999975e-56 < lambda1 < 0.00145

   1. Initial program 78.9%

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

      1. associate-*l* 78.9%

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

   3. Simplified 78.9%

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

      1. div-sub 78.9%

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

      2. sin-diff 80.1%

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

      3. div-inv 80.1%

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

      4. metadata-eval 80.1%

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

      5. div-inv 80.1%

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

      6. metadata-eval 80.1%

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

      7. div-inv 80.1%

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

      8. metadata-eval 80.1%

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

      9. div-inv 80.1%

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

      10. metadata-eval 80.1%

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

   6. Applied egg-rr 80.1%

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

      1. div-sub 78.9%

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

      2. sin-diff 80.1%

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

      3. div-inv 80.1%

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

      4. metadata-eval 80.1%

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

      5. div-inv 80.1%

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

      6. metadata-eval 80.1%

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

      7. div-inv 80.1%

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

      8. metadata-eval 80.1%

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

      9. div-inv 80.1%

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

      10. metadata-eval 80.1%

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

   8. Applied egg-rr 97.6%

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

      1. *-commutative 97.6%

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

      2. *-commutative 97.6%

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

      3. fmm-def 97.6%

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

      4. *-commutative 97.6%

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

      5. *-commutative 97.6%

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

      6. *-commutative 97.6%

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

      7. distribute-lft-neg-in 97.6%

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

      8. sin-neg 97.6%

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

      9. distribute-rgt-neg-in 97.6%

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

      10. metadata-eval 97.6%

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

      11. *-commutative 97.6%

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

      12. *-commutative 97.6%

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

   10. Simplified 97.6%

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

   11. Taylor expanded in lambda1 around 0 97.6%

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

   if 0.00145 < lambda1

   1. Initial program 44.3%

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

      1. associate-*l* 44.3%

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

   3. Simplified 44.3%

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

      1. div-sub 44.3%

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

      2. sin-diff 45.4%

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

      3. div-inv 45.4%

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

      4. metadata-eval 45.4%

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

      5. div-inv 45.4%

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

      6. metadata-eval 45.4%

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

      7. div-inv 45.4%

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

      8. metadata-eval 45.4%

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

      9. div-inv 45.4%

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

      10. metadata-eval 45.4%

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

   6. Applied egg-rr 45.4%

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

   7. Taylor expanded in lambda2 around 0 45.6%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -6.2 \cdot 10^{-56}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}}}{\sqrt{\left(1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\\ \mathbf{elif}\;\lambda_1 \leq 0.00145:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}\right) + {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) + \sin \left(\phi_2 \cdot -0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}^{2}}}{\sqrt{\left(1 - {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{\left(1 - {\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(0.5 \cdot \lambda_1\right)}^{2}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    t_1 = ((cos((phi2 * 0.5d0)) * sin((0.5d0 * phi1))) - (cos((0.5d0 * phi1)) * sin((phi2 * 0.5d0)))) ** 2.0d0
    t_2 = (cos(phi1) * cos(phi2)) * (t_0 * t_0)
    code = r * (2.0d0 * atan2(sqrt((t_2 + t_1)), sqrt(((1.0d0 - t_1) - t_2))))
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.cos((phi2 * 0.5)) * Math.sin((0.5 * phi1))) - (Math.cos((0.5 * phi1)) * Math.sin((phi2 * 0.5)))), 2.0);
	double t_2 = (Math.cos(phi1) * Math.cos(phi2)) * (t_0 * t_0);
	return R * (2.0 * Math.atan2(Math.sqrt((t_2 + t_1)), Math.sqrt(((1.0 - t_1) - t_2))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	t_1 = math.pow(((math.cos((phi2 * 0.5)) * math.sin((0.5 * phi1))) - (math.cos((0.5 * phi1)) * math.sin((phi2 * 0.5)))), 2.0)
	t_2 = (math.cos(phi1) * math.cos(phi2)) * (t_0 * t_0)
	return R * (2.0 * math.atan2(math.sqrt((t_2 + t_1)), math.sqrt(((1.0 - t_1) - t_2))))

Julia

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

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = ((cos((phi2 * 0.5)) * sin((0.5 * phi1))) - (cos((0.5 * phi1)) * sin((phi2 * 0.5)))) ^ 2.0;
	t_2 = (cos(phi1) * cos(phi2)) * (t_0 * t_0);
	tmp = R * (2.0 * atan2(sqrt((t_2 + t_1)), sqrt(((1.0 - t_1) - t_2))));
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[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$2 + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 62.9%

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

   1. associate-*l* 62.9%

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

3. Simplified 63.0%

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

   1. div-sub 63.0%

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

   2. sin-diff 63.9%

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

   3. div-inv 63.9%

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

   4. metadata-eval 63.9%

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

   5. div-inv 63.9%

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

   6. metadata-eval 63.9%

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

   7. div-inv 63.9%

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

   8. metadata-eval 63.9%

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

   9. div-inv 63.9%

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

   10. metadata-eval 63.9%

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

6. Applied egg-rr 63.9%

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

   1. div-sub 63.0%

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

   2. sin-diff 63.9%

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

   3. div-inv 63.9%

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

   4. metadata-eval 63.9%

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

   5. div-inv 63.9%

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

   6. metadata-eval 63.9%

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

   7. div-inv 63.9%

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

   8. metadata-eval 63.9%

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

   9. div-inv 63.9%

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

   10. metadata-eval 63.9%

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

8. Applied egg-rr 77.0%

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

9. Final simplification 77.0%

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

Alternative 8: 78.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

   1. associate-*l* 62.9%

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

3. Simplified 63.0%

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

   1. div-sub 63.0%

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

   2. sin-diff 63.9%

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

   3. div-inv 63.9%

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

   4. metadata-eval 63.9%

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

   5. div-inv 63.9%

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

   6. metadata-eval 63.9%

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

   7. div-inv 63.9%

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

   8. metadata-eval 63.9%

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

   9. div-inv 63.9%

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

   10. metadata-eval 63.9%

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

6. Applied egg-rr 63.9%

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

   1. div-sub 63.0%

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

   2. sin-diff 63.9%

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

   3. div-inv 63.9%

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

   4. metadata-eval 63.9%

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

   5. div-inv 63.9%

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

   6. metadata-eval 63.9%

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

   7. div-inv 63.9%

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

   8. metadata-eval 63.9%

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

   9. div-inv 63.9%

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

   10. metadata-eval 63.9%

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

8. Applied egg-rr 77.0%

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

   1. *-commutative 77.0%

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

   2. *-commutative 77.0%

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

   3. fmm-def 77.0%

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

   4. *-commutative 77.0%

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

   5. *-commutative 77.0%

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

   6. *-commutative 77.0%

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

   7. distribute-lft-neg-in 77.0%

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

   8. sin-neg 77.0%

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

   9. distribute-rgt-neg-in 77.0%

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

   10. metadata-eval 77.0%

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

   11. *-commutative 77.0%

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

   12. *-commutative 77.0%

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

10. Simplified 77.0%

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

    1. sin-mult 77.0%

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

    2. clear-num 77.0%

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

    3. +-inverses 77.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \sin \left(\phi_1 \cdot 0.5\right), \sin \left(-0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{1}{\frac{2}{\cos \color{blue}{0} - \cos \left(\frac{\lambda_1 - \lambda_2}{2} + \frac{\lambda_1 - \lambda_2}{2}\right)}}}}\right) \]

    4. add-log-exp 28.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \sin \left(\phi_1 \cdot 0.5\right), \sin \left(-0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{1}{\frac{2}{\cos 0 - \cos \left(\color{blue}{\log \left(e^{\frac{\lambda_1 - \lambda_2}{2}}\right)} + \frac{\lambda_1 - \lambda_2}{2}\right)}}}}\right) \]

    5. add-log-exp 28.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \sin \left(\phi_1 \cdot 0.5\right), \sin \left(-0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{1}{\frac{2}{\cos 0 - \cos \left(\log \left(e^{\frac{\lambda_1 - \lambda_2}{2}}\right) + \color{blue}{\log \left(e^{\frac{\lambda_1 - \lambda_2}{2}}\right)}\right)}}}}\right) \]

    6. sum-log 28.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \sin \left(\phi_1 \cdot 0.5\right), \sin \left(-0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{1}{\frac{2}{\cos 0 - \cos \color{blue}{\log \left(e^{\frac{\lambda_1 - \lambda_2}{2}} \cdot e^{\frac{\lambda_1 - \lambda_2}{2}}\right)}}}}}\right) \]

    7. exp-sqrt 28.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \sin \left(\phi_1 \cdot 0.5\right), \sin \left(-0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{1}{\frac{2}{\cos 0 - \cos \log \left(\color{blue}{\sqrt{e^{\lambda_1 - \lambda_2}}} \cdot e^{\frac{\lambda_1 - \lambda_2}{2}}\right)}}}}\right) \]

    8. exp-sqrt 28.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \sin \left(\phi_1 \cdot 0.5\right), \sin \left(-0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{1}{\frac{2}{\cos 0 - \cos \log \left(\sqrt{e^{\lambda_1 - \lambda_2}} \cdot \color{blue}{\sqrt{e^{\lambda_1 - \lambda_2}}}\right)}}}}\right) \]

    9. add-sqr-sqrt 28.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \sin \left(\phi_1 \cdot 0.5\right), \sin \left(-0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{1}{\frac{2}{\cos 0 - \cos \log \color{blue}{\left(e^{\lambda_1 - \lambda_2}\right)}}}}}\right) \]

    10. add-log-exp 77.0%

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \sin \left(\phi_1 \cdot 0.5\right), \sin \left(-0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \frac{1}{\frac{2}{\cos 0 - \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}}}}}\right) \]

12. Applied egg-rr 77.0%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \sin \left(\phi_1 \cdot 0.5\right), \sin \left(-0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\frac{1}{\frac{2}{\cos 0 - \cos \left(\lambda_1 - \lambda_2\right)}}}}}\right) \]

13. Final simplification 77.0%

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

Alternative 9: 62.7% accurate, 0.8× speedup

Math

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

FPCore

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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 = (cos(phi1) * cos(phi2)) * (t_0 * t_0)
    code = r * (2.0d0 * atan2(sqrt((t_1 + (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0))), sqrt(((1.0d0 - (((cos((phi2 * 0.5d0)) * sin((0.5d0 * phi1))) - (cos((0.5d0 * phi1)) * sin((phi2 * 0.5d0)))) ** 2.0d0)) - 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.cos(phi1) * Math.cos(phi2)) * (t_0 * t_0);
	return R * (2.0 * Math.atan2(Math.sqrt((t_1 + Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0))), Math.sqrt(((1.0 - Math.pow(((Math.cos((phi2 * 0.5)) * Math.sin((0.5 * phi1))) - (Math.cos((0.5 * phi1)) * Math.sin((phi2 * 0.5)))), 2.0)) - t_1))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	t_1 = (math.cos(phi1) * math.cos(phi2)) * (t_0 * t_0)
	return R * (2.0 * math.atan2(math.sqrt((t_1 + math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0))), math.sqrt(((1.0 - math.pow(((math.cos((phi2 * 0.5)) * math.sin((0.5 * phi1))) - (math.cos((0.5 * phi1)) * math.sin((phi2 * 0.5)))), 2.0)) - t_1))))

Julia

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

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = (cos(phi1) * cos(phi2)) * (t_0 * t_0);
	tmp = R * (2.0 * atan2(sqrt((t_1 + (sin(((phi1 - phi2) / 2.0)) ^ 2.0))), sqrt(((1.0 - (((cos((phi2 * 0.5)) * sin((0.5 * phi1))) - (cos((0.5 * phi1)) * sin((phi2 * 0.5)))) ^ 2.0)) - 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[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$1 + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - N[Power[N[(N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 62.9%

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

   1. associate-*l* 62.9%

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

3. Simplified 63.0%

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

   1. div-sub 63.0%

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

   2. sin-diff 63.9%

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

   3. div-inv 63.9%

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

   4. metadata-eval 63.9%

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

   5. div-inv 63.9%

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

   6. metadata-eval 63.9%

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

   7. div-inv 63.9%

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

   8. metadata-eval 63.9%

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

   9. div-inv 63.9%

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

   10. metadata-eval 63.9%

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

6. Applied egg-rr 63.9%

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

7. Final simplification 63.9%

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

Alternative 10: 62.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.9%

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

   1. associate-*l* 62.9%

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

3. Simplified 63.0%

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

6. Applied egg-rr 63.3%

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

   1. unpow1/2 63.3%

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

   2. unpow2 63.3%

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

   3. rem-sqrt-square 63.3%

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

   4. metadata-eval 63.3%

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

   5. cancel-sign-sub-inv 63.3%

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

   6. *-commutative 63.3%

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

8. Simplified 63.3%

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

9. Final simplification 63.3%

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

Alternative 11: 61.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := \cos \phi_2 \cdot t\_1\\ t_3 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t\_0 \cdot t\_0\right)\\ t_4 := {\cos \left(\phi_2 \cdot 0.5\right)}^{2}\\ t_5 := \sqrt{t\_3 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}\\ \mathbf{if}\;\phi_2 \leq -52000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{t\_4 - t\_3}}\right)\\ \mathbf{elif}\;\phi_2 \leq 3.3:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_5}{\sqrt{1 - \left(\cos \phi_1 \cdot t\_1 + {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_5}{\sqrt{t\_4 - t\_2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_2 (* (cos phi2) t_1))
        (t_3 (* (* (cos phi1) (cos phi2)) (* t_0 t_0)))
        (t_4 (pow (cos (* phi2 0.5)) 2.0))
        (t_5 (sqrt (+ t_3 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))))
   (if (<= phi2 -52000000.0)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_2 (pow (sin (* phi2 -0.5)) 2.0)))
        (sqrt (- t_4 t_3)))))
     (if (<= phi2 3.3)
       (*
        R
        (*
         2.0
         (atan2
          t_5
          (sqrt (- 1.0 (+ (* (cos phi1) t_1) (pow (sin (* 0.5 phi1)) 2.0)))))))
       (* R (* 2.0 (atan2 t_5 (sqrt (- t_4 t_2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_2 = cos(phi2) * t_1;
	double t_3 = (cos(phi1) * cos(phi2)) * (t_0 * t_0);
	double t_4 = pow(cos((phi2 * 0.5)), 2.0);
	double t_5 = sqrt((t_3 + pow(sin(((phi1 - phi2) / 2.0)), 2.0)));
	double tmp;
	if (phi2 <= -52000000.0) {
		tmp = R * (2.0 * atan2(sqrt((t_2 + pow(sin((phi2 * -0.5)), 2.0))), sqrt((t_4 - t_3))));
	} else if (phi2 <= 3.3) {
		tmp = R * (2.0 * atan2(t_5, sqrt((1.0 - ((cos(phi1) * t_1) + pow(sin((0.5 * phi1)), 2.0))))));
	} else {
		tmp = R * (2.0 * atan2(t_5, sqrt((t_4 - t_2))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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 = sin((0.5d0 * (lambda1 - lambda2))) ** 2.0d0
    t_2 = cos(phi2) * t_1
    t_3 = (cos(phi1) * cos(phi2)) * (t_0 * t_0)
    t_4 = cos((phi2 * 0.5d0)) ** 2.0d0
    t_5 = sqrt((t_3 + (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0)))
    if (phi2 <= (-52000000.0d0)) then
        tmp = r * (2.0d0 * atan2(sqrt((t_2 + (sin((phi2 * (-0.5d0))) ** 2.0d0))), sqrt((t_4 - t_3))))
    else if (phi2 <= 3.3d0) then
        tmp = r * (2.0d0 * atan2(t_5, sqrt((1.0d0 - ((cos(phi1) * t_1) + (sin((0.5d0 * phi1)) ** 2.0d0))))))
    else
        tmp = r * (2.0d0 * atan2(t_5, sqrt((t_4 - 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 = Math.pow(Math.sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_2 = Math.cos(phi2) * t_1;
	double t_3 = (Math.cos(phi1) * Math.cos(phi2)) * (t_0 * t_0);
	double t_4 = Math.pow(Math.cos((phi2 * 0.5)), 2.0);
	double t_5 = Math.sqrt((t_3 + Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0)));
	double tmp;
	if (phi2 <= -52000000.0) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_2 + Math.pow(Math.sin((phi2 * -0.5)), 2.0))), Math.sqrt((t_4 - t_3))));
	} else if (phi2 <= 3.3) {
		tmp = R * (2.0 * Math.atan2(t_5, Math.sqrt((1.0 - ((Math.cos(phi1) * t_1) + Math.pow(Math.sin((0.5 * phi1)), 2.0))))));
	} else {
		tmp = R * (2.0 * Math.atan2(t_5, Math.sqrt((t_4 - t_2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	t_1 = math.pow(math.sin((0.5 * (lambda1 - lambda2))), 2.0)
	t_2 = math.cos(phi2) * t_1
	t_3 = (math.cos(phi1) * math.cos(phi2)) * (t_0 * t_0)
	t_4 = math.pow(math.cos((phi2 * 0.5)), 2.0)
	t_5 = math.sqrt((t_3 + math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0)))
	tmp = 0
	if phi2 <= -52000000.0:
		tmp = R * (2.0 * math.atan2(math.sqrt((t_2 + math.pow(math.sin((phi2 * -0.5)), 2.0))), math.sqrt((t_4 - t_3))))
	elif phi2 <= 3.3:
		tmp = R * (2.0 * math.atan2(t_5, math.sqrt((1.0 - ((math.cos(phi1) * t_1) + math.pow(math.sin((0.5 * phi1)), 2.0))))))
	else:
		tmp = R * (2.0 * math.atan2(t_5, math.sqrt((t_4 - t_2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_2 = Float64(cos(phi2) * t_1)
	t_3 = Float64(Float64(cos(phi1) * cos(phi2)) * Float64(t_0 * t_0))
	t_4 = cos(Float64(phi2 * 0.5)) ^ 2.0
	t_5 = sqrt(Float64(t_3 + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0)))
	tmp = 0.0
	if (phi2 <= -52000000.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_2 + (sin(Float64(phi2 * -0.5)) ^ 2.0))), sqrt(Float64(t_4 - t_3)))));
	elseif (phi2 <= 3.3)
		tmp = Float64(R * Float64(2.0 * atan(t_5, sqrt(Float64(1.0 - Float64(Float64(cos(phi1) * t_1) + (sin(Float64(0.5 * phi1)) ^ 2.0)))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(t_5, sqrt(Float64(t_4 - 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 = sin((0.5 * (lambda1 - lambda2))) ^ 2.0;
	t_2 = cos(phi2) * t_1;
	t_3 = (cos(phi1) * cos(phi2)) * (t_0 * t_0);
	t_4 = cos((phi2 * 0.5)) ^ 2.0;
	t_5 = sqrt((t_3 + (sin(((phi1 - phi2) / 2.0)) ^ 2.0)));
	tmp = 0.0;
	if (phi2 <= -52000000.0)
		tmp = R * (2.0 * atan2(sqrt((t_2 + (sin((phi2 * -0.5)) ^ 2.0))), sqrt((t_4 - t_3))));
	elseif (phi2 <= 3.3)
		tmp = R * (2.0 * atan2(t_5, sqrt((1.0 - ((cos(phi1) * t_1) + (sin((0.5 * phi1)) ^ 2.0))))));
	else
		tmp = R * (2.0 * atan2(t_5, sqrt((t_4 - 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[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(t$95$3 + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -52000000.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$2 + N[Power[N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(t$95$4 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 3.3], N[(R * N[(2.0 * N[ArcTan[t$95$5 / N[Sqrt[N[(1.0 - N[(N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[t$95$5 / N[Sqrt[N[(t$95$4 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -5.2e7

   1. Initial program 54.0%

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

      1. associate-*l* 54.0%

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

   3. Simplified 54.0%

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

   5. Taylor expanded in phi1 around 0 54.0%

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

      1. unpow2 54.0%

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

      2. 1-sub-sin 54.2%

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

      3. unpow2 54.2%

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

      4. *-commutative 54.2%

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

      5. metadata-eval 54.2%

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

      6. distribute-rgt-neg-in 54.2%

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

      7. cos-neg 54.2%

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

   7. Simplified 54.2%

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

   8. Taylor expanded in phi1 around 0 55.3%

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

   if -5.2e7 < phi2 < 3.2999999999999998

   1. Initial program 73.1%

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

      1. associate-*l* 73.1%

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

   3. Simplified 73.2%

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

   5. Taylor expanded in phi2 around 0 73.3%

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

   if 3.2999999999999998 < 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. Step-by-step derivation

      1. associate-*l* 50.2%

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

   3. Simplified 50.2%

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

   5. Taylor expanded in phi1 around 0 50.2%

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

      1. unpow2 50.2%

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

      2. 1-sub-sin 50.3%

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

      3. unpow2 50.3%

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

      4. *-commutative 50.3%

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

      5. metadata-eval 50.3%

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

      6. distribute-rgt-neg-in 50.3%

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

      7. cos-neg 50.3%

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

   7. Simplified 50.3%

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

   8. Taylor expanded in phi1 around 0 51.4%

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

      1. *-commutative 51.4%

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

   10. Simplified 51.4%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -52000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot 0.5\right)}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\\ \mathbf{elif}\;\phi_2 \leq 3.3:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - \left(\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot 0.5\right)}^{2} - \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.7% accurate, 1.1× speedup

Math

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

FPCore

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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 = (cos(phi1) * cos(phi2)) * (t_0 * t_0)
    code = r * (2.0d0 * atan2(sqrt((t_1 + (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0))), sqrt(((1.0d0 + ((cos((phi1 - phi2)) / 2.0d0) - 0.5d0)) - 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.cos(phi1) * Math.cos(phi2)) * (t_0 * t_0);
	return R * (2.0 * Math.atan2(Math.sqrt((t_1 + Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0))), Math.sqrt(((1.0 + ((Math.cos((phi1 - phi2)) / 2.0) - 0.5)) - t_1))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	t_1 = (math.cos(phi1) * math.cos(phi2)) * (t_0 * t_0)
	return R * (2.0 * math.atan2(math.sqrt((t_1 + math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0))), math.sqrt(((1.0 + ((math.cos((phi1 - phi2)) / 2.0) - 0.5)) - t_1))))

Julia

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

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = (cos(phi1) * cos(phi2)) * (t_0 * t_0);
	tmp = R * (2.0 * atan2(sqrt((t_1 + (sin(((phi1 - phi2) / 2.0)) ^ 2.0))), sqrt(((1.0 + ((cos((phi1 - phi2)) / 2.0) - 0.5)) - 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[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$1 + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(1.0 + N[(N[(N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 62.9%

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

   1. associate-*l* 62.9%

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

3. Simplified 63.0%

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

   1. unpow2 63.0%

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

   2. sin-mult 63.0%

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

   3. div-inv 63.0%

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

   4. metadata-eval 63.0%

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

   5. div-inv 63.0%

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

   6. metadata-eval 63.0%

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

   7. div-inv 63.0%

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

   8. metadata-eval 63.0%

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

   9. div-inv 63.0%

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

   10. metadata-eval 63.0%

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

6. Applied egg-rr 63.0%

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

   1. div-sub 63.0%

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

   2. +-inverses 63.0%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(\left(\phi_1 - \phi_2\right) \cdot 0.5 + \left(\phi_1 - \phi_2\right) \cdot 0.5\right)}{2}\right)\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. cos-0 63.0%

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

   4. metadata-eval 63.0%

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

   5. distribute-lft-out 63.0%

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

   6. metadata-eval 63.0%

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

   7. *-rgt-identity 63.0%

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

8. Simplified 63.0%

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

9. Final simplification 63.0%

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

Alternative 13: 60.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t\_0 \cdot t\_0\right)\\ \mathbf{if}\;\phi_2 \leq -52000000 \lor \neg \left(\phi_2 \leq 4 \cdot 10^{-14}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot t\_1 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot 0.5\right)}^{2} - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2 + {\sin \left(\frac{\phi_1}{2}\right)}^{2}}}{\sqrt{1 - \left(\cos \phi_1 \cdot t\_1 + {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_2 (* (* (cos phi1) (cos phi2)) (* t_0 t_0))))
   (if (or (<= phi2 -52000000.0) (not (<= phi2 4e-14)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ (* (cos phi2) t_1) (pow (sin (* phi2 -0.5)) 2.0)))
        (sqrt (- (pow (cos (* phi2 0.5)) 2.0) t_2)))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_2 (pow (sin (/ phi1 2.0)) 2.0)))
        (sqrt
         (- 1.0 (+ (* (cos phi1) t_1) (pow (sin (* 0.5 phi1)) 2.0))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_2 = (cos(phi1) * cos(phi2)) * (t_0 * t_0);
	double tmp;
	if ((phi2 <= -52000000.0) || !(phi2 <= 4e-14)) {
		tmp = R * (2.0 * atan2(sqrt(((cos(phi2) * t_1) + pow(sin((phi2 * -0.5)), 2.0))), sqrt((pow(cos((phi2 * 0.5)), 2.0) - t_2))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((t_2 + pow(sin((phi1 / 2.0)), 2.0))), sqrt((1.0 - ((cos(phi1) * t_1) + pow(sin((0.5 * phi1)), 2.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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 = sin(((lambda1 - lambda2) / 2.0d0))
    t_1 = sin((0.5d0 * (lambda1 - lambda2))) ** 2.0d0
    t_2 = (cos(phi1) * cos(phi2)) * (t_0 * t_0)
    if ((phi2 <= (-52000000.0d0)) .or. (.not. (phi2 <= 4d-14))) then
        tmp = r * (2.0d0 * atan2(sqrt(((cos(phi2) * t_1) + (sin((phi2 * (-0.5d0))) ** 2.0d0))), sqrt(((cos((phi2 * 0.5d0)) ** 2.0d0) - t_2))))
    else
        tmp = r * (2.0d0 * atan2(sqrt((t_2 + (sin((phi1 / 2.0d0)) ** 2.0d0))), sqrt((1.0d0 - ((cos(phi1) * t_1) + (sin((0.5d0 * phi1)) ** 2.0d0))))))
    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 = Math.pow(Math.sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_2 = (Math.cos(phi1) * Math.cos(phi2)) * (t_0 * t_0);
	double tmp;
	if ((phi2 <= -52000000.0) || !(phi2 <= 4e-14)) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(((Math.cos(phi2) * t_1) + Math.pow(Math.sin((phi2 * -0.5)), 2.0))), Math.sqrt((Math.pow(Math.cos((phi2 * 0.5)), 2.0) - t_2))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_2 + Math.pow(Math.sin((phi1 / 2.0)), 2.0))), Math.sqrt((1.0 - ((Math.cos(phi1) * t_1) + Math.pow(Math.sin((0.5 * phi1)), 2.0))))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	t_1 = math.pow(math.sin((0.5 * (lambda1 - lambda2))), 2.0)
	t_2 = (math.cos(phi1) * math.cos(phi2)) * (t_0 * t_0)
	tmp = 0
	if (phi2 <= -52000000.0) or not (phi2 <= 4e-14):
		tmp = R * (2.0 * math.atan2(math.sqrt(((math.cos(phi2) * t_1) + math.pow(math.sin((phi2 * -0.5)), 2.0))), math.sqrt((math.pow(math.cos((phi2 * 0.5)), 2.0) - t_2))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt((t_2 + math.pow(math.sin((phi1 / 2.0)), 2.0))), math.sqrt((1.0 - ((math.cos(phi1) * t_1) + math.pow(math.sin((0.5 * phi1)), 2.0))))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_2 = Float64(Float64(cos(phi1) * cos(phi2)) * Float64(t_0 * t_0))
	tmp = 0.0
	if ((phi2 <= -52000000.0) || !(phi2 <= 4e-14))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(cos(phi2) * t_1) + (sin(Float64(phi2 * -0.5)) ^ 2.0))), sqrt(Float64((cos(Float64(phi2 * 0.5)) ^ 2.0) - t_2)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_2 + (sin(Float64(phi1 / 2.0)) ^ 2.0))), sqrt(Float64(1.0 - Float64(Float64(cos(phi1) * t_1) + (sin(Float64(0.5 * phi1)) ^ 2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = sin((0.5 * (lambda1 - lambda2))) ^ 2.0;
	t_2 = (cos(phi1) * cos(phi2)) * (t_0 * t_0);
	tmp = 0.0;
	if ((phi2 <= -52000000.0) || ~((phi2 <= 4e-14)))
		tmp = R * (2.0 * atan2(sqrt(((cos(phi2) * t_1) + (sin((phi2 * -0.5)) ^ 2.0))), sqrt(((cos((phi2 * 0.5)) ^ 2.0) - t_2))));
	else
		tmp = R * (2.0 * atan2(sqrt((t_2 + (sin((phi1 / 2.0)) ^ 2.0))), sqrt((1.0 - ((cos(phi1) * t_1) + (sin((0.5 * phi1)) ^ 2.0))))));
	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[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi2, -52000000.0], N[Not[LessEqual[phi2, 4e-14]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision] + N[Power[N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$2 + N[Power[N[Sin[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -5.2e7 or 4e-14 < phi2

   1. Initial program 53.0%

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

      1. associate-*l* 53.0%

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

   3. Simplified 53.0%

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

   5. Taylor expanded in phi1 around 0 52.8%

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

      1. unpow2 52.8%

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

      2. 1-sub-sin 52.9%

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

      3. unpow2 52.9%

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

      4. *-commutative 52.9%

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

      5. metadata-eval 52.9%

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

      6. distribute-rgt-neg-in 52.9%

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

      7. cos-neg 52.9%

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

   7. Simplified 52.9%

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

   8. Taylor expanded in phi1 around 0 53.7%

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

   if -5.2e7 < phi2 < 4e-14

   1. Initial program 72.9%

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

      1. associate-*l* 72.9%

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

   3. Simplified 73.0%

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

      1. div-sub 73.0%

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

      2. sin-diff 73.2%

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

      3. div-inv 73.2%

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

      4. metadata-eval 73.2%

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

      5. div-inv 73.2%

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

      6. metadata-eval 73.2%

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

      7. div-inv 73.2%

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

      8. metadata-eval 73.2%

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

      9. div-inv 73.2%

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

      10. metadata-eval 73.2%

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

   6. Applied egg-rr 73.2%

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

   7. Taylor expanded in phi1 around inf 72.4%

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

   8. Taylor expanded in phi2 around 0 72.3%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -52000000 \lor \neg \left(\phi_2 \leq 4 \cdot 10^{-14}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot 0.5\right)}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1}{2}\right)}^{2}}}{\sqrt{1 - \left(\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 60.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\cos \left(\phi_2 \cdot 0.5\right)}^{2}\\ t_2 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_3 := \cos \phi_2 \cdot t\_2\\ t_4 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t\_0 \cdot t\_0\right)\\ \mathbf{if}\;\phi_2 \leq -52000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{t\_1 - t\_4}}\right)\\ \mathbf{elif}\;\phi_2 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4 + {\sin \left(\frac{\phi_1}{2}\right)}^{2}}}{\sqrt{1 - \left(\cos \phi_1 \cdot t\_2 + {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4 + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{t\_1 - t\_3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (pow (cos (* phi2 0.5)) 2.0))
        (t_2 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_3 (* (cos phi2) t_2))
        (t_4 (* (* (cos phi1) (cos phi2)) (* t_0 t_0))))
   (if (<= phi2 -52000000.0)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_3 (pow (sin (* phi2 -0.5)) 2.0)))
        (sqrt (- t_1 t_4)))))
     (if (<= phi2 2e-13)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ t_4 (pow (sin (/ phi1 2.0)) 2.0)))
          (sqrt (- 1.0 (+ (* (cos phi1) t_2) (pow (sin (* 0.5 phi1)) 2.0)))))))
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ t_4 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
          (sqrt (- t_1 t_3)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = pow(cos((phi2 * 0.5)), 2.0);
	double t_2 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_3 = cos(phi2) * t_2;
	double t_4 = (cos(phi1) * cos(phi2)) * (t_0 * t_0);
	double tmp;
	if (phi2 <= -52000000.0) {
		tmp = R * (2.0 * atan2(sqrt((t_3 + pow(sin((phi2 * -0.5)), 2.0))), sqrt((t_1 - t_4))));
	} else if (phi2 <= 2e-13) {
		tmp = R * (2.0 * atan2(sqrt((t_4 + pow(sin((phi1 / 2.0)), 2.0))), sqrt((1.0 - ((cos(phi1) * t_2) + pow(sin((0.5 * phi1)), 2.0))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((t_4 + pow(sin(((phi1 - phi2) / 2.0)), 2.0))), sqrt((t_1 - t_3))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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 = sin(((lambda1 - lambda2) / 2.0d0))
    t_1 = cos((phi2 * 0.5d0)) ** 2.0d0
    t_2 = sin((0.5d0 * (lambda1 - lambda2))) ** 2.0d0
    t_3 = cos(phi2) * t_2
    t_4 = (cos(phi1) * cos(phi2)) * (t_0 * t_0)
    if (phi2 <= (-52000000.0d0)) then
        tmp = r * (2.0d0 * atan2(sqrt((t_3 + (sin((phi2 * (-0.5d0))) ** 2.0d0))), sqrt((t_1 - t_4))))
    else if (phi2 <= 2d-13) then
        tmp = r * (2.0d0 * atan2(sqrt((t_4 + (sin((phi1 / 2.0d0)) ** 2.0d0))), sqrt((1.0d0 - ((cos(phi1) * t_2) + (sin((0.5d0 * phi1)) ** 2.0d0))))))
    else
        tmp = r * (2.0d0 * atan2(sqrt((t_4 + (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0))), sqrt((t_1 - 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 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_1 = Math.pow(Math.cos((phi2 * 0.5)), 2.0);
	double t_2 = Math.pow(Math.sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_3 = Math.cos(phi2) * t_2;
	double t_4 = (Math.cos(phi1) * Math.cos(phi2)) * (t_0 * t_0);
	double tmp;
	if (phi2 <= -52000000.0) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_3 + Math.pow(Math.sin((phi2 * -0.5)), 2.0))), Math.sqrt((t_1 - t_4))));
	} else if (phi2 <= 2e-13) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_4 + Math.pow(Math.sin((phi1 / 2.0)), 2.0))), Math.sqrt((1.0 - ((Math.cos(phi1) * t_2) + Math.pow(Math.sin((0.5 * phi1)), 2.0))))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_4 + Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0))), Math.sqrt((t_1 - t_3))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	t_1 = math.pow(math.cos((phi2 * 0.5)), 2.0)
	t_2 = math.pow(math.sin((0.5 * (lambda1 - lambda2))), 2.0)
	t_3 = math.cos(phi2) * t_2
	t_4 = (math.cos(phi1) * math.cos(phi2)) * (t_0 * t_0)
	tmp = 0
	if phi2 <= -52000000.0:
		tmp = R * (2.0 * math.atan2(math.sqrt((t_3 + math.pow(math.sin((phi2 * -0.5)), 2.0))), math.sqrt((t_1 - t_4))))
	elif phi2 <= 2e-13:
		tmp = R * (2.0 * math.atan2(math.sqrt((t_4 + math.pow(math.sin((phi1 / 2.0)), 2.0))), math.sqrt((1.0 - ((math.cos(phi1) * t_2) + math.pow(math.sin((0.5 * phi1)), 2.0))))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt((t_4 + math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0))), math.sqrt((t_1 - t_3))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = cos(Float64(phi2 * 0.5)) ^ 2.0
	t_2 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_3 = Float64(cos(phi2) * t_2)
	t_4 = Float64(Float64(cos(phi1) * cos(phi2)) * Float64(t_0 * t_0))
	tmp = 0.0
	if (phi2 <= -52000000.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_3 + (sin(Float64(phi2 * -0.5)) ^ 2.0))), sqrt(Float64(t_1 - t_4)))));
	elseif (phi2 <= 2e-13)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_4 + (sin(Float64(phi1 / 2.0)) ^ 2.0))), sqrt(Float64(1.0 - Float64(Float64(cos(phi1) * t_2) + (sin(Float64(0.5 * phi1)) ^ 2.0)))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_4 + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))), sqrt(Float64(t_1 - t_3)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = cos((phi2 * 0.5)) ^ 2.0;
	t_2 = sin((0.5 * (lambda1 - lambda2))) ^ 2.0;
	t_3 = cos(phi2) * t_2;
	t_4 = (cos(phi1) * cos(phi2)) * (t_0 * t_0);
	tmp = 0.0;
	if (phi2 <= -52000000.0)
		tmp = R * (2.0 * atan2(sqrt((t_3 + (sin((phi2 * -0.5)) ^ 2.0))), sqrt((t_1 - t_4))));
	elseif (phi2 <= 2e-13)
		tmp = R * (2.0 * atan2(sqrt((t_4 + (sin((phi1 / 2.0)) ^ 2.0))), sqrt((1.0 - ((cos(phi1) * t_2) + (sin((0.5 * phi1)) ^ 2.0))))));
	else
		tmp = R * (2.0 * atan2(sqrt((t_4 + (sin(((phi1 - phi2) / 2.0)) ^ 2.0))), sqrt((t_1 - t_3))));
	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[Power[N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -52000000.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$3 + N[Power[N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(t$95$1 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 2e-13], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$4 + N[Power[N[Sin[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(N[Cos[phi1], $MachinePrecision] * t$95$2), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$4 + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(t$95$1 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -5.2e7

   1. Initial program 54.0%

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

      1. associate-*l* 54.0%

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

   3. Simplified 54.0%

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

   5. Taylor expanded in phi1 around 0 54.0%

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

      1. unpow2 54.0%

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

      2. 1-sub-sin 54.2%

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

      3. unpow2 54.2%

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

      4. *-commutative 54.2%

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

      5. metadata-eval 54.2%

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

      6. distribute-rgt-neg-in 54.2%

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

      7. cos-neg 54.2%

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

   7. Simplified 54.2%

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

   8. Taylor expanded in phi1 around 0 55.3%

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

   if -5.2e7 < phi2 < 2.0000000000000001e-13

   1. Initial program 72.9%

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

      1. associate-*l* 72.9%

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

   3. Simplified 73.0%

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

      1. div-sub 73.0%

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

      2. sin-diff 73.2%

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

      3. div-inv 73.2%

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

      4. metadata-eval 73.2%

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

      5. div-inv 73.2%

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

      6. metadata-eval 73.2%

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

      7. div-inv 73.2%

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

      8. metadata-eval 73.2%

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

      9. div-inv 73.2%

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

      10. metadata-eval 73.2%

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

   6. Applied egg-rr 73.2%

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

   7. Taylor expanded in phi1 around inf 72.4%

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

   8. Taylor expanded in phi2 around 0 72.3%

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

   if 2.0000000000000001e-13 < phi2

   1. Initial program 52.0%

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

      1. associate-*l* 52.0%

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

   3. Simplified 52.0%

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

   5. Taylor expanded in phi1 around 0 51.7%

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

      1. unpow2 51.7%

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

      2. 1-sub-sin 51.7%

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

      3. unpow2 51.7%

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

      4. *-commutative 51.7%

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

      5. metadata-eval 51.7%

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

      6. distribute-rgt-neg-in 51.7%

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

      7. cos-neg 51.7%

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

   7. Simplified 51.7%

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

   8. Taylor expanded in phi1 around 0 53.1%

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

      1. *-commutative 53.1%

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

   10. Simplified 53.1%

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

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -52000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot 0.5\right)}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\\ \mathbf{elif}\;\phi_2 \leq 2 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1}{2}\right)}^{2}}}{\sqrt{1 - \left(\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot 0.5\right)}^{2} - \cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 52.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ t_3 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -3.2 \cdot 10^{-32} \lor \neg \left(\phi_2 \leq 3.3\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {t\_2}^{2} + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot 0.5\right)}^{2} - t\_3 \cdot \left(t\_1 \cdot t\_1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(t\_0, \sqrt{t\_3} \cdot t\_2\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(1.5 + -0.5 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) + -1\right), {t\_0}^{2}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 (- phi1 phi2))))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (sin (* 0.5 (- lambda1 lambda2))))
        (t_3 (* (cos phi1) (cos phi2))))
   (if (or (<= phi2 -3.2e-32) (not (<= phi2 3.3)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ (* (cos phi2) (pow t_2 2.0)) (pow (sin (* phi2 -0.5)) 2.0)))
        (sqrt (- (pow (cos (* phi2 0.5)) 2.0) (* t_3 (* t_1 t_1)))))))
     (*
      (* R 2.0)
      (atan2
       (hypot t_0 (* (sqrt t_3) t_2))
       (sqrt
        (-
         1.0
         (fma
          (cos phi1)
          (* (cos phi2) (+ (+ 1.5 (* -0.5 (cos (- lambda2 lambda1)))) -1.0))
          (pow t_0 2.0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * (phi1 - phi2)));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = sin((0.5 * (lambda1 - lambda2)));
	double t_3 = cos(phi1) * cos(phi2);
	double tmp;
	if ((phi2 <= -3.2e-32) || !(phi2 <= 3.3)) {
		tmp = R * (2.0 * atan2(sqrt(((cos(phi2) * pow(t_2, 2.0)) + pow(sin((phi2 * -0.5)), 2.0))), sqrt((pow(cos((phi2 * 0.5)), 2.0) - (t_3 * (t_1 * t_1))))));
	} else {
		tmp = (R * 2.0) * atan2(hypot(t_0, (sqrt(t_3) * t_2)), sqrt((1.0 - fma(cos(phi1), (cos(phi2) * ((1.5 + (-0.5 * cos((lambda2 - lambda1)))) + -1.0)), pow(t_0, 2.0)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(phi1 - phi2)))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = sin(Float64(0.5 * Float64(lambda1 - lambda2)))
	t_3 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if ((phi2 <= -3.2e-32) || !(phi2 <= 3.3))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(cos(phi2) * (t_2 ^ 2.0)) + (sin(Float64(phi2 * -0.5)) ^ 2.0))), sqrt(Float64((cos(Float64(phi2 * 0.5)) ^ 2.0) - Float64(t_3 * Float64(t_1 * t_1)))))));
	else
		tmp = Float64(Float64(R * 2.0) * atan(hypot(t_0, Float64(sqrt(t_3) * t_2)), sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * Float64(Float64(1.5 + Float64(-0.5 * cos(Float64(lambda2 - lambda1)))) + -1.0)), (t_0 ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi2, -3.2e-32], N[Not[LessEqual[phi2, 3.3]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - N[(t$95$3 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[t$95$0 ^ 2 + N[(N[Sqrt[t$95$3], $MachinePrecision] * t$95$2), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(N[(1.5 + N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -3.2000000000000002e-32 or 3.2999999999999998 < phi2

   1. Initial program 52.1%

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

      1. associate-*l* 52.1%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in phi1 around 0 50.8%

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

      1. unpow2 50.8%

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

      2. 1-sub-sin 50.9%

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

      3. unpow2 50.9%

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

      4. *-commutative 50.9%

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

      5. metadata-eval 50.9%

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

      6. distribute-rgt-neg-in 50.9%

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

      7. cos-neg 50.9%

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

   7. Simplified 50.9%

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

   8. Taylor expanded in phi1 around 0 51.6%

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

   if -3.2000000000000002e-32 < phi2 < 3.2999999999999998

   1. Initial program 74.3%

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

      1. associate-*r* 74.3%

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

      2. *-commutative 74.3%

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

   3. Simplified 74.3%

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

   6. Applied egg-rr 56.5%

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

      1. *-lft-identity 56.5%

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

      2. *-commutative 56.5%

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

      3. *-commutative 56.5%

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

      4. metadata-eval 56.5%

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

      5. cancel-sign-sub-inv 56.5%

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

   8. Simplified 56.5%

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

      1. expm1-log1p-u 56.4%

         \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

      2. cancel-sign-sub-inv 56.4%

         \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{0.5 + \left(-0.5\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

      3. metadata-eval 56.4%

         \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(0.5 + \color{blue}{-0.5} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

   10. Applied egg-rr 56.4%

       \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]
   11. Step-by-step derivation

       1. expm1-undefine 56.5%

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

       2. sub-neg 56.5%

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

       3. log1p-undefine 56.5%

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

       4. rem-exp-log 56.5%

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

       5. associate-+r+ 56.5%

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

       6. metadata-eval 56.5%

          \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(\color{blue}{1.5} + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + \left(-1\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

       7. sub-neg 56.5%

          \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(1.5 + -0.5 \cdot \cos \color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right) + \left(-1\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

       8. remove-double-neg 56.5%

          \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(1.5 + -0.5 \cdot \cos \left(\color{blue}{\left(-\left(-\lambda_1\right)\right)} + \left(-\lambda_2\right)\right)\right) + \left(-1\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

       9. mul-1-neg 56.5%

          \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(1.5 + -0.5 \cdot \cos \left(\left(-\color{blue}{-1 \cdot \lambda_1}\right) + \left(-\lambda_2\right)\right)\right) + \left(-1\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

       10. distribute-neg-in 56.5%

           \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(1.5 + -0.5 \cdot \cos \color{blue}{\left(-\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)}\right) + \left(-1\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

       11. +-commutative 56.5%

           \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(1.5 + -0.5 \cdot \cos \left(-\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) + \left(-1\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

       12. cos-neg 56.5%

           \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(1.5 + -0.5 \cdot \color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}\right) + \left(-1\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

       13. mul-1-neg 56.5%

           \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(1.5 + -0.5 \cdot \cos \left(\lambda_2 + \color{blue}{\left(-\lambda_1\right)}\right)\right) + \left(-1\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

       14. unsub-neg 56.5%

           \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(1.5 + -0.5 \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) + \left(-1\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

       15. metadata-eval 56.5%

           \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(1.5 + -0.5 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) + \color{blue}{-1}\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

   12. Simplified 56.5%

       \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(\left(1.5 + -0.5 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) + -1\right)}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.2 \cdot 10^{-32} \lor \neg \left(\phi_2 \leq 3.3\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot 0.5\right)}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sqrt{\cos \phi_1 \cdot \cos \phi_2} \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(\left(1.5 + -0.5 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) + -1\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 52.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -2.6 \cdot 10^{-34} \lor \neg \left(\phi_2 \leq 3.3\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {t\_1}^{2} + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot 0.5\right)}^{2} - t\_2 \cdot \left(t\_0 \cdot t\_0\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sqrt{t\_2} \cdot t\_1\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (sin (* 0.5 (- lambda1 lambda2))))
        (t_2 (* (cos phi1) (cos phi2))))
   (if (or (<= phi2 -2.6e-34) (not (<= phi2 3.3)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ (* (cos phi2) (pow t_1 2.0)) (pow (sin (* phi2 -0.5)) 2.0)))
        (sqrt (- (pow (cos (* phi2 0.5)) 2.0) (* t_2 (* t_0 t_0)))))))
     (*
      (* R 2.0)
      (atan2
       (hypot (sin (* 0.5 (- phi1 phi2))) (* (sqrt t_2) t_1))
       (sqrt
        (-
         1.0
         (fma
          (cos phi1)
          (* (cos phi2) (- 0.5 (* 0.5 (cos (- lambda1 lambda2)))))
          (pow (sin (* 0.5 phi1)) 2.0)))))))))

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 * (lambda1 - lambda2)));
	double t_2 = cos(phi1) * cos(phi2);
	double tmp;
	if ((phi2 <= -2.6e-34) || !(phi2 <= 3.3)) {
		tmp = R * (2.0 * atan2(sqrt(((cos(phi2) * pow(t_1, 2.0)) + pow(sin((phi2 * -0.5)), 2.0))), sqrt((pow(cos((phi2 * 0.5)), 2.0) - (t_2 * (t_0 * t_0))))));
	} else {
		tmp = (R * 2.0) * atan2(hypot(sin((0.5 * (phi1 - phi2))), (sqrt(t_2) * t_1)), sqrt((1.0 - fma(cos(phi1), (cos(phi2) * (0.5 - (0.5 * cos((lambda1 - lambda2))))), pow(sin((0.5 * phi1)), 2.0)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sin(Float64(0.5 * Float64(lambda1 - lambda2)))
	t_2 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if ((phi2 <= -2.6e-34) || !(phi2 <= 3.3))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(cos(phi2) * (t_1 ^ 2.0)) + (sin(Float64(phi2 * -0.5)) ^ 2.0))), sqrt(Float64((cos(Float64(phi2 * 0.5)) ^ 2.0) - Float64(t_2 * Float64(t_0 * t_0)))))));
	else
		tmp = Float64(Float64(R * 2.0) * atan(hypot(sin(Float64(0.5 * Float64(phi1 - phi2))), Float64(sqrt(t_2) * t_1)), sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2))))), (sin(Float64(0.5 * phi1)) ^ 2.0))))));
	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 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi2, -2.6e-34], N[Not[LessEqual[phi2, 3.3]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - N[(t$95$2 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2 + N[(N[Sqrt[t$95$2], $MachinePrecision] * t$95$1), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.5999999999999999e-34 or 3.2999999999999998 < phi2

   1. Initial program 52.1%

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

      1. associate-*l* 52.1%

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

   3. Simplified 52.1%

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

   5. Taylor expanded in phi1 around 0 50.8%

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

      1. unpow2 50.8%

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

      2. 1-sub-sin 50.9%

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

      3. unpow2 50.9%

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

      4. *-commutative 50.9%

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

      5. metadata-eval 50.9%

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

      6. distribute-rgt-neg-in 50.9%

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

      7. cos-neg 50.9%

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

   7. Simplified 50.9%

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

   8. Taylor expanded in phi1 around 0 51.6%

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

   if -2.5999999999999999e-34 < phi2 < 3.2999999999999998

   1. Initial program 74.3%

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

      1. associate-*r* 74.3%

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

      2. *-commutative 74.3%

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

   3. Simplified 74.3%

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

   6. Applied egg-rr 56.5%

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

      1. *-lft-identity 56.5%

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

      2. *-commutative 56.5%

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

      3. *-commutative 56.5%

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

      4. metadata-eval 56.5%

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

      5. cancel-sign-sub-inv 56.5%

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

   8. Simplified 56.5%

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

   9. Taylor expanded in phi2 around 0 56.5%

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

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.6 \cdot 10^{-34} \lor \neg \left(\phi_2 \leq 3.3\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot 0.5\right)}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sqrt{\cos \phi_1 \cdot \cos \phi_2} \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 47.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\phi_2 \cdot -0.5\right)}^{2}\\ t_1 := \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ t_2 := \sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\\ \mathbf{if}\;\phi_2 \leq -1.02 \cdot 10^{+52} \lor \neg \left(\phi_2 \leq 52000\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0 + \cos \phi_2 \cdot \left(t\_1 \cdot \left|t\_1\right|\right)}}{\sqrt{1 - \left(\cos \phi_2 \cdot {t\_1}^{2} + t\_0\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(t\_2, \sqrt{\cos \phi_1 \cdot \cos \phi_2} \cdot t\_1\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {t\_2}^{2}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* phi2 -0.5)) 2.0))
        (t_1 (sin (* 0.5 (- lambda1 lambda2))))
        (t_2 (sin (* 0.5 (- phi1 phi2)))))
   (if (or (<= phi2 -1.02e+52) (not (<= phi2 52000.0)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_0 (* (cos phi2) (* t_1 (fabs t_1)))))
        (sqrt (- 1.0 (+ (* (cos phi2) (pow t_1 2.0)) t_0))))))
     (*
      (* R 2.0)
      (atan2
       (hypot t_2 (* (sqrt (* (cos phi1) (cos phi2))) t_1))
       (sqrt
        (-
         1.0
         (fma
          (cos phi1)
          (* (cos phi2) (- 0.5 (* 0.5 (cos (- lambda1 lambda2)))))
          (pow t_2 2.0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((phi2 * -0.5)), 2.0);
	double t_1 = sin((0.5 * (lambda1 - lambda2)));
	double t_2 = sin((0.5 * (phi1 - phi2)));
	double tmp;
	if ((phi2 <= -1.02e+52) || !(phi2 <= 52000.0)) {
		tmp = R * (2.0 * atan2(sqrt((t_0 + (cos(phi2) * (t_1 * fabs(t_1))))), sqrt((1.0 - ((cos(phi2) * pow(t_1, 2.0)) + t_0)))));
	} else {
		tmp = (R * 2.0) * atan2(hypot(t_2, (sqrt((cos(phi1) * cos(phi2))) * t_1)), sqrt((1.0 - fma(cos(phi1), (cos(phi2) * (0.5 - (0.5 * cos((lambda1 - lambda2))))), pow(t_2, 2.0)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(phi2 * -0.5)) ^ 2.0
	t_1 = sin(Float64(0.5 * Float64(lambda1 - lambda2)))
	t_2 = sin(Float64(0.5 * Float64(phi1 - phi2)))
	tmp = 0.0
	if ((phi2 <= -1.02e+52) || !(phi2 <= 52000.0))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_0 + Float64(cos(phi2) * Float64(t_1 * abs(t_1))))), sqrt(Float64(1.0 - Float64(Float64(cos(phi2) * (t_1 ^ 2.0)) + t_0))))));
	else
		tmp = Float64(Float64(R * 2.0) * atan(hypot(t_2, Float64(sqrt(Float64(cos(phi1) * cos(phi2))) * t_1)), sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2))))), (t_2 ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[phi2, -1.02e+52], N[Not[LessEqual[phi2, 52000.0]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 + N[(N[Cos[phi2], $MachinePrecision] * N[(t$95$1 * N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[t$95$2 ^ 2 + N[(N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.02000000000000002e52 or 52000 < phi2

   1. Initial program 51.3%

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

      1. add-sqr-sqrt 25.0%

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

      2. sqrt-prod 44.2%

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

      3. rem-sqrt-square 44.2%

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

      4. div-inv 44.2%

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

      5. metadata-eval 44.2%

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

   4. Applied egg-rr 44.2%

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

   5. Taylor expanded in phi1 around 0 44.3%

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

   6. Taylor expanded in phi1 around 0 44.1%

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

   7. Taylor expanded in phi1 around 0 44.6%

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

   if -1.02000000000000002e52 < phi2 < 52000

   1. Initial program 72.5%

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

      1. associate-*r* 72.5%

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

      2. *-commutative 72.5%

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

   3. Simplified 72.5%

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

   6. Applied egg-rr 54.4%

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

      1. *-lft-identity 54.4%

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

      2. *-commutative 54.4%

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

      3. *-commutative 54.4%

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

      4. metadata-eval 54.4%

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

      5. cancel-sign-sub-inv 54.4%

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

   8. Simplified 54.4%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.02 \cdot 10^{+52} \lor \neg \left(\phi_2 \leq 52000\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_2 \cdot -0.5\right)}^{2} + \cos \phi_2 \cdot \left(\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left|\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right|\right)}}{\sqrt{1 - \left(\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sqrt{\cos \phi_1 \cdot \cos \phi_2} \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 61.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    t_1 = sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0
    t_2 = cos(phi1) * cos(phi2)
    code = r * (2.0d0 * atan2(sqrt(((t_2 * (t_0 * t_0)) + t_1)), sqrt(((1.0d0 - t_1) + (t_2 * ((cos((lambda1 - lambda2)) / 2.0d0) - 0.5d0))))))
end function

Java

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

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)
	t_2 = math.cos(phi1) * math.cos(phi2)
	return R * (2.0 * math.atan2(math.sqrt(((t_2 * (t_0 * t_0)) + t_1)), math.sqrt(((1.0 - t_1) + (t_2 * ((math.cos((lambda1 - lambda2)) / 2.0) - 0.5))))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_2 = Float64(cos(phi1) * cos(phi2))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(t_2 * Float64(t_0 * t_0)) + t_1)), sqrt(Float64(Float64(1.0 - t_1) + Float64(t_2 * Float64(Float64(cos(Float64(lambda1 - lambda2)) / 2.0) - 0.5)))))))
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;
	t_2 = cos(phi1) * cos(phi2);
	tmp = R * (2.0 * atan2(sqrt(((t_2 * (t_0 * t_0)) + t_1)), sqrt(((1.0 - t_1) + (t_2 * ((cos((lambda1 - lambda2)) / 2.0) - 0.5))))));
end

Wolfram

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

Derivation

1. Initial program 62.9%

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

   1. associate-*l* 62.9%

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

3. Simplified 63.0%

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

   1. sin-mult 74.6%

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

   2. cos-sum 74.5%

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

   3. cos-2 74.6%

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

   4. div-sub 74.6%

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

   5. +-inverses 74.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \sin \left(\phi_1 \cdot 0.5\right), \sin \left(-0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(2 \cdot \frac{\lambda_1 - \lambda_2}{2}\right)}{2}\right)}}{\sqrt{\left(1 - {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

6. Applied egg-rr 63.0%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\frac{\cos 0}{2} - \frac{\cos \left(\lambda_1 - \lambda_2\right)}{2}\right)}}}\right) \]
7. Step-by-step derivation

   1. cos-0 74.6%

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

   2. metadata-eval 74.6%

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

8. Simplified 63.0%

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

9. Final simplification 63.0%

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

Alternative 19: 40.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\\ t_1 := \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ t_2 := {t\_0}^{2}\\ \mathbf{if}\;\lambda_2 \leq -3.4 \cdot 10^{-25} \lor \neg \left(\lambda_2 \leq 2.75 \cdot 10^{-8}\right):\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \phi_1\right), t\_1 \cdot \sqrt{\cos \phi_1}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), t\_2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(t\_0, \sqrt{\cos \phi_1 \cdot \cos \phi_2} \cdot t\_1\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \lambda_1\right), t\_2\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 (- phi1 phi2))))
        (t_1 (sin (* 0.5 (- lambda1 lambda2))))
        (t_2 (pow t_0 2.0)))
   (if (or (<= lambda2 -3.4e-25) (not (<= lambda2 2.75e-8)))
     (*
      (* R 2.0)
      (atan2
       (hypot (sin (* 0.5 phi1)) (* t_1 (sqrt (cos phi1))))
       (sqrt
        (-
         1.0
         (fma
          (cos phi1)
          (* (cos phi2) (- 0.5 (* 0.5 (cos (- lambda1 lambda2)))))
          t_2)))))
     (*
      (* R 2.0)
      (atan2
       (hypot t_0 (* (sqrt (* (cos phi1) (cos phi2))) t_1))
       (sqrt
        (-
         1.0
         (fma
          (cos phi1)
          (* (cos phi2) (- 0.5 (* 0.5 (cos lambda1))))
          t_2))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * (phi1 - phi2)));
	double t_1 = sin((0.5 * (lambda1 - lambda2)));
	double t_2 = pow(t_0, 2.0);
	double tmp;
	if ((lambda2 <= -3.4e-25) || !(lambda2 <= 2.75e-8)) {
		tmp = (R * 2.0) * atan2(hypot(sin((0.5 * phi1)), (t_1 * sqrt(cos(phi1)))), sqrt((1.0 - fma(cos(phi1), (cos(phi2) * (0.5 - (0.5 * cos((lambda1 - lambda2))))), t_2))));
	} else {
		tmp = (R * 2.0) * atan2(hypot(t_0, (sqrt((cos(phi1) * cos(phi2))) * t_1)), sqrt((1.0 - fma(cos(phi1), (cos(phi2) * (0.5 - (0.5 * cos(lambda1)))), t_2))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(phi1 - phi2)))
	t_1 = sin(Float64(0.5 * Float64(lambda1 - lambda2)))
	t_2 = t_0 ^ 2.0
	tmp = 0.0
	if ((lambda2 <= -3.4e-25) || !(lambda2 <= 2.75e-8))
		tmp = Float64(Float64(R * 2.0) * atan(hypot(sin(Float64(0.5 * phi1)), Float64(t_1 * sqrt(cos(phi1)))), sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2))))), t_2)))));
	else
		tmp = Float64(Float64(R * 2.0) * atan(hypot(t_0, Float64(sqrt(Float64(cos(phi1) * cos(phi2))) * t_1)), sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(lambda1)))), t_2)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$0, 2.0], $MachinePrecision]}, If[Or[LessEqual[lambda2, -3.4e-25], N[Not[LessEqual[lambda2, 2.75e-8]], $MachinePrecision]], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] ^ 2 + N[(t$95$1 * N[Sqrt[N[Cos[phi1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[t$95$0 ^ 2 + N[(N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -3.40000000000000002e-25 or 2.7500000000000001e-8 < lambda2

   1. Initial program 49.4%

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

      1. associate-*r* 49.4%

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

      2. *-commutative 49.4%

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

   3. Simplified 49.4%

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

   6. Applied egg-rr 32.6%

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

      1. *-lft-identity 32.6%

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

      2. *-commutative 32.6%

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

      3. *-commutative 32.6%

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

      4. metadata-eval 32.6%

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

      5. cancel-sign-sub-inv 32.6%

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

   8. Simplified 32.6%

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

   9. Taylor expanded in phi2 around 0 27.5%

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

   10. Taylor expanded in phi2 around 0 30.9%

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

   if -3.40000000000000002e-25 < lambda2 < 2.7500000000000001e-8

   1. Initial program 79.3%

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

      1. associate-*r* 79.3%

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

      2. *-commutative 79.3%

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

   3. Simplified 79.3%

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

   6. Applied egg-rr 52.0%

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

      1. *-lft-identity 52.0%

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

      2. *-commutative 52.0%

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

      3. *-commutative 52.0%

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

      4. metadata-eval 52.0%

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

      5. cancel-sign-sub-inv 52.0%

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

   8. Simplified 52.0%

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

   9. Taylor expanded in lambda2 around 0 52.0%

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

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -3.4 \cdot 10^{-25} \lor \neg \left(\lambda_2 \leq 2.75 \cdot 10^{-8}\right):\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sqrt{\cos \phi_1 \cdot \cos \phi_2} \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \lambda_1\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 39.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\\ t_1 := \sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {t\_0}^{2}\right)}\\ \mathbf{if}\;\lambda_2 \leq -3.4 \cdot 10^{-25} \lor \neg \left(\lambda_2 \leq 3 \cdot 10^{-15}\right):\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1}\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(t\_0, \sqrt{\cos \phi_1 \cdot \cos \phi_2} \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 (- phi1 phi2))))
        (t_1
         (sqrt
          (-
           1.0
           (fma
            (cos phi1)
            (* (cos phi2) (- 0.5 (* 0.5 (cos (- lambda1 lambda2)))))
            (pow t_0 2.0))))))
   (if (or (<= lambda2 -3.4e-25) (not (<= lambda2 3e-15)))
     (*
      (* R 2.0)
      (atan2
       (hypot
        (sin (* 0.5 phi1))
        (* (sin (* 0.5 (- lambda1 lambda2))) (sqrt (cos phi1))))
       t_1))
     (*
      (* R 2.0)
      (atan2
       (hypot t_0 (* (sqrt (* (cos phi1) (cos phi2))) (sin (* 0.5 lambda1))))
       t_1)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * (phi1 - phi2)));
	double t_1 = sqrt((1.0 - fma(cos(phi1), (cos(phi2) * (0.5 - (0.5 * cos((lambda1 - lambda2))))), pow(t_0, 2.0))));
	double tmp;
	if ((lambda2 <= -3.4e-25) || !(lambda2 <= 3e-15)) {
		tmp = (R * 2.0) * atan2(hypot(sin((0.5 * phi1)), (sin((0.5 * (lambda1 - lambda2))) * sqrt(cos(phi1)))), t_1);
	} else {
		tmp = (R * 2.0) * atan2(hypot(t_0, (sqrt((cos(phi1) * cos(phi2))) * sin((0.5 * lambda1)))), t_1);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(phi1 - phi2)))
	t_1 = sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2))))), (t_0 ^ 2.0))))
	tmp = 0.0
	if ((lambda2 <= -3.4e-25) || !(lambda2 <= 3e-15))
		tmp = Float64(Float64(R * 2.0) * atan(hypot(sin(Float64(0.5 * phi1)), Float64(sin(Float64(0.5 * Float64(lambda1 - lambda2))) * sqrt(cos(phi1)))), t_1));
	else
		tmp = Float64(Float64(R * 2.0) * atan(hypot(t_0, Float64(sqrt(Float64(cos(phi1) * cos(phi2))) * sin(Float64(0.5 * lambda1)))), t_1));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[lambda2, -3.4e-25], N[Not[LessEqual[lambda2, 3e-15]], $MachinePrecision]], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] ^ 2 + N[(N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Cos[phi1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / t$95$1], $MachinePrecision]), $MachinePrecision], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[t$95$0 ^ 2 + N[(N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / t$95$1], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -3.40000000000000002e-25 or 3e-15 < lambda2

   1. Initial program 49.4%

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

      1. associate-*r* 49.4%

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

      2. *-commutative 49.4%

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

   3. Simplified 49.4%

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

   6. Applied egg-rr 32.6%

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

      1. *-lft-identity 32.6%

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

      2. *-commutative 32.6%

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

      3. *-commutative 32.6%

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

      4. metadata-eval 32.6%

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

      5. cancel-sign-sub-inv 32.6%

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

   8. Simplified 32.6%

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

   9. Taylor expanded in phi2 around 0 27.5%

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

   10. Taylor expanded in phi2 around 0 30.9%

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

   if -3.40000000000000002e-25 < lambda2 < 3e-15

   1. Initial program 79.3%

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

      1. associate-*r* 79.3%

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

      2. *-commutative 79.3%

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

   3. Simplified 79.3%

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

   6. Applied egg-rr 52.0%

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

      1. *-lft-identity 52.0%

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

      2. *-commutative 52.0%

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

      3. *-commutative 52.0%

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

      4. metadata-eval 52.0%

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

      5. cancel-sign-sub-inv 52.0%

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

   8. Simplified 52.0%

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

   9. Taylor expanded in lambda2 around 0 51.2%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -3.4 \cdot 10^{-25} \lor \neg \left(\lambda_2 \leq 3 \cdot 10^{-15}\right):\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sqrt{\cos \phi_1 \cdot \cos \phi_2} \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 40.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ t_2 := \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ t_3 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_4 := \sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\\ \mathbf{if}\;\phi_1 \leq -6.5 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0 \cdot \left(t\_3 \cdot t\_3\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - \cos \phi_1 \cdot {t\_1}^{2}}}\right)\\ \mathbf{elif}\;\phi_1 \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(t\_4, \sqrt{t\_0} \cdot t\_1\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, t\_2, {\sin \left(\phi_2 \cdot -0.5\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \phi_1\right), t\_1 \cdot \sqrt{\cos \phi_1}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, t\_2, {t\_4}^{2}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2)))
        (t_1 (sin (* 0.5 (- lambda1 lambda2))))
        (t_2 (* (cos phi2) (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))))
        (t_3 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_4 (sin (* 0.5 (- phi1 phi2)))))
   (if (<= phi1 -6.5e-6)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ (* t_0 (* t_3 t_3)) (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
        (sqrt (- 1.0 (* (cos phi1) (pow t_1 2.0)))))))
     (if (<= phi1 4e-5)
       (*
        (* R 2.0)
        (atan2
         (hypot t_4 (* (sqrt t_0) t_1))
         (sqrt (- 1.0 (fma (cos phi1) t_2 (pow (sin (* phi2 -0.5)) 2.0))))))
       (*
        (* R 2.0)
        (atan2
         (hypot (sin (* 0.5 phi1)) (* t_1 (sqrt (cos phi1))))
         (sqrt (- 1.0 (fma (cos phi1) t_2 (pow t_4 2.0))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * cos(phi2);
	double t_1 = sin((0.5 * (lambda1 - lambda2)));
	double t_2 = cos(phi2) * (0.5 - (0.5 * cos((lambda1 - lambda2))));
	double t_3 = sin(((lambda1 - lambda2) / 2.0));
	double t_4 = sin((0.5 * (phi1 - phi2)));
	double tmp;
	if (phi1 <= -6.5e-6) {
		tmp = R * (2.0 * atan2(sqrt(((t_0 * (t_3 * t_3)) + pow(sin(((phi1 - phi2) / 2.0)), 2.0))), sqrt((1.0 - (cos(phi1) * pow(t_1, 2.0))))));
	} else if (phi1 <= 4e-5) {
		tmp = (R * 2.0) * atan2(hypot(t_4, (sqrt(t_0) * t_1)), sqrt((1.0 - fma(cos(phi1), t_2, pow(sin((phi2 * -0.5)), 2.0)))));
	} else {
		tmp = (R * 2.0) * atan2(hypot(sin((0.5 * phi1)), (t_1 * sqrt(cos(phi1)))), sqrt((1.0 - fma(cos(phi1), t_2, pow(t_4, 2.0)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	t_1 = sin(Float64(0.5 * Float64(lambda1 - lambda2)))
	t_2 = Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))))
	t_3 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_4 = sin(Float64(0.5 * Float64(phi1 - phi2)))
	tmp = 0.0
	if (phi1 <= -6.5e-6)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(t_0 * Float64(t_3 * t_3)) + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))), sqrt(Float64(1.0 - Float64(cos(phi1) * (t_1 ^ 2.0)))))));
	elseif (phi1 <= 4e-5)
		tmp = Float64(Float64(R * 2.0) * atan(hypot(t_4, Float64(sqrt(t_0) * t_1)), sqrt(Float64(1.0 - fma(cos(phi1), t_2, (sin(Float64(phi2 * -0.5)) ^ 2.0))))));
	else
		tmp = Float64(Float64(R * 2.0) * atan(hypot(sin(Float64(0.5 * phi1)), Float64(t_1 * sqrt(cos(phi1)))), sqrt(Float64(1.0 - fma(cos(phi1), t_2, (t_4 ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -6.5e-6], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(t$95$0 * N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 4e-5], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[t$95$4 ^ 2 + N[(N[Sqrt[t$95$0], $MachinePrecision] * t$95$1), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * t$95$2 + N[Power[N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] ^ 2 + N[(t$95$1 * N[Sqrt[N[Cos[phi1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * t$95$2 + N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -6.4999999999999996e-6

   1. Initial program 46.5%

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

      1. associate-*l* 46.5%

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

   3. Simplified 46.5%

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

   5. Taylor expanded in phi1 around 0 19.1%

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

      1. unpow2 19.1%

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

      2. 1-sub-sin 19.1%

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

      3. unpow2 19.1%

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

      4. *-commutative 19.1%

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

      5. metadata-eval 19.1%

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

      6. distribute-rgt-neg-in 19.1%

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

      7. cos-neg 19.1%

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

   7. Simplified 19.1%

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

   8. Taylor expanded in phi2 around 0 20.3%

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

   if -6.4999999999999996e-6 < phi1 < 4.00000000000000033e-5

   1. Initial program 75.1%

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

      1. associate-*r* 75.1%

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

      2. *-commutative 75.1%

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

   3. Simplified 75.1%

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

   6. Applied egg-rr 56.8%

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

      1. *-lft-identity 56.8%

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

      2. *-commutative 56.8%

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

      3. *-commutative 56.8%

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

      4. metadata-eval 56.8%

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

      5. cancel-sign-sub-inv 56.8%

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

   8. Simplified 56.8%

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

   9. Taylor expanded in phi1 around 0 56.6%

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

       1. *-commutative 56.6%

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

   11. Simplified 56.6%

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

   if 4.00000000000000033e-5 < phi1

   1. Initial program 50.9%

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

      1. associate-*r* 50.9%

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

      2. *-commutative 50.9%

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

   3. Simplified 50.9%

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

   6. Applied egg-rr 27.8%

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

      1. *-lft-identity 27.8%

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

      2. *-commutative 27.8%

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

      3. *-commutative 27.8%

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

      4. metadata-eval 27.8%

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

      5. cancel-sign-sub-inv 27.8%

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

   8. Simplified 27.8%

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

   9. Taylor expanded in phi2 around 0 27.8%

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

   10. Taylor expanded in phi2 around 0 28.9%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6.5 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}\right)\\ \mathbf{elif}\;\phi_1 \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sqrt{\cos \phi_1 \cdot \cos \phi_2} \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\phi_2 \cdot -0.5\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 40.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\\ t_1 := \sqrt{\cos \phi_1 \cdot \cos \phi_2}\\ t_2 := \mathsf{hypot}\left(t\_0, t\_1 \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\\ t_3 := {t\_0}^{2}\\ \mathbf{if}\;\lambda_1 \leq -8.4 \cdot 10^{+55}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(t\_0, t\_1 \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), t\_3\right)}}\\ \mathbf{elif}\;\lambda_1 \leq 1.05 \cdot 10^{-6}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{t\_2}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \lambda_2\right), t\_3\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{t\_2}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \lambda_1\right), t\_3\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 (- phi1 phi2))))
        (t_1 (sqrt (* (cos phi1) (cos phi2))))
        (t_2 (hypot t_0 (* t_1 (sin (* 0.5 (- lambda1 lambda2))))))
        (t_3 (pow t_0 2.0)))
   (if (<= lambda1 -8.4e+55)
     (*
      (* R 2.0)
      (atan2
       (hypot t_0 (* t_1 (sin (* 0.5 lambda1))))
       (sqrt
        (-
         1.0
         (fma
          (cos phi1)
          (* (cos phi2) (- 0.5 (* 0.5 (cos (- lambda1 lambda2)))))
          t_3)))))
     (if (<= lambda1 1.05e-6)
       (*
        (* R 2.0)
        (atan2
         t_2
         (sqrt
          (-
           1.0
           (fma
            (cos phi1)
            (* (cos phi2) (- 0.5 (* 0.5 (cos lambda2))))
            t_3)))))
       (*
        (* R 2.0)
        (atan2
         t_2
         (sqrt
          (-
           1.0
           (fma
            (cos phi1)
            (* (cos phi2) (- 0.5 (* 0.5 (cos lambda1))))
            t_3)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * (phi1 - phi2)));
	double t_1 = sqrt((cos(phi1) * cos(phi2)));
	double t_2 = hypot(t_0, (t_1 * sin((0.5 * (lambda1 - lambda2)))));
	double t_3 = pow(t_0, 2.0);
	double tmp;
	if (lambda1 <= -8.4e+55) {
		tmp = (R * 2.0) * atan2(hypot(t_0, (t_1 * sin((0.5 * lambda1)))), sqrt((1.0 - fma(cos(phi1), (cos(phi2) * (0.5 - (0.5 * cos((lambda1 - lambda2))))), t_3))));
	} else if (lambda1 <= 1.05e-6) {
		tmp = (R * 2.0) * atan2(t_2, sqrt((1.0 - fma(cos(phi1), (cos(phi2) * (0.5 - (0.5 * cos(lambda2)))), t_3))));
	} else {
		tmp = (R * 2.0) * atan2(t_2, sqrt((1.0 - fma(cos(phi1), (cos(phi2) * (0.5 - (0.5 * cos(lambda1)))), t_3))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(phi1 - phi2)))
	t_1 = sqrt(Float64(cos(phi1) * cos(phi2)))
	t_2 = hypot(t_0, Float64(t_1 * sin(Float64(0.5 * Float64(lambda1 - lambda2)))))
	t_3 = t_0 ^ 2.0
	tmp = 0.0
	if (lambda1 <= -8.4e+55)
		tmp = Float64(Float64(R * 2.0) * atan(hypot(t_0, Float64(t_1 * sin(Float64(0.5 * lambda1)))), sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2))))), t_3)))));
	elseif (lambda1 <= 1.05e-6)
		tmp = Float64(Float64(R * 2.0) * atan(t_2, sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(lambda2)))), t_3)))));
	else
		tmp = Float64(Float64(R * 2.0) * atan(t_2, sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(lambda1)))), t_3)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[t$95$0 ^ 2 + N[(t$95$1 * N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$0, 2.0], $MachinePrecision]}, If[LessEqual[lambda1, -8.4e+55], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[t$95$0 ^ 2 + N[(t$95$1 * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, 1.05e-6], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[t$95$2 / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[t$95$2 / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -8.4000000000000002e55

   1. Initial program 48.7%

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

      1. associate-*r* 48.7%

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

      2. *-commutative 48.7%

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

   3. Simplified 48.8%

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

   6. Applied egg-rr 35.9%

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

      1. *-lft-identity 35.9%

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

      2. *-commutative 35.9%

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

      3. *-commutative 35.9%

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

      4. metadata-eval 35.9%

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

      5. cancel-sign-sub-inv 35.9%

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

   8. Simplified 35.9%

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

   9. Taylor expanded in lambda2 around 0 36.1%

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

   if -8.4000000000000002e55 < lambda1 < 1.0499999999999999e-6

   1. Initial program 76.6%

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

      1. associate-*r* 76.6%

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

      2. *-commutative 76.6%

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

   3. Simplified 76.6%

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

   6. Applied egg-rr 47.3%

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

      1. *-lft-identity 47.3%

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

      2. *-commutative 47.3%

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

      3. *-commutative 47.3%

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

      4. metadata-eval 47.3%

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

      5. cancel-sign-sub-inv 47.3%

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

   8. Simplified 47.3%

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

   9. Taylor expanded in lambda1 around 0 47.3%

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

       1. cos-neg 47.3%

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

   11. Simplified 47.3%

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

   if 1.0499999999999999e-6 < lambda1

   1. Initial program 43.9%

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

      1. associate-*r* 43.9%

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

      2. *-commutative 43.9%

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

   3. Simplified 43.9%

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

   6. Applied egg-rr 32.7%

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

      1. *-lft-identity 32.7%

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

      2. *-commutative 32.7%

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

      3. *-commutative 32.7%

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

      4. metadata-eval 32.7%

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

      5. cancel-sign-sub-inv 32.7%

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

   8. Simplified 32.7%

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

   9. Taylor expanded in lambda2 around 0 32.8%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -8.4 \cdot 10^{+55}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sqrt{\cos \phi_1 \cdot \cos \phi_2} \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\\ \mathbf{elif}\;\lambda_1 \leq 1.05 \cdot 10^{-6}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sqrt{\cos \phi_1 \cdot \cos \phi_2} \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \lambda_2\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sqrt{\cos \phi_1 \cdot \cos \phi_2} \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \lambda_1\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 41.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * (phi1 - phi2)));
	return (R * 2.0) * atan2(hypot(t_0, (sqrt((cos(phi1) * cos(phi2))) * sin((0.5 * (lambda1 - lambda2))))), sqrt((1.0 - fma(cos(phi1), (cos(phi2) * (0.5 - (0.5 * cos((lambda1 - lambda2))))), pow(t_0, 2.0)))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(phi1 - phi2)))
	return Float64(Float64(R * 2.0) * atan(hypot(t_0, Float64(sqrt(Float64(cos(phi1) * cos(phi2))) * sin(Float64(0.5 * Float64(lambda1 - lambda2))))), sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2))))), (t_0 ^ 2.0))))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[t$95$0 ^ 2 + N[(N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.9%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Step-by-step derivation

   1. associate-*r* 62.9%

      \[\leadsto \color{blue}{\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}} \]

   2. *-commutative 62.9%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}} \cdot \left(R \cdot 2\right)} \]

3. Simplified 63.0%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}} \cdot \left(R \cdot 2\right)} \]
4. Add Preprocessing

6. Applied egg-rr 41.4%

   \[\leadsto \color{blue}{\left(1 \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right), \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}}\right)} \cdot \left(R \cdot 2\right) \]
7. Step-by-step derivation

   1. *-lft-identity 41.4%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right), \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}}} \cdot \left(R \cdot 2\right) \]

   2. *-commutative 41.4%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \color{blue}{\left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}, \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

   3. *-commutative 41.4%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \color{blue}{\left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

   4. metadata-eval 41.4%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + \color{blue}{\left(-0.5\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

   5. cancel-sign-sub-inv 41.4%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

8. Simplified 41.4%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}} \cdot \left(R \cdot 2\right) \]

9. Final simplification 41.4%

   \[\leadsto \left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sqrt{\cos \phi_1 \cdot \cos \phi_2} \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \]
10. Add Preprocessing

Alternative 24: 37.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\phi_2 \leq -8.5 \cdot 10^{-36} \lor \neg \left(\phi_2 \leq 1.4 \cdot 10^{-15}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t\_1 \cdot t\_1\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - \cos \phi_1 \cdot {t\_0}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \phi_1\right), t\_0 \cdot \sqrt{\cos \phi_1}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 (- lambda1 lambda2))))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0))))
   (if (or (<= phi2 -8.5e-36) (not (<= phi2 1.4e-15)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (* (* (cos phi1) (cos phi2)) (* t_1 t_1))
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
        (sqrt (- 1.0 (* (cos phi1) (pow t_0 2.0)))))))
     (*
      (* R 2.0)
      (atan2
       (hypot (sin (* 0.5 phi1)) (* t_0 (sqrt (cos phi1))))
       (sqrt
        (-
         1.0
         (fma
          (cos phi1)
          (* (cos phi2) (- 0.5 (* 0.5 (cos (- lambda1 lambda2)))))
          (pow (sin (* 0.5 (- phi1 phi2))) 2.0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * (lambda1 - lambda2)));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if ((phi2 <= -8.5e-36) || !(phi2 <= 1.4e-15)) {
		tmp = R * (2.0 * atan2(sqrt((((cos(phi1) * cos(phi2)) * (t_1 * t_1)) + pow(sin(((phi1 - phi2) / 2.0)), 2.0))), sqrt((1.0 - (cos(phi1) * pow(t_0, 2.0))))));
	} else {
		tmp = (R * 2.0) * atan2(hypot(sin((0.5 * phi1)), (t_0 * sqrt(cos(phi1)))), sqrt((1.0 - fma(cos(phi1), (cos(phi2) * (0.5 - (0.5 * cos((lambda1 - lambda2))))), pow(sin((0.5 * (phi1 - phi2))), 2.0)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2)))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	tmp = 0.0
	if ((phi2 <= -8.5e-36) || !(phi2 <= 1.4e-15))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(Float64(cos(phi1) * cos(phi2)) * Float64(t_1 * t_1)) + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))), sqrt(Float64(1.0 - Float64(cos(phi1) * (t_0 ^ 2.0)))))));
	else
		tmp = Float64(Float64(R * 2.0) * atan(hypot(sin(Float64(0.5 * phi1)), Float64(t_0 * sqrt(cos(phi1)))), sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2))))), (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[phi2, -8.5e-36], N[Not[LessEqual[phi2, 1.4e-15]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] ^ 2 + N[(t$95$0 * N[Sqrt[N[Cos[phi1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -8.5000000000000007e-36 or 1.40000000000000007e-15 < phi2

   1. Initial program 53.0%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. associate-*l* 52.9%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Simplified 52.9%

      \[\leadsto \color{blue}{R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around 0 51.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\color{blue}{\left(1 - {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. unpow2 51.5%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - \color{blue}{\sin \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. 1-sub-sin 51.6%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\color{blue}{\cos \left(-0.5 \cdot \phi_2\right) \cdot \cos \left(-0.5 \cdot \phi_2\right)} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. unpow2 51.6%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\color{blue}{{\cos \left(-0.5 \cdot \phi_2\right)}^{2}} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. *-commutative 51.6%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{{\cos \color{blue}{\left(\phi_2 \cdot -0.5\right)}}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. metadata-eval 51.6%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{{\cos \left(\phi_2 \cdot \color{blue}{\left(-0.5\right)}\right)}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. distribute-rgt-neg-in 51.6%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{{\cos \color{blue}{\left(-\phi_2 \cdot 0.5\right)}}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. cos-neg 51.6%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{{\color{blue}{\cos \left(\phi_2 \cdot 0.5\right)}}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Simplified 51.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\color{blue}{{\cos \left(\phi_2 \cdot 0.5\right)}^{2}} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. Taylor expanded in phi2 around 0 22.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\color{blue}{1 - \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}\right) \]

   if -8.5000000000000007e-36 < phi2 < 1.40000000000000007e-15

   1. Initial program 74.1%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. associate-*r* 74.1%

         \[\leadsto \color{blue}{\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}} \]

      2. *-commutative 74.1%

         \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}} \cdot \left(R \cdot 2\right)} \]

   3. Simplified 74.1%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}} \cdot \left(R \cdot 2\right)} \]
   4. Add Preprocessing

   6. Applied egg-rr 55.7%

      \[\leadsto \color{blue}{\left(1 \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right), \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}}\right)} \cdot \left(R \cdot 2\right) \]
   7. Step-by-step derivation

      1. *-lft-identity 55.7%

         \[\leadsto \color{blue}{\tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right), \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}}} \cdot \left(R \cdot 2\right) \]

      2. *-commutative 55.7%

         \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \color{blue}{\left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}, \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

      3. *-commutative 55.7%

         \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \color{blue}{\left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

      4. metadata-eval 55.7%

         \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + \color{blue}{\left(-0.5\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

      5. cancel-sign-sub-inv 55.7%

         \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

   8. Simplified 55.7%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}} \cdot \left(R \cdot 2\right) \]

   9. Taylor expanded in phi2 around 0 55.7%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\color{blue}{\sin \left(0.5 \cdot \phi_1\right)}, \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

   10. Taylor expanded in phi2 around 0 55.7%

       \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \phi_1\right), \color{blue}{\sqrt{\cos \phi_1} \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -8.5 \cdot 10^{-36} \lor \neg \left(\phi_2 \leq 1.4 \cdot 10^{-15}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 34.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t\_0 \cdot t\_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+
        (* (* (cos phi1) (cos phi2)) (* t_0 t_0))
        (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
      (sqrt
       (- 1.0 (* (cos phi1) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	return R * (2.0 * atan2(sqrt((((cos(phi1) * cos(phi2)) * (t_0 * t_0)) + pow(sin(((phi1 - phi2) / 2.0)), 2.0))), sqrt((1.0 - (cos(phi1) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0))))));
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    code = r * (2.0d0 * atan2(sqrt((((cos(phi1) * cos(phi2)) * (t_0 * t_0)) + (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0))), sqrt((1.0d0 - (cos(phi1) * (sin((0.5d0 * (lambda1 - lambda2))) ** 2.0d0))))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(((lambda1 - lambda2) / 2.0));
	return R * (2.0 * Math.atan2(Math.sqrt((((Math.cos(phi1) * Math.cos(phi2)) * (t_0 * t_0)) + Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0))), Math.sqrt((1.0 - (Math.cos(phi1) * Math.pow(Math.sin((0.5 * (lambda1 - lambda2))), 2.0))))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	return R * (2.0 * math.atan2(math.sqrt((((math.cos(phi1) * math.cos(phi2)) * (t_0 * t_0)) + math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0))), math.sqrt((1.0 - (math.cos(phi1) * math.pow(math.sin((0.5 * (lambda1 - lambda2))), 2.0))))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(Float64(cos(phi1) * cos(phi2)) * Float64(t_0 * t_0)) + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))), sqrt(Float64(1.0 - Float64(cos(phi1) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)))))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	tmp = R * (2.0 * atan2(sqrt((((cos(phi1) * cos(phi2)) * (t_0 * t_0)) + (sin(((phi1 - phi2) / 2.0)) ^ 2.0))), sqrt((1.0 - (cos(phi1) * (sin((0.5 * (lambda1 - lambda2))) ^ 2.0))))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.9%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Step-by-step derivation

   1. associate-*l* 62.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

3. Simplified 63.0%

   \[\leadsto \color{blue}{R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in phi1 around 0 49.4%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\color{blue}{\left(1 - {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
6. Step-by-step derivation

   1. unpow2 49.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - \color{blue}{\sin \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(-0.5 \cdot \phi_2\right)}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. 1-sub-sin 49.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\color{blue}{\cos \left(-0.5 \cdot \phi_2\right) \cdot \cos \left(-0.5 \cdot \phi_2\right)} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. unpow2 49.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\color{blue}{{\cos \left(-0.5 \cdot \phi_2\right)}^{2}} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. *-commutative 49.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{{\cos \color{blue}{\left(\phi_2 \cdot -0.5\right)}}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. metadata-eval 49.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{{\cos \left(\phi_2 \cdot \color{blue}{\left(-0.5\right)}\right)}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. distribute-rgt-neg-in 49.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{{\cos \color{blue}{\left(-\phi_2 \cdot 0.5\right)}}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. cos-neg 49.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{{\color{blue}{\cos \left(\phi_2 \cdot 0.5\right)}}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

7. Simplified 49.4%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\color{blue}{{\cos \left(\phi_2 \cdot 0.5\right)}^{2}} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

8. Taylor expanded in phi2 around 0 34.3%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\color{blue}{1 - \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}\right) \]

9. Final simplification 34.3%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}\right) \]
10. Add Preprocessing

Alternative 26: 21.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_2 \cdot -0.5\right)}^{2} + \cos \phi_2 \cdot \left(t\_0 \cdot \left|t\_0\right|\right)}}{\sqrt{1 - \cos \phi_1 \cdot {t\_0}^{2}}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 (- lambda1 lambda2)))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+ (pow (sin (* phi2 -0.5)) 2.0) (* (cos phi2) (* t_0 (fabs t_0)))))
      (sqrt (- 1.0 (* (cos phi1) (pow t_0 2.0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * (lambda1 - lambda2)));
	return R * (2.0 * atan2(sqrt((pow(sin((phi2 * -0.5)), 2.0) + (cos(phi2) * (t_0 * fabs(t_0))))), sqrt((1.0 - (cos(phi1) * pow(t_0, 2.0))))));
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = sin((0.5d0 * (lambda1 - lambda2)))
    code = r * (2.0d0 * atan2(sqrt(((sin((phi2 * (-0.5d0))) ** 2.0d0) + (cos(phi2) * (t_0 * abs(t_0))))), sqrt((1.0d0 - (cos(phi1) * (t_0 ** 2.0d0))))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((0.5 * (lambda1 - lambda2)));
	return R * (2.0 * Math.atan2(Math.sqrt((Math.pow(Math.sin((phi2 * -0.5)), 2.0) + (Math.cos(phi2) * (t_0 * Math.abs(t_0))))), Math.sqrt((1.0 - (Math.cos(phi1) * Math.pow(t_0, 2.0))))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((0.5 * (lambda1 - lambda2)))
	return R * (2.0 * math.atan2(math.sqrt((math.pow(math.sin((phi2 * -0.5)), 2.0) + (math.cos(phi2) * (t_0 * math.fabs(t_0))))), math.sqrt((1.0 - (math.cos(phi1) * math.pow(t_0, 2.0))))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2)))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(phi2 * -0.5)) ^ 2.0) + Float64(cos(phi2) * Float64(t_0 * abs(t_0))))), sqrt(Float64(1.0 - Float64(cos(phi1) * (t_0 ^ 2.0)))))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin((0.5 * (lambda1 - lambda2)));
	tmp = R * (2.0 * atan2(sqrt(((sin((phi2 * -0.5)) ^ 2.0) + (cos(phi2) * (t_0 * abs(t_0))))), sqrt((1.0 - (cos(phi1) * (t_0 ^ 2.0))))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(t$95$0 * N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.9%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 29.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \color{blue}{\left(\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)} \cdot \sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. sqrt-prod 45.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \color{blue}{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. rem-sqrt-square 45.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \color{blue}{\left|\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right|}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. div-inv 45.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \left|\sin \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}\right|}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. metadata-eval 45.1%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \left|\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{0.5}\right)\right|}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

4. Applied egg-rr 45.1%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \color{blue}{\left|\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right|}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Taylor expanded in phi1 around 0 36.4%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \left|\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right|}}{\sqrt{1 - \left(\color{blue}{{\sin \left(-0.5 \cdot \phi_2\right)}^{2}} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

6. Taylor expanded in phi1 around 0 32.2%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_2 \cdot \left(\left|\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right| \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + {\sin \left(-0.5 \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(-0.5 \cdot \phi_2\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

7. Taylor expanded in phi2 around 0 20.9%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot \left(\left|\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right| \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + {\sin \left(-0.5 \cdot \phi_2\right)}^{2}}}{\color{blue}{\sqrt{1 - \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}\right) \]

8. Final simplification 20.9%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_2 \cdot -0.5\right)}^{2} + \cos \phi_2 \cdot \left(\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left|\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right|\right)}}{\sqrt{1 - \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}\right) \]
9. Add Preprocessing

Alternative 27: 17.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(1 + {\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot -0.010416666666666666 - 0.25\right)\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (* R 2.0)
  (atan2
   (*
    (sin (* 0.5 (- lambda1 lambda2)))
    (+
     1.0
     (* (pow phi2 2.0) (- (* (pow phi2 2.0) -0.010416666666666666) 0.25))))
   (sqrt
    (-
     1.0
     (fma
      (cos phi1)
      (* (cos phi2) (- 0.5 (* 0.5 (cos (- lambda1 lambda2)))))
      (pow (sin (* 0.5 (- phi1 phi2))) 2.0)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return (R * 2.0) * atan2((sin((0.5 * (lambda1 - lambda2))) * (1.0 + (pow(phi2, 2.0) * ((pow(phi2, 2.0) * -0.010416666666666666) - 0.25)))), sqrt((1.0 - fma(cos(phi1), (cos(phi2) * (0.5 - (0.5 * cos((lambda1 - lambda2))))), pow(sin((0.5 * (phi1 - phi2))), 2.0)))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(Float64(R * 2.0) * atan(Float64(sin(Float64(0.5 * Float64(lambda1 - lambda2))) * Float64(1.0 + Float64((phi2 ^ 2.0) * Float64(Float64((phi2 ^ 2.0) * -0.010416666666666666) - 0.25)))), sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2))))), (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[(N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[Power[phi2, 2.0], $MachinePrecision] * N[(N[(N[Power[phi2, 2.0], $MachinePrecision] * -0.010416666666666666), $MachinePrecision] - 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.9%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Step-by-step derivation

   1. associate-*r* 62.9%

      \[\leadsto \color{blue}{\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}} \]

   2. *-commutative 62.9%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}} \cdot \left(R \cdot 2\right)} \]

3. Simplified 63.0%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}} \cdot \left(R \cdot 2\right)} \]
4. Add Preprocessing

6. Applied egg-rr 41.4%

   \[\leadsto \color{blue}{\left(1 \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right), \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}}\right)} \cdot \left(R \cdot 2\right) \]
7. Step-by-step derivation

   1. *-lft-identity 41.4%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right), \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}}} \cdot \left(R \cdot 2\right) \]

   2. *-commutative 41.4%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \color{blue}{\left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}, \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

   3. *-commutative 41.4%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \color{blue}{\left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

   4. metadata-eval 41.4%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + \color{blue}{\left(-0.5\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

   5. cancel-sign-sub-inv 41.4%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

8. Simplified 41.4%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}} \cdot \left(R \cdot 2\right) \]

9. Taylor expanded in phi2 around 0 31.8%

   \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\color{blue}{\sin \left(0.5 \cdot \phi_1\right)}, \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

10. Taylor expanded in phi1 around 0 13.8%

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_2} \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

11. Taylor expanded in phi2 around 0 16.3%

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(1 + {\phi_2}^{2} \cdot \left(-0.010416666666666666 \cdot {\phi_2}^{2} - 0.25\right)\right)} \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

12. Final simplification 16.3%

    \[\leadsto \left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(1 + {\phi_2}^{2} \cdot \left({\phi_2}^{2} \cdot -0.010416666666666666 - 0.25\right)\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \]
13. Add Preprocessing

Alternative 28: 16.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(1 + {\phi_2}^{2} \cdot -0.25\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (* R 2.0)
  (atan2
   (* (sin (* 0.5 (- lambda1 lambda2))) (+ 1.0 (* (pow phi2 2.0) -0.25)))
   (sqrt
    (-
     1.0
     (fma
      (cos phi1)
      (* (cos phi2) (- 0.5 (* 0.5 (cos (- lambda1 lambda2)))))
      (pow (sin (* 0.5 (- phi1 phi2))) 2.0)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return (R * 2.0) * atan2((sin((0.5 * (lambda1 - lambda2))) * (1.0 + (pow(phi2, 2.0) * -0.25))), sqrt((1.0 - fma(cos(phi1), (cos(phi2) * (0.5 - (0.5 * cos((lambda1 - lambda2))))), pow(sin((0.5 * (phi1 - phi2))), 2.0)))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(Float64(R * 2.0) * atan(Float64(sin(Float64(0.5 * Float64(lambda1 - lambda2))) * Float64(1.0 + Float64((phi2 ^ 2.0) * -0.25))), sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2))))), (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[(N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[Power[phi2, 2.0], $MachinePrecision] * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.9%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Step-by-step derivation

   1. associate-*r* 62.9%

      \[\leadsto \color{blue}{\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}} \]

   2. *-commutative 62.9%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}} \cdot \left(R \cdot 2\right)} \]

3. Simplified 63.0%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}} \cdot \left(R \cdot 2\right)} \]
4. Add Preprocessing

6. Applied egg-rr 41.4%

   \[\leadsto \color{blue}{\left(1 \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right), \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}}\right)} \cdot \left(R \cdot 2\right) \]
7. Step-by-step derivation

   1. *-lft-identity 41.4%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right), \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}}} \cdot \left(R \cdot 2\right) \]

   2. *-commutative 41.4%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \color{blue}{\left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}, \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

   3. *-commutative 41.4%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \color{blue}{\left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

   4. metadata-eval 41.4%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + \color{blue}{\left(-0.5\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

   5. cancel-sign-sub-inv 41.4%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

8. Simplified 41.4%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}} \cdot \left(R \cdot 2\right) \]

9. Taylor expanded in phi2 around 0 31.8%

   \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\color{blue}{\sin \left(0.5 \cdot \phi_1\right)}, \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

10. Taylor expanded in phi1 around 0 13.8%

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_2} \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

11. Taylor expanded in phi2 around 0 16.3%

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(1 + -0.25 \cdot {\phi_2}^{2}\right)} \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]
12. Step-by-step derivation

    1. *-commutative 16.3%

       \[\leadsto \tan^{-1}_* \frac{\left(1 + \color{blue}{{\phi_2}^{2} \cdot -0.25}\right) \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

13. Simplified 16.3%

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(1 + {\phi_2}^{2} \cdot -0.25\right)} \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

14. Final simplification 16.3%

    \[\leadsto \left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(1 + {\phi_2}^{2} \cdot -0.25\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \]
15. Add Preprocessing

Alternative 29: 16.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (* R 2.0)
  (atan2
   (sin (* 0.5 (- lambda1 lambda2)))
   (sqrt
    (-
     1.0
     (fma
      (cos phi1)
      (* (cos phi2) (- 0.5 (* 0.5 (cos (- lambda1 lambda2)))))
      (pow (sin (* 0.5 (- phi1 phi2))) 2.0)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return (R * 2.0) * atan2(sin((0.5 * (lambda1 - lambda2))), sqrt((1.0 - fma(cos(phi1), (cos(phi2) * (0.5 - (0.5 * cos((lambda1 - lambda2))))), pow(sin((0.5 * (phi1 - phi2))), 2.0)))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(Float64(R * 2.0) * atan(sin(Float64(0.5 * Float64(lambda1 - lambda2))), sqrt(Float64(1.0 - fma(cos(phi1), Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2))))), (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.9%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Step-by-step derivation

   1. associate-*r* 62.9%

      \[\leadsto \color{blue}{\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}} \]

   2. *-commutative 62.9%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}} \cdot \left(R \cdot 2\right)} \]

3. Simplified 63.0%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}} \cdot \left(R \cdot 2\right)} \]
4. Add Preprocessing

6. Applied egg-rr 41.4%

   \[\leadsto \color{blue}{\left(1 \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right), \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}}\right)} \cdot \left(R \cdot 2\right) \]
7. Step-by-step derivation

   1. *-lft-identity 41.4%

      \[\leadsto \color{blue}{\tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right), \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}}} \cdot \left(R \cdot 2\right) \]

   2. *-commutative 41.4%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \color{blue}{\left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}, \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

   3. *-commutative 41.4%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \color{blue}{\left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)} \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

   4. metadata-eval 41.4%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 + \color{blue}{\left(-0.5\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

   5. cancel-sign-sub-inv 41.4%

      \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

8. Simplified 41.4%

   \[\leadsto \color{blue}{\tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}} \cdot \left(R \cdot 2\right) \]

9. Taylor expanded in phi2 around 0 31.8%

   \[\leadsto \tan^{-1}_* \frac{\mathsf{hypot}\left(\color{blue}{\sin \left(0.5 \cdot \phi_1\right)}, \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

10. Taylor expanded in phi1 around 0 13.8%

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_2} \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

11. Taylor expanded in phi2 around 0 16.2%

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

12. Final simplification 16.2%

    \[\leadsto \left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024185 
(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))))))))))