Spherical law of cosines

Percentage Accurate: 74.3% → 96.8%

Time: 47.1s

Alternatives: 30

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (acos
   (+
    (* (sin phi1) (sin phi2))
    (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
  R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R;
}

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
    code = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * r
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2))))) * R;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos(((math.sin(phi1) * math.sin(phi2)) + ((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2))))) * R

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) * R)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Initial Program: 74.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (acos
   (+
    (* (sin phi1) (sin phi2))
    (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
  R))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R;
}

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
    code = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * r
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2))))) * R;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return math.acos(((math.sin(phi1) * math.sin(phi2)) + ((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2))))) * R

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) * R)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Alternative 1: 96.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R_s \cdot \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0:\\ \;\;\;\;\left|R_m \cdot \left(\phi_1 - \phi_2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<=
       (acos
        (+
         (* (sin phi1) (sin phi2))
         (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
       0.0)
    (fabs (* R_m (- phi1 phi2)))
    (*
     R_m
     (acos
      (fma
       (sin phi1)
       (sin phi2)
       (*
        (cos phi1)
        (*
         (cos phi2)
         (fma
          (sin lambda1)
          (sin lambda2)
          (* (cos lambda2) (cos lambda1)))))))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) <= 0.0) {
		tmp = fabs((R_m * (phi1 - phi2)));
	} else {
		tmp = R_m * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * (cos(phi2) * fma(sin(lambda1), sin(lambda2), (cos(lambda2) * cos(lambda1)))))));
	}
	return R_s * tmp;
}

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) <= 0.0)
		tmp = abs(Float64(R_m * Float64(phi1 - phi2)));
	else
		tmp = Float64(R_m * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * Float64(cos(phi2) * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda2) * cos(lambda1))))))));
	end
	return Float64(R_s * tmp)
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[Abs[N[(R$95$m * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))) < 0.0

   1. Initial program 3.8%

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

   3. Simplified 3.8%

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

   5. Taylor expanded in lambda2 around 0 3.8%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \left(-\lambda_1\right)}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg 3.8%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]

   7. Simplified 3.8%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]

   8. Taylor expanded in lambda1 around 0 3.8%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 3.8%

         \[\leadsto R \cdot \color{blue}{\left(\sqrt{\cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot \sqrt{\cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)}\right)} \]

      2. pow2 3.8%

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

      3. cos-diff 3.8%

         \[\leadsto R \cdot {\left(\sqrt{\cos^{-1} \color{blue}{\cos \left(\phi_1 - \phi_2\right)}}\right)}^{2} \]

   10. Applied egg-rr 3.8%

       \[\leadsto R \cdot \color{blue}{{\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}} \]
   11. Step-by-step derivation

       1. rem-square-sqrt 3.8%

          \[\leadsto \color{blue}{\sqrt{R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}} \cdot \sqrt{R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}}} \]

       2. sqrt-unprod 3.8%

          \[\leadsto \color{blue}{\sqrt{\left(R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}\right) \cdot \left(R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}\right)}} \]

       3. pow2 3.8%

          \[\leadsto \sqrt{\color{blue}{{\left(R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}\right)}^{2}}} \]

       4. unpow2 3.8%

          \[\leadsto \sqrt{{\left(R \cdot \color{blue}{\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)} \cdot \sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}\right)}^{2}} \]

       5. add-sqr-sqrt 3.8%

          \[\leadsto \sqrt{{\left(R \cdot \color{blue}{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}} \]

       6. acos-cos-s 12.0%

          \[\leadsto \sqrt{{\left(R \cdot \color{blue}{\left(\phi_1 - \phi_2\right)}\right)}^{2}} \]

   12. Applied egg-rr 12.0%

       \[\leadsto \color{blue}{\sqrt{{\left(R \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}} \]
   13. Step-by-step derivation

       1. unpow2 12.0%

          \[\leadsto \sqrt{\color{blue}{\left(R \cdot \left(\phi_1 - \phi_2\right)\right) \cdot \left(R \cdot \left(\phi_1 - \phi_2\right)\right)}} \]

       2. rem-sqrt-square 20.4%

          \[\leadsto \color{blue}{\left|R \cdot \left(\phi_1 - \phi_2\right)\right|} \]

   14. Simplified 20.4%

       \[\leadsto \color{blue}{\left|R \cdot \left(\phi_1 - \phi_2\right)\right|} \]

   if 0.0 < (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))))

   1. Initial program 75.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   3. Simplified 75.3%

      \[\leadsto \color{blue}{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot R} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. cos-diff 98.9%

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

      2. distribute-lft-in 98.9%

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

   6. Applied egg-rr 98.9%

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

      1. distribute-lft-out 98.9%

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

      2. +-commutative 98.9%

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

      3. fma-def 98.9%

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

      4. *-commutative 98.9%

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

   8. Simplified 98.9%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0:\\ \;\;\;\;\left|R \cdot \left(\phi_1 - \phi_2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ R_s \cdot \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(t_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0:\\ \;\;\;\;\left|R_m \cdot \left(\phi_1 - \phi_2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (*
    R_s
    (if (<=
         (acos (+ t_0 (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
         0.0)
      (fabs (* R_m (- phi1 phi2)))
      (*
       R_m
       (acos
        (+
         t_0
         (*
          (cos phi2)
          (*
           (cos phi1)
           (fma
            (cos lambda2)
            (cos lambda1)
            (* (sin lambda1) (sin lambda2))))))))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (acos((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) <= 0.0) {
		tmp = fabs((R_m * (phi1 - phi2)));
	} else {
		tmp = R_m * acos((t_0 + (cos(phi2) * (cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2)))))));
	}
	return R_s * tmp;
}

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) <= 0.0)
		tmp = abs(Float64(R_m * Float64(phi1 - phi2)));
	else
		tmp = Float64(R_m * acos(Float64(t_0 + Float64(cos(phi2) * Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))))))));
	end
	return Float64(R_s * tmp)
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[N[ArcCos[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[Abs[N[(R$95$m * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(R$95$m * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))) < 0.0

   1. Initial program 3.8%

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

   3. Simplified 3.8%

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

   5. Taylor expanded in lambda2 around 0 3.8%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \left(-\lambda_1\right)}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg 3.8%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]

   7. Simplified 3.8%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]

   8. Taylor expanded in lambda1 around 0 3.8%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 3.8%

         \[\leadsto R \cdot \color{blue}{\left(\sqrt{\cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot \sqrt{\cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)}\right)} \]

      2. pow2 3.8%

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

      3. cos-diff 3.8%

         \[\leadsto R \cdot {\left(\sqrt{\cos^{-1} \color{blue}{\cos \left(\phi_1 - \phi_2\right)}}\right)}^{2} \]

   10. Applied egg-rr 3.8%

       \[\leadsto R \cdot \color{blue}{{\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}} \]
   11. Step-by-step derivation

       1. rem-square-sqrt 3.8%

          \[\leadsto \color{blue}{\sqrt{R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}} \cdot \sqrt{R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}}} \]

       2. sqrt-unprod 3.8%

          \[\leadsto \color{blue}{\sqrt{\left(R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}\right) \cdot \left(R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}\right)}} \]

       3. pow2 3.8%

          \[\leadsto \sqrt{\color{blue}{{\left(R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}\right)}^{2}}} \]

       4. unpow2 3.8%

          \[\leadsto \sqrt{{\left(R \cdot \color{blue}{\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)} \cdot \sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}\right)}^{2}} \]

       5. add-sqr-sqrt 3.8%

          \[\leadsto \sqrt{{\left(R \cdot \color{blue}{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}} \]

       6. acos-cos-s 12.0%

          \[\leadsto \sqrt{{\left(R \cdot \color{blue}{\left(\phi_1 - \phi_2\right)}\right)}^{2}} \]

   12. Applied egg-rr 12.0%

       \[\leadsto \color{blue}{\sqrt{{\left(R \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}} \]
   13. Step-by-step derivation

       1. unpow2 12.0%

          \[\leadsto \sqrt{\color{blue}{\left(R \cdot \left(\phi_1 - \phi_2\right)\right) \cdot \left(R \cdot \left(\phi_1 - \phi_2\right)\right)}} \]

       2. rem-sqrt-square 20.4%

          \[\leadsto \color{blue}{\left|R \cdot \left(\phi_1 - \phi_2\right)\right|} \]

   14. Simplified 20.4%

       \[\leadsto \color{blue}{\left|R \cdot \left(\phi_1 - \phi_2\right)\right|} \]

   if 0.0 < (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))))

   1. Initial program 75.3%

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

      1. cos-diff 98.9%

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

      2. distribute-lft-in 98.9%

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

   4. Applied egg-rr 98.9%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right) \cdot R \]
   5. Step-by-step derivation

      1. distribute-lft-out 98.9%

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

      2. *-commutative 98.9%

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

      3. associate-*l* 98.9%

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

      4. *-commutative 98.9%

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

      5. fma-udef 98.9%

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

      6. *-commutative 98.9%

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

   6. Simplified 98.9%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0:\\ \;\;\;\;\left|R \cdot \left(\phi_1 - \phi_2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ R_s \cdot \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(t_1 + t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0:\\ \;\;\;\;\left|R_m \cdot \left(\phi_1 - \phi_2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(t_1 + t_0 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2))) (t_1 (* (sin phi1) (sin phi2))))
   (*
    R_s
    (if (<= (acos (+ t_1 (* t_0 (cos (- lambda1 lambda2))))) 0.0)
      (fabs (* R_m (- phi1 phi2)))
      (*
       R_m
       (acos
        (+
         t_1
         (*
          t_0
          (fma
           (sin lambda1)
           (sin lambda2)
           (* (cos lambda2) (cos lambda1)))))))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * cos(phi2);
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if (acos((t_1 + (t_0 * cos((lambda1 - lambda2))))) <= 0.0) {
		tmp = fabs((R_m * (phi1 - phi2)));
	} else {
		tmp = R_m * acos((t_1 + (t_0 * fma(sin(lambda1), sin(lambda2), (cos(lambda2) * cos(lambda1))))));
	}
	return R_s * tmp;
}

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (acos(Float64(t_1 + Float64(t_0 * cos(Float64(lambda1 - lambda2))))) <= 0.0)
		tmp = abs(Float64(R_m * Float64(phi1 - phi2)));
	else
		tmp = Float64(R_m * acos(Float64(t_1 + Float64(t_0 * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda2) * cos(lambda1)))))));
	end
	return Float64(R_s * tmp)
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[N[ArcCos[N[(t$95$1 + N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[Abs[N[(R$95$m * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(R$95$m * N[ArcCos[N[(t$95$1 + N[(t$95$0 * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))) < 0.0

   1. Initial program 3.8%

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

   3. Simplified 3.8%

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

   5. Taylor expanded in lambda2 around 0 3.8%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \left(-\lambda_1\right)}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg 3.8%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]

   7. Simplified 3.8%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]

   8. Taylor expanded in lambda1 around 0 3.8%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 3.8%

         \[\leadsto R \cdot \color{blue}{\left(\sqrt{\cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot \sqrt{\cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)}\right)} \]

      2. pow2 3.8%

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

      3. cos-diff 3.8%

         \[\leadsto R \cdot {\left(\sqrt{\cos^{-1} \color{blue}{\cos \left(\phi_1 - \phi_2\right)}}\right)}^{2} \]

   10. Applied egg-rr 3.8%

       \[\leadsto R \cdot \color{blue}{{\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}} \]
   11. Step-by-step derivation

       1. rem-square-sqrt 3.8%

          \[\leadsto \color{blue}{\sqrt{R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}} \cdot \sqrt{R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}}} \]

       2. sqrt-unprod 3.8%

          \[\leadsto \color{blue}{\sqrt{\left(R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}\right) \cdot \left(R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}\right)}} \]

       3. pow2 3.8%

          \[\leadsto \sqrt{\color{blue}{{\left(R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}\right)}^{2}}} \]

       4. unpow2 3.8%

          \[\leadsto \sqrt{{\left(R \cdot \color{blue}{\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)} \cdot \sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}\right)}^{2}} \]

       5. add-sqr-sqrt 3.8%

          \[\leadsto \sqrt{{\left(R \cdot \color{blue}{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}} \]

       6. acos-cos-s 12.0%

          \[\leadsto \sqrt{{\left(R \cdot \color{blue}{\left(\phi_1 - \phi_2\right)}\right)}^{2}} \]

   12. Applied egg-rr 12.0%

       \[\leadsto \color{blue}{\sqrt{{\left(R \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}} \]
   13. Step-by-step derivation

       1. unpow2 12.0%

          \[\leadsto \sqrt{\color{blue}{\left(R \cdot \left(\phi_1 - \phi_2\right)\right) \cdot \left(R \cdot \left(\phi_1 - \phi_2\right)\right)}} \]

       2. rem-sqrt-square 20.4%

          \[\leadsto \color{blue}{\left|R \cdot \left(\phi_1 - \phi_2\right)\right|} \]

   14. Simplified 20.4%

       \[\leadsto \color{blue}{\left|R \cdot \left(\phi_1 - \phi_2\right)\right|} \]

   if 0.0 < (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))))

   1. Initial program 75.3%

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

      1. cos-diff 98.9%

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

      2. distribute-lft-in 98.9%

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

   4. Applied egg-rr 98.9%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right) \cdot R \]
   5. Step-by-step derivation

      1. distribute-lft-out 98.9%

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

      2. +-commutative 98.9%

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

      3. fma-def 98.9%

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

      4. *-commutative 98.9%

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

   6. Simplified 98.9%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0:\\ \;\;\;\;\left|R \cdot \left(\phi_1 - \phi_2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ R_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.33:\\ \;\;\;\;R_m \cdot \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\right)}\right)\\ \mathbf{elif}\;\phi_2 \leq 150000000000:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t_0, \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (*
    R_s
    (if (<= phi2 -0.33)
      (*
       R_m
       (log
        (exp
         (acos
          (fma (sin phi1) (sin phi2) (* (* (cos phi1) (cos phi2)) t_0))))))
      (if (<= phi2 150000000000.0)
        (*
         R_m
         (acos
          (+
           (*
            (cos phi2)
            (*
             (cos phi1)
             (fma
              (cos lambda2)
              (cos lambda1)
              (* (sin lambda1) (sin lambda2)))))
           (* (sin phi1) phi2))))
        (*
         R_m
         (-
          (* PI 0.5)
          (asin
           (fma
            (cos phi1)
            (* (cos phi2) t_0)
            (* (sin phi1) (sin phi2)))))))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= -0.33) {
		tmp = R_m * log(exp(acos(fma(sin(phi1), sin(phi2), ((cos(phi1) * cos(phi2)) * t_0)))));
	} else if (phi2 <= 150000000000.0) {
		tmp = R_m * acos(((cos(phi2) * (cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2))))) + (sin(phi1) * phi2)));
	} else {
		tmp = R_m * ((((double) M_PI) * 0.5) - asin(fma(cos(phi1), (cos(phi2) * t_0), (sin(phi1) * sin(phi2)))));
	}
	return R_s * tmp;
}

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi2 <= -0.33)
		tmp = Float64(R_m * log(exp(acos(fma(sin(phi1), sin(phi2), Float64(Float64(cos(phi1) * cos(phi2)) * t_0))))));
	elseif (phi2 <= 150000000000.0)
		tmp = Float64(R_m * acos(Float64(Float64(cos(phi2) * Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))))) + Float64(sin(phi1) * phi2))));
	else
		tmp = Float64(R_m * Float64(Float64(pi * 0.5) - asin(fma(cos(phi1), Float64(cos(phi2) * t_0), Float64(sin(phi1) * sin(phi2))))));
	end
	return Float64(R_s * tmp)
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi2, -0.33], N[(R$95$m * N[Log[N[Exp[N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 150000000000.0], N[(R$95$m * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.330000000000000016

   1. Initial program 75.3%

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

      1. cos-diff 98.5%

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

      2. distribute-lft-in 98.4%

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

   4. Applied egg-rr 98.4%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right) \cdot R \]
   5. Step-by-step derivation

      1. distribute-lft-out 98.5%

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

      2. *-commutative 98.5%

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

      3. associate-*l* 98.4%

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

      4. *-commutative 98.4%

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

      5. fma-udef 98.4%

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

      6. *-commutative 98.4%

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

   6. Simplified 98.4%

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

      1. add-log-exp 98.4%

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

      2. fma-def 98.4%

         \[\leadsto \log \left(e^{\cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right)\right)}}\right) \cdot R \]

      3. associate-*r* 98.4%

         \[\leadsto \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}\right)\right)}\right) \cdot R \]

      4. fma-udef 98.5%

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

      5. cos-diff 75.3%

         \[\leadsto \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right)\right)}\right) \cdot R \]

   8. Applied egg-rr 75.3%

      \[\leadsto \color{blue}{\log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)}\right)} \cdot R \]

   if -0.330000000000000016 < phi2 < 1.5e11

   1. Initial program 66.6%

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

      1. cos-diff 90.5%

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

      2. distribute-lft-in 90.6%

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

   4. Applied egg-rr 90.6%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right) \cdot R \]
   5. Step-by-step derivation

      1. distribute-lft-out 90.5%

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

      2. *-commutative 90.5%

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

      3. associate-*l* 90.6%

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

      4. *-commutative 90.6%

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

      5. fma-udef 90.6%

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

      6. *-commutative 90.6%

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

   6. Simplified 90.6%

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

   7. Taylor expanded in phi2 around 0 89.1%

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

      1. *-commutative 89.1%

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

   9. Simplified 89.1%

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

   if 1.5e11 < phi2

   1. Initial program 79.9%

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

   3. Simplified 79.9%

      \[\leadsto \color{blue}{R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. cos-diff 99.2%

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

      2. *-commutative 99.2%

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

      3. *-commutative 99.2%

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

      4. cos-diff 79.9%

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

      5. fma-def 79.9%

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

      6. +-commutative 79.9%

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

      7. acos-asin 80.0%

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

      8. sub-neg 80.0%

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

      9. div-inv 80.0%

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

      10. metadata-eval 80.0%

          \[\leadsto R \cdot \left(\pi \cdot \color{blue}{0.5} + \left(-\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \]

      11. +-commutative 80.0%

          \[\leadsto R \cdot \left(\pi \cdot 0.5 + \left(-\sin^{-1} \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)}\right)\right) \]

   6. Applied egg-rr 80.0%

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

      1. sub-neg 80.0%

         \[\leadsto R \cdot \color{blue}{\left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)} \]

   8. Simplified 80.0%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.33:\\ \;\;\;\;R \cdot \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)}\right)\\ \mathbf{elif}\;\phi_2 \leq 150000000000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ R_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.45 \cdot 10^{-6}:\\ \;\;\;\;R_m \cdot \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\right)}\right)\\ \mathbf{elif}\;\phi_2 \leq 0.00155:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t_0, \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (*
    R_s
    (if (<= phi2 -1.45e-6)
      (*
       R_m
       (log
        (exp
         (acos
          (fma (sin phi1) (sin phi2) (* (* (cos phi1) (cos phi2)) t_0))))))
      (if (<= phi2 0.00155)
        (*
         R_m
         (acos
          (fma
           (sin phi1)
           (sin phi2)
           (*
            (cos phi1)
            (+
             (* (cos lambda2) (cos lambda1))
             (* (sin lambda1) (sin lambda2)))))))
        (*
         R_m
         (-
          (* PI 0.5)
          (asin
           (fma
            (cos phi1)
            (* (cos phi2) t_0)
            (* (sin phi1) (sin phi2)))))))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= -1.45e-6) {
		tmp = R_m * log(exp(acos(fma(sin(phi1), sin(phi2), ((cos(phi1) * cos(phi2)) * t_0)))));
	} else if (phi2 <= 0.00155) {
		tmp = R_m * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2))))));
	} else {
		tmp = R_m * ((((double) M_PI) * 0.5) - asin(fma(cos(phi1), (cos(phi2) * t_0), (sin(phi1) * sin(phi2)))));
	}
	return R_s * tmp;
}

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi2 <= -1.45e-6)
		tmp = Float64(R_m * log(exp(acos(fma(sin(phi1), sin(phi2), Float64(Float64(cos(phi1) * cos(phi2)) * t_0))))));
	elseif (phi2 <= 0.00155)
		tmp = Float64(R_m * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda1) * sin(lambda2)))))));
	else
		tmp = Float64(R_m * Float64(Float64(pi * 0.5) - asin(fma(cos(phi1), Float64(cos(phi2) * t_0), Float64(sin(phi1) * sin(phi2))))));
	end
	return Float64(R_s * tmp)
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi2, -1.45e-6], N[(R$95$m * N[Log[N[Exp[N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 0.00155], N[(R$95$m * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.4500000000000001e-6

   1. Initial program 75.6%

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

      1. cos-diff 98.5%

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

      2. distribute-lft-in 98.4%

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

   4. Applied egg-rr 98.4%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right) \cdot R \]
   5. Step-by-step derivation

      1. distribute-lft-out 98.5%

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

      2. *-commutative 98.5%

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

      3. associate-*l* 98.4%

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

      4. *-commutative 98.4%

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

      5. fma-udef 98.4%

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

      6. *-commutative 98.4%

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

   6. Simplified 98.4%

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

      1. add-log-exp 98.4%

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

      2. fma-def 98.4%

         \[\leadsto \log \left(e^{\cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right)\right)}}\right) \cdot R \]

      3. associate-*r* 98.4%

         \[\leadsto \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}\right)\right)}\right) \cdot R \]

      4. fma-udef 98.5%

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

      5. cos-diff 75.6%

         \[\leadsto \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right)\right)}\right) \cdot R \]

   8. Applied egg-rr 75.6%

      \[\leadsto \color{blue}{\log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)}\right)} \cdot R \]

   if -1.4500000000000001e-6 < phi2 < 0.00154999999999999995

   1. Initial program 66.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   3. Simplified 66.8%

      \[\leadsto \color{blue}{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot R} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. cos-diff 90.2%

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

      2. distribute-lft-in 90.2%

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

   6. Applied egg-rr 90.2%

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

      1. distribute-lft-out 90.2%

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

      2. +-commutative 90.2%

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

      3. fma-def 90.2%

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

      4. *-commutative 90.2%

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

   8. Simplified 90.2%

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

   9. Taylor expanded in phi2 around 0 90.2%

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

   if 0.00154999999999999995 < phi2

   1. Initial program 78.3%

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

   3. Simplified 78.3%

      \[\leadsto \color{blue}{R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. cos-diff 99.2%

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

      2. *-commutative 99.2%

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

      3. *-commutative 99.2%

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

      4. cos-diff 78.3%

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

      5. fma-def 78.3%

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

      6. +-commutative 78.3%

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

      7. acos-asin 78.4%

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

      8. sub-neg 78.4%

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

      9. div-inv 78.4%

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

      10. metadata-eval 78.4%

          \[\leadsto R \cdot \left(\pi \cdot \color{blue}{0.5} + \left(-\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \]

      11. +-commutative 78.4%

          \[\leadsto R \cdot \left(\pi \cdot 0.5 + \left(-\sin^{-1} \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)}\right)\right) \]

   6. Applied egg-rr 78.4%

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

      1. sub-neg 78.4%

         \[\leadsto R \cdot \color{blue}{\left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)} \]

   8. Simplified 78.4%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.45 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)}\right)\\ \mathbf{elif}\;\phi_2 \leq 0.00155:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ R_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.3 \cdot 10^{-8}:\\ \;\;\;\;R_m \cdot \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\right)}\right)\\ \mathbf{elif}\;\phi_2 \leq 0.00155:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t_0, \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (*
    R_s
    (if (<= phi2 -2.3e-8)
      (*
       R_m
       (log
        (exp
         (acos
          (fma (sin phi1) (sin phi2) (* (* (cos phi1) (cos phi2)) t_0))))))
      (if (<= phi2 0.00155)
        (*
         R_m
         (acos
          (*
           (cos phi1)
           (fma (sin lambda1) (sin lambda2) (* (cos lambda2) (cos lambda1))))))
        (*
         R_m
         (-
          (* PI 0.5)
          (asin
           (fma
            (cos phi1)
            (* (cos phi2) t_0)
            (* (sin phi1) (sin phi2)))))))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= -2.3e-8) {
		tmp = R_m * log(exp(acos(fma(sin(phi1), sin(phi2), ((cos(phi1) * cos(phi2)) * t_0)))));
	} else if (phi2 <= 0.00155) {
		tmp = R_m * acos((cos(phi1) * fma(sin(lambda1), sin(lambda2), (cos(lambda2) * cos(lambda1)))));
	} else {
		tmp = R_m * ((((double) M_PI) * 0.5) - asin(fma(cos(phi1), (cos(phi2) * t_0), (sin(phi1) * sin(phi2)))));
	}
	return R_s * tmp;
}

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi2 <= -2.3e-8)
		tmp = Float64(R_m * log(exp(acos(fma(sin(phi1), sin(phi2), Float64(Float64(cos(phi1) * cos(phi2)) * t_0))))));
	elseif (phi2 <= 0.00155)
		tmp = Float64(R_m * acos(Float64(cos(phi1) * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda2) * cos(lambda1))))));
	else
		tmp = Float64(R_m * Float64(Float64(pi * 0.5) - asin(fma(cos(phi1), Float64(cos(phi2) * t_0), Float64(sin(phi1) * sin(phi2))))));
	end
	return Float64(R_s * tmp)
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi2, -2.3e-8], N[(R$95$m * N[Log[N[Exp[N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 0.00155], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -2.3000000000000001e-8

   1. Initial program 75.6%

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

      1. cos-diff 98.5%

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

      2. distribute-lft-in 98.4%

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

   4. Applied egg-rr 98.4%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right) \cdot R \]
   5. Step-by-step derivation

      1. distribute-lft-out 98.5%

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

      2. *-commutative 98.5%

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

      3. associate-*l* 98.4%

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

      4. *-commutative 98.4%

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

      5. fma-udef 98.4%

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

      6. *-commutative 98.4%

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

   6. Simplified 98.4%

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

      1. add-log-exp 98.4%

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

      2. fma-def 98.4%

         \[\leadsto \log \left(e^{\cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right)\right)}}\right) \cdot R \]

      3. associate-*r* 98.4%

         \[\leadsto \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)}\right)\right)}\right) \cdot R \]

      4. fma-udef 98.5%

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

      5. cos-diff 75.6%

         \[\leadsto \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right)\right)}\right) \cdot R \]

   8. Applied egg-rr 75.6%

      \[\leadsto \color{blue}{\log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)}\right)} \cdot R \]

   if -2.3000000000000001e-8 < phi2 < 0.00154999999999999995

   1. Initial program 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in phi2 around 0 66.7%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]
   6. Step-by-step derivation

      1. cos-diff 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)}\right) \]

      2. *-commutative 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\color{blue}{\cos \lambda_1 \cdot \cos \lambda_2} + \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \]

      3. *-commutative 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \color{blue}{\sin \lambda_1 \cdot \sin \lambda_2}\right)\right) \]

   7. Applied egg-rr 89.6%

      \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \]

      2. fma-def 89.6%

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

      3. *-commutative 89.6%

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

   9. Simplified 89.6%

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

   if 0.00154999999999999995 < phi2

   1. Initial program 78.3%

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

   3. Simplified 78.3%

      \[\leadsto \color{blue}{R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. cos-diff 99.2%

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

      2. *-commutative 99.2%

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

      3. *-commutative 99.2%

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

      4. cos-diff 78.3%

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

      5. fma-def 78.3%

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

      6. +-commutative 78.3%

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

      7. acos-asin 78.4%

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

      8. sub-neg 78.4%

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

      9. div-inv 78.4%

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

      10. metadata-eval 78.4%

          \[\leadsto R \cdot \left(\pi \cdot \color{blue}{0.5} + \left(-\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \]

      11. +-commutative 78.4%

          \[\leadsto R \cdot \left(\pi \cdot 0.5 + \left(-\sin^{-1} \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)}\right)\right) \]

   6. Applied egg-rr 78.4%

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

      1. sub-neg 78.4%

         \[\leadsto R \cdot \color{blue}{\left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)} \]

   8. Simplified 78.4%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.3 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)}\right)\\ \mathbf{elif}\;\phi_2 \leq 0.00155:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ R_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.4 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(t_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R_m\\ \mathbf{elif}\;\phi_2 \leq 0.00155:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right), t_0\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (*
    R_s
    (if (<= phi2 -3.4e-8)
      (*
       (acos (+ t_0 (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
       R_m)
      (if (<= phi2 0.00155)
        (*
         R_m
         (acos
          (*
           (cos phi1)
           (fma (sin lambda1) (sin lambda2) (* (cos lambda2) (cos lambda1))))))
        (*
         R_m
         (-
          (* PI 0.5)
          (asin
           (fma
            (cos phi1)
            (* (cos phi2) (cos (- lambda2 lambda1)))
            t_0)))))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi2 <= -3.4e-8) {
		tmp = acos((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R_m;
	} else if (phi2 <= 0.00155) {
		tmp = R_m * acos((cos(phi1) * fma(sin(lambda1), sin(lambda2), (cos(lambda2) * cos(lambda1)))));
	} else {
		tmp = R_m * ((((double) M_PI) * 0.5) - asin(fma(cos(phi1), (cos(phi2) * cos((lambda2 - lambda1))), t_0)));
	}
	return R_s * tmp;
}

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi2 <= -3.4e-8)
		tmp = Float64(acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) * R_m);
	elseif (phi2 <= 0.00155)
		tmp = Float64(R_m * acos(Float64(cos(phi1) * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda2) * cos(lambda1))))));
	else
		tmp = Float64(R_m * Float64(Float64(pi * 0.5) - asin(fma(cos(phi1), Float64(cos(phi2) * cos(Float64(lambda2 - lambda1))), t_0))));
	end
	return Float64(R_s * tmp)
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi2, -3.4e-8], N[(N[ArcCos[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R$95$m), $MachinePrecision], If[LessEqual[phi2, 0.00155], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -3.4e-8

   1. Initial program 75.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   if -3.4e-8 < phi2 < 0.00154999999999999995

   1. Initial program 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in phi2 around 0 66.7%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]
   6. Step-by-step derivation

      1. cos-diff 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)}\right) \]

      2. *-commutative 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\color{blue}{\cos \lambda_1 \cdot \cos \lambda_2} + \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \]

      3. *-commutative 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \color{blue}{\sin \lambda_1 \cdot \sin \lambda_2}\right)\right) \]

   7. Applied egg-rr 89.6%

      \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \]

      2. fma-def 89.6%

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

      3. *-commutative 89.6%

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

   9. Simplified 89.6%

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

   if 0.00154999999999999995 < phi2

   1. Initial program 78.3%

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

   3. Simplified 78.3%

      \[\leadsto \color{blue}{R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. cos-diff 99.2%

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

      2. *-commutative 99.2%

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

      3. *-commutative 99.2%

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

      4. cos-diff 78.3%

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

      5. fma-def 78.3%

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

      6. +-commutative 78.3%

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

      7. acos-asin 78.4%

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

      8. sub-neg 78.4%

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

      9. div-inv 78.4%

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

      10. metadata-eval 78.4%

          \[\leadsto R \cdot \left(\pi \cdot \color{blue}{0.5} + \left(-\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \]

      11. +-commutative 78.4%

          \[\leadsto R \cdot \left(\pi \cdot 0.5 + \left(-\sin^{-1} \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)}\right)\right) \]

   6. Applied egg-rr 78.4%

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

      1. sub-neg 78.4%

         \[\leadsto R \cdot \color{blue}{\left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)} \]

   8. Simplified 78.4%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.4 \cdot 10^{-8}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 0.00155:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ R_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;\cos^{-1} \left(t_1 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right) \cdot R_m\\ \mathbf{elif}\;\phi_2 \leq 0.00155:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t_0, t_1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))) (t_1 (* (sin phi1) (sin phi2))))
   (*
    R_s
    (if (<= phi2 -5.2e-9)
      (* (acos (+ t_1 (* (* (cos phi1) (cos phi2)) t_0))) R_m)
      (if (<= phi2 0.00155)
        (*
         R_m
         (acos
          (*
           (cos phi1)
           (fma (sin lambda1) (sin lambda2) (* (cos lambda2) (cos lambda1))))))
        (* R_m (acos (fma (cos phi1) (* (cos phi2) t_0) t_1))))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi2 <= -5.2e-9) {
		tmp = acos((t_1 + ((cos(phi1) * cos(phi2)) * t_0))) * R_m;
	} else if (phi2 <= 0.00155) {
		tmp = R_m * acos((cos(phi1) * fma(sin(lambda1), sin(lambda2), (cos(lambda2) * cos(lambda1)))));
	} else {
		tmp = R_m * acos(fma(cos(phi1), (cos(phi2) * t_0), t_1));
	}
	return R_s * tmp;
}

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi2 <= -5.2e-9)
		tmp = Float64(acos(Float64(t_1 + Float64(Float64(cos(phi1) * cos(phi2)) * t_0))) * R_m);
	elseif (phi2 <= 0.00155)
		tmp = Float64(R_m * acos(Float64(cos(phi1) * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda2) * cos(lambda1))))));
	else
		tmp = Float64(R_m * acos(fma(cos(phi1), Float64(cos(phi2) * t_0), t_1)));
	end
	return Float64(R_s * tmp)
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi2, -5.2e-9], N[(N[ArcCos[N[(t$95$1 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R$95$m), $MachinePrecision], If[LessEqual[phi2, 0.00155], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -5.2000000000000002e-9

   1. Initial program 75.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   if -5.2000000000000002e-9 < phi2 < 0.00154999999999999995

   1. Initial program 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in phi2 around 0 66.7%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]
   6. Step-by-step derivation

      1. cos-diff 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)}\right) \]

      2. *-commutative 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\color{blue}{\cos \lambda_1 \cdot \cos \lambda_2} + \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \]

      3. *-commutative 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \color{blue}{\sin \lambda_1 \cdot \sin \lambda_2}\right)\right) \]

   7. Applied egg-rr 89.6%

      \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \]

      2. fma-def 89.6%

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

      3. *-commutative 89.6%

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

   9. Simplified 89.6%

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

   if 0.00154999999999999995 < phi2

   1. Initial program 78.3%

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

      1. +-commutative 78.3%

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

      2. associate-*l* 78.3%

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

      3. fma-def 78.4%

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

   3. Simplified 78.4%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -5.2 \cdot 10^{-9}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 0.00155:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.2 \cdot 10^{-8} \lor \neg \left(\phi_2 \leq 0.00155\right):\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R_m\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (or (<= phi2 -1.2e-8) (not (<= phi2 0.00155)))
    (*
     (acos
      (+
       (* (sin phi1) (sin phi2))
       (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
     R_m)
    (*
     R_m
     (acos
      (*
       (cos phi1)
       (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2)))))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -1.2e-8) || !(phi2 <= 0.00155)) {
		tmp = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R_m;
	} else {
		tmp = R_m * acos((cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2)))));
	}
	return R_s * tmp;
}

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi2 <= -1.2e-8) || !(phi2 <= 0.00155))
		tmp = Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) * R_m);
	else
		tmp = Float64(R_m * acos(Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))))));
	end
	return Float64(R_s * tmp)
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[Or[LessEqual[phi2, -1.2e-8], N[Not[LessEqual[phi2, 0.00155]], $MachinePrecision]], N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R$95$m), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.19999999999999999e-8 or 0.00154999999999999995 < phi2

   1. Initial program 77.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   if -1.19999999999999999e-8 < phi2 < 0.00154999999999999995

   1. Initial program 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in phi2 around 0 66.7%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]
   6. Step-by-step derivation

      1. cos-diff 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)}\right) \]

      2. *-commutative 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\color{blue}{\cos \lambda_1 \cdot \cos \lambda_2} + \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \]

      3. *-commutative 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \color{blue}{\sin \lambda_1 \cdot \sin \lambda_2}\right)\right) \]

   7. Applied egg-rr 89.6%

      \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \]
   8. Step-by-step derivation

      1. *-commutative 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\color{blue}{\cos \lambda_2 \cdot \cos \lambda_1} + \sin \lambda_1 \cdot \sin \lambda_2\right)\right) \]

      2. fma-udef 89.6%

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

      3. *-commutative 89.6%

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

   9. Simplified 89.6%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.2 \cdot 10^{-8} \lor \neg \left(\phi_2 \leq 0.00155\right):\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq -8 \cdot 10^{-9} \lor \neg \left(\phi_2 \leq 0.00155\right):\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R_m\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (or (<= phi2 -8e-9) (not (<= phi2 0.00155)))
    (*
     (acos
      (+
       (* (sin phi1) (sin phi2))
       (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
     R_m)
    (*
     R_m
     (acos
      (*
       (cos phi1)
       (fma (sin lambda1) (sin lambda2) (* (cos lambda2) (cos lambda1)))))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -8e-9) || !(phi2 <= 0.00155)) {
		tmp = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R_m;
	} else {
		tmp = R_m * acos((cos(phi1) * fma(sin(lambda1), sin(lambda2), (cos(lambda2) * cos(lambda1)))));
	}
	return R_s * tmp;
}

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi2 <= -8e-9) || !(phi2 <= 0.00155))
		tmp = Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) * R_m);
	else
		tmp = Float64(R_m * acos(Float64(cos(phi1) * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda2) * cos(lambda1))))));
	end
	return Float64(R_s * tmp)
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[Or[LessEqual[phi2, -8e-9], N[Not[LessEqual[phi2, 0.00155]], $MachinePrecision]], N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R$95$m), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi2 < -8.0000000000000005e-9 or 0.00154999999999999995 < phi2

   1. Initial program 77.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   if -8.0000000000000005e-9 < phi2 < 0.00154999999999999995

   1. Initial program 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in phi2 around 0 66.7%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]
   6. Step-by-step derivation

      1. cos-diff 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)}\right) \]

      2. *-commutative 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\color{blue}{\cos \lambda_1 \cdot \cos \lambda_2} + \sin \lambda_2 \cdot \sin \lambda_1\right)\right) \]

      3. *-commutative 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \color{blue}{\sin \lambda_1 \cdot \sin \lambda_2}\right)\right) \]

   7. Applied egg-rr 89.6%

      \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \]

      2. fma-def 89.6%

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

      3. *-commutative 89.6%

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

   9. Simplified 89.6%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -8 \cdot 10^{-9} \lor \neg \left(\phi_2 \leq 0.00155\right):\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.55 \cdot 10^{-7} \lor \neg \left(\phi_2 \leq 0.00155\right):\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R_m\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (or (<= phi2 -1.55e-7) (not (<= phi2 0.00155)))
    (*
     (acos
      (+
       (* (sin phi1) (sin phi2))
       (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
     R_m)
    (*
     R_m
     (acos
      (*
       (cos phi1)
       (+
        (* (cos lambda2) (cos lambda1))
        (* (sin lambda1) (sin lambda2)))))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -1.55e-7) || !(phi2 <= 0.00155)) {
		tmp = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R_m;
	} else {
		tmp = R_m * acos((cos(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))));
	}
	return R_s * tmp;
}

Fortran

R_m = abs(R)
R_s = copysign(1.0d0, R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if ((phi2 <= (-1.55d-7)) .or. (.not. (phi2 <= 0.00155d0))) then
        tmp = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * r_m
    else
        tmp = r_m * acos((cos(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))))
    end if
    code = r_s * tmp
end function

Java

R_m = Math.abs(R);
R_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -1.55e-7) || !(phi2 <= 0.00155)) {
		tmp = Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2))))) * R_m;
	} else {
		tmp = R_m * Math.acos((Math.cos(phi1) * ((Math.cos(lambda2) * Math.cos(lambda1)) + (Math.sin(lambda1) * Math.sin(lambda2)))));
	}
	return R_s * tmp;
}

Python

R_m = math.fabs(R)
R_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (phi2 <= -1.55e-7) or not (phi2 <= 0.00155):
		tmp = math.acos(((math.sin(phi1) * math.sin(phi2)) + ((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2))))) * R_m
	else:
		tmp = R_m * math.acos((math.cos(phi1) * ((math.cos(lambda2) * math.cos(lambda1)) + (math.sin(lambda1) * math.sin(lambda2)))))
	return R_s * tmp

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi2 <= -1.55e-7) || !(phi2 <= 0.00155))
		tmp = Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) * R_m);
	else
		tmp = Float64(R_m * acos(Float64(cos(phi1) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda1) * sin(lambda2))))));
	end
	return Float64(R_s * tmp)
end

MATLAB

R_m = abs(R);
R_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((phi2 <= -1.55e-7) || ~((phi2 <= 0.00155)))
		tmp = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R_m;
	else
		tmp = R_m * acos((cos(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[Or[LessEqual[phi2, -1.55e-7], N[Not[LessEqual[phi2, 0.00155]], $MachinePrecision]], N[(N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R$95$m), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.55e-7 or 0.00154999999999999995 < phi2

   1. Initial program 77.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   if -1.55e-7 < phi2 < 0.00154999999999999995

   1. Initial program 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in phi2 around 0 66.7%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]
   6. Step-by-step derivation

      1. cos-diff 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)}\right) \]

      2. +-commutative 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)}\right) \]

      3. *-commutative 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\color{blue}{\sin \lambda_1 \cdot \sin \lambda_2} + \cos \lambda_2 \cdot \cos \lambda_1\right)\right) \]

      4. *-commutative 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \color{blue}{\cos \lambda_1 \cdot \cos \lambda_2}\right)\right) \]

   7. Applied egg-rr 89.6%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.55 \cdot 10^{-7} \lor \neg \left(\phi_2 \leq 0.00155\right):\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.00037 \lor \neg \left(\phi_2 \leq 0.00155\right):\\ \;\;\;\;R_m \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (or (<= phi2 -0.00037) (not (<= phi2 0.00155)))
    (*
     R_m
     (acos
      (+
       (* (sin phi1) (sin phi2))
       (* (* (cos phi1) (cos phi2)) (cos lambda2)))))
    (*
     R_m
     (acos
      (*
       (cos phi1)
       (+
        (* (cos lambda2) (cos lambda1))
        (* (sin lambda1) (sin lambda2)))))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -0.00037) || !(phi2 <= 0.00155)) {
		tmp = R_m * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos(lambda2))));
	} else {
		tmp = R_m * acos((cos(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))));
	}
	return R_s * tmp;
}

Fortran

R_m = abs(R)
R_s = copysign(1.0d0, R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if ((phi2 <= (-0.00037d0)) .or. (.not. (phi2 <= 0.00155d0))) then
        tmp = r_m * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos(lambda2))))
    else
        tmp = r_m * acos((cos(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))))
    end if
    code = r_s * tmp
end function

Java

R_m = Math.abs(R);
R_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -0.00037) || !(phi2 <= 0.00155)) {
		tmp = R_m * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos(lambda2))));
	} else {
		tmp = R_m * Math.acos((Math.cos(phi1) * ((Math.cos(lambda2) * Math.cos(lambda1)) + (Math.sin(lambda1) * Math.sin(lambda2)))));
	}
	return R_s * tmp;
}

Python

R_m = math.fabs(R)
R_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (phi2 <= -0.00037) or not (phi2 <= 0.00155):
		tmp = R_m * math.acos(((math.sin(phi1) * math.sin(phi2)) + ((math.cos(phi1) * math.cos(phi2)) * math.cos(lambda2))))
	else:
		tmp = R_m * math.acos((math.cos(phi1) * ((math.cos(lambda2) * math.cos(lambda1)) + (math.sin(lambda1) * math.sin(lambda2)))))
	return R_s * tmp

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi2 <= -0.00037) || !(phi2 <= 0.00155))
		tmp = Float64(R_m * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(lambda2)))));
	else
		tmp = Float64(R_m * acos(Float64(cos(phi1) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda1) * sin(lambda2))))));
	end
	return Float64(R_s * tmp)
end

MATLAB

R_m = abs(R);
R_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((phi2 <= -0.00037) || ~((phi2 <= 0.00155)))
		tmp = R_m * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos(lambda2))));
	else
		tmp = R_m * acos((cos(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[Or[LessEqual[phi2, -0.00037], N[Not[LessEqual[phi2, 0.00155]], $MachinePrecision]], N[(R$95$m * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi2 < -3.6999999999999999e-4 or 0.00154999999999999995 < phi2

   1. Initial program 77.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around 0 57.4%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(-\lambda_2\right)\right)}\right) \cdot R \]
   4. Step-by-step derivation

      1. cos-neg 57.4%

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{\cos \lambda_2}\right)\right) \cdot R \]

      2. associate-*r* 57.4%

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2}\right) \cdot R \]

      3. *-commutative 57.4%

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \lambda_2 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}\right) \cdot R \]

   5. Simplified 57.4%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \lambda_2 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}\right) \cdot R \]

   if -3.6999999999999999e-4 < phi2 < 0.00154999999999999995

   1. Initial program 66.8%

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

   3. Simplified 66.8%

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

   5. Taylor expanded in phi2 around 0 66.7%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]
   6. Step-by-step derivation

      1. cos-diff 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)}\right) \]

      2. +-commutative 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)}\right) \]

      3. *-commutative 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\color{blue}{\sin \lambda_1 \cdot \sin \lambda_2} + \cos \lambda_2 \cdot \cos \lambda_1\right)\right) \]

      4. *-commutative 89.6%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \color{blue}{\cos \lambda_1 \cdot \cos \lambda_2}\right)\right) \]

   7. Applied egg-rr 89.6%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.00037 \lor \neg \left(\phi_2 \leq 0.00155\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ R_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.45 \cdot 10^{+156}:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq -2.25 \cdot 10^{-5}:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(t_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (*
    R_s
    (if (<= lambda1 -1.45e+156)
      (*
       R_m
       (acos
        (*
         (cos phi1)
         (+ (* (cos lambda2) (cos lambda1)) (* (sin lambda1) (sin lambda2))))))
      (if (<= lambda1 -2.25e-5)
        (* R_m (acos (+ t_0 (* (cos phi1) (* (cos phi2) (cos lambda1))))))
        (* R_m (acos (+ t_0 (* (* (cos phi1) (cos phi2)) (cos lambda2))))))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (lambda1 <= -1.45e+156) {
		tmp = R_m * acos((cos(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))));
	} else if (lambda1 <= -2.25e-5) {
		tmp = R_m * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda1)))));
	} else {
		tmp = R_m * acos((t_0 + ((cos(phi1) * cos(phi2)) * cos(lambda2))));
	}
	return R_s * tmp;
}

Fortran

R_m = abs(R)
R_s = copysign(1.0d0, R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    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) :: tmp
    t_0 = sin(phi1) * sin(phi2)
    if (lambda1 <= (-1.45d+156)) then
        tmp = r_m * acos((cos(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))))
    else if (lambda1 <= (-2.25d-5)) then
        tmp = r_m * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda1)))))
    else
        tmp = r_m * acos((t_0 + ((cos(phi1) * cos(phi2)) * cos(lambda2))))
    end if
    code = r_s * tmp
end function

Java

R_m = Math.abs(R);
R_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (lambda1 <= -1.45e+156) {
		tmp = R_m * Math.acos((Math.cos(phi1) * ((Math.cos(lambda2) * Math.cos(lambda1)) + (Math.sin(lambda1) * Math.sin(lambda2)))));
	} else if (lambda1 <= -2.25e-5) {
		tmp = R_m * Math.acos((t_0 + (Math.cos(phi1) * (Math.cos(phi2) * Math.cos(lambda1)))));
	} else {
		tmp = R_m * Math.acos((t_0 + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos(lambda2))));
	}
	return R_s * tmp;
}

Python

R_m = math.fabs(R)
R_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if lambda1 <= -1.45e+156:
		tmp = R_m * math.acos((math.cos(phi1) * ((math.cos(lambda2) * math.cos(lambda1)) + (math.sin(lambda1) * math.sin(lambda2)))))
	elif lambda1 <= -2.25e-5:
		tmp = R_m * math.acos((t_0 + (math.cos(phi1) * (math.cos(phi2) * math.cos(lambda1)))))
	else:
		tmp = R_m * math.acos((t_0 + ((math.cos(phi1) * math.cos(phi2)) * math.cos(lambda2))))
	return R_s * tmp

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda1 <= -1.45e+156)
		tmp = Float64(R_m * acos(Float64(cos(phi1) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda1) * sin(lambda2))))));
	elseif (lambda1 <= -2.25e-5)
		tmp = Float64(R_m * acos(Float64(t_0 + Float64(cos(phi1) * Float64(cos(phi2) * cos(lambda1))))));
	else
		tmp = Float64(R_m * acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(lambda2)))));
	end
	return Float64(R_s * tmp)
end

MATLAB

R_m = abs(R);
R_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if (lambda1 <= -1.45e+156)
		tmp = R_m * acos((cos(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))));
	elseif (lambda1 <= -2.25e-5)
		tmp = R_m * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda1)))));
	else
		tmp = R_m * acos((t_0 + ((cos(phi1) * cos(phi2)) * cos(lambda2))));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[lambda1, -1.45e+156], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, -2.25e-5], N[(R$95$m * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -1.45000000000000005e156

   1. Initial program 50.7%

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

   3. Simplified 50.8%

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

   5. Taylor expanded in phi2 around 0 33.5%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]
   6. Step-by-step derivation

      1. cos-diff 54.4%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)}\right) \]

      2. +-commutative 54.4%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)}\right) \]

      3. *-commutative 54.4%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\color{blue}{\sin \lambda_1 \cdot \sin \lambda_2} + \cos \lambda_2 \cdot \cos \lambda_1\right)\right) \]

      4. *-commutative 54.4%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \color{blue}{\cos \lambda_1 \cdot \cos \lambda_2}\right)\right) \]

   7. Applied egg-rr 54.4%

      \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \]

   if -1.45000000000000005e156 < lambda1 < -2.25000000000000014e-5

   1. Initial program 68.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0 64.9%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}\right) \cdot R \]
   4. Step-by-step derivation

      1. *-commutative 64.9%

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1}\right) \cdot R \]

      2. associate-*r* 64.9%

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)}\right) \cdot R \]

   5. Simplified 64.9%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)}\right) \cdot R \]

   if -2.25000000000000014e-5 < lambda1

   1. Initial program 75.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around 0 59.2%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(-\lambda_2\right)\right)}\right) \cdot R \]
   4. Step-by-step derivation

      1. cos-neg 59.2%

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{\cos \lambda_2}\right)\right) \cdot R \]

      2. associate-*r* 59.2%

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2}\right) \cdot R \]

      3. *-commutative 59.2%

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \lambda_2 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}\right) \cdot R \]

   5. Simplified 59.2%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \lambda_2 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}\right) \cdot R \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.45 \cdot 10^{+156}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq -2.25 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.00155:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi2 0.00155)
    (*
     R_m
     (acos
      (*
       (cos phi1)
       (+ (* (cos lambda2) (cos lambda1)) (* (sin lambda1) (sin lambda2))))))
    (* R_m (acos (* (cos phi2) (cos (- lambda2 lambda1))))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.00155) {
		tmp = R_m * acos((cos(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))));
	} else {
		tmp = R_m * acos((cos(phi2) * cos((lambda2 - lambda1))));
	}
	return R_s * tmp;
}

Fortran

R_m = abs(R)
R_s = copysign(1.0d0, R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 0.00155d0) then
        tmp = r_m * acos((cos(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))))
    else
        tmp = r_m * acos((cos(phi2) * cos((lambda2 - lambda1))))
    end if
    code = r_s * tmp
end function

Java

R_m = Math.abs(R);
R_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.00155) {
		tmp = R_m * Math.acos((Math.cos(phi1) * ((Math.cos(lambda2) * Math.cos(lambda1)) + (Math.sin(lambda1) * Math.sin(lambda2)))));
	} else {
		tmp = R_m * Math.acos((Math.cos(phi2) * Math.cos((lambda2 - lambda1))));
	}
	return R_s * tmp;
}

Python

R_m = math.fabs(R)
R_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 0.00155:
		tmp = R_m * math.acos((math.cos(phi1) * ((math.cos(lambda2) * math.cos(lambda1)) + (math.sin(lambda1) * math.sin(lambda2)))))
	else:
		tmp = R_m * math.acos((math.cos(phi2) * math.cos((lambda2 - lambda1))))
	return R_s * tmp

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 0.00155)
		tmp = Float64(R_m * acos(Float64(cos(phi1) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = Float64(R_m * acos(Float64(cos(phi2) * cos(Float64(lambda2 - lambda1)))));
	end
	return Float64(R_s * tmp)
end

MATLAB

R_m = abs(R);
R_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 0.00155)
		tmp = R_m * acos((cos(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))));
	else
		tmp = R_m * acos((cos(phi2) * cos((lambda2 - lambda1))));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi2, 0.00155], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.00154999999999999995

   1. Initial program 69.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in phi2 around 0 49.8%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]
   6. Step-by-step derivation

      1. cos-diff 64.7%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)}\right) \]

      2. +-commutative 64.7%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)}\right) \]

      3. *-commutative 64.7%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\color{blue}{\sin \lambda_1 \cdot \sin \lambda_2} + \cos \lambda_2 \cdot \cos \lambda_1\right)\right) \]

      4. *-commutative 64.7%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \color{blue}{\cos \lambda_1 \cdot \cos \lambda_2}\right)\right) \]

   7. Applied egg-rr 64.7%

      \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)}\right) \]

   if 0.00154999999999999995 < phi2

   1. Initial program 78.3%

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

   3. Simplified 78.3%

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

   5. Taylor expanded in phi1 around 0 47.0%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.00155:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 58.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ R_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.3 \cdot 10^{+27}:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (*
    R_s
    (if (<= phi1 -1.3e+27)
      (* R_m (acos (* (cos phi1) (log1p (expm1 t_0)))))
      (* R_m (acos (* (cos phi2) t_0)))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -1.3e+27) {
		tmp = R_m * acos((cos(phi1) * log1p(expm1(t_0))));
	} else {
		tmp = R_m * acos((cos(phi2) * t_0));
	}
	return R_s * tmp;
}

Java

R_m = Math.abs(R);
R_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -1.3e+27) {
		tmp = R_m * Math.acos((Math.cos(phi1) * Math.log1p(Math.expm1(t_0))));
	} else {
		tmp = R_m * Math.acos((Math.cos(phi2) * t_0));
	}
	return R_s * tmp;
}

Python

R_m = math.fabs(R)
R_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	tmp = 0
	if phi1 <= -1.3e+27:
		tmp = R_m * math.acos((math.cos(phi1) * math.log1p(math.expm1(t_0))))
	else:
		tmp = R_m * math.acos((math.cos(phi2) * t_0))
	return R_s * tmp

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi1 <= -1.3e+27)
		tmp = Float64(R_m * acos(Float64(cos(phi1) * log1p(expm1(t_0)))));
	else
		tmp = Float64(R_m * acos(Float64(cos(phi2) * t_0)));
	end
	return Float64(R_s * tmp)
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi1, -1.3e+27], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Log[1 + N[(Exp[t$95$0] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.30000000000000004e27

   1. Initial program 72.2%

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

   3. Simplified 72.2%

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

   5. Taylor expanded in phi2 around 0 44.1%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]
   6. Step-by-step derivation

      1. log1p-expm1-u 44.2%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)\right)}\right) \]

   7. Applied egg-rr 44.2%

      \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)\right)}\right) \]

   if -1.30000000000000004e27 < phi1

   1. Initial program 72.3%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in phi1 around 0 50.9%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.3 \cdot 10^{+27}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 58.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ R_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.3 \cdot 10^{+27}:\\ \;\;\;\;R_m \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(\cos \phi_1 \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (*
    R_s
    (if (<= phi1 -1.3e+27)
      (* R_m (- (/ PI 2.0) (asin (* (cos phi1) t_0))))
      (* R_m (acos (* (cos phi2) t_0)))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -1.3e+27) {
		tmp = R_m * ((((double) M_PI) / 2.0) - asin((cos(phi1) * t_0)));
	} else {
		tmp = R_m * acos((cos(phi2) * t_0));
	}
	return R_s * tmp;
}

Java

R_m = Math.abs(R);
R_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -1.3e+27) {
		tmp = R_m * ((Math.PI / 2.0) - Math.asin((Math.cos(phi1) * t_0)));
	} else {
		tmp = R_m * Math.acos((Math.cos(phi2) * t_0));
	}
	return R_s * tmp;
}

Python

R_m = math.fabs(R)
R_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	tmp = 0
	if phi1 <= -1.3e+27:
		tmp = R_m * ((math.pi / 2.0) - math.asin((math.cos(phi1) * t_0)))
	else:
		tmp = R_m * math.acos((math.cos(phi2) * t_0))
	return R_s * tmp

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi1 <= -1.3e+27)
		tmp = Float64(R_m * Float64(Float64(pi / 2.0) - asin(Float64(cos(phi1) * t_0))));
	else
		tmp = Float64(R_m * acos(Float64(cos(phi2) * t_0)));
	end
	return Float64(R_s * tmp)
end

MATLAB

R_m = abs(R);
R_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	tmp = 0.0;
	if (phi1 <= -1.3e+27)
		tmp = R_m * ((pi / 2.0) - asin((cos(phi1) * t_0)));
	else
		tmp = R_m * acos((cos(phi2) * t_0));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi1, -1.3e+27], N[(R$95$m * N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcSin[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.30000000000000004e27

   1. Initial program 72.2%

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

   3. Simplified 72.2%

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

   5. Taylor expanded in phi2 around 0 44.1%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]
   6. Step-by-step derivation

      1. acos-asin 44.1%

         \[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)} \]

   7. Applied egg-rr 44.1%

      \[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)} \]

   if -1.30000000000000004e27 < phi1

   1. Initial program 72.3%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in phi1 around 0 50.9%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.3 \cdot 10^{+27}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 49.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.014:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \cos^{-1} \cos \left(\phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi2 0.014)
    (* R_m (acos (* (cos phi1) (cos (- lambda2 lambda1)))))
    (* R_m (acos (cos (- phi1 phi2)))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.014) {
		tmp = R_m * acos((cos(phi1) * cos((lambda2 - lambda1))));
	} else {
		tmp = R_m * acos(cos((phi1 - phi2)));
	}
	return R_s * tmp;
}

Fortran

R_m = abs(R)
R_s = copysign(1.0d0, R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 0.014d0) then
        tmp = r_m * acos((cos(phi1) * cos((lambda2 - lambda1))))
    else
        tmp = r_m * acos(cos((phi1 - phi2)))
    end if
    code = r_s * tmp
end function

Java

R_m = Math.abs(R);
R_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.014) {
		tmp = R_m * Math.acos((Math.cos(phi1) * Math.cos((lambda2 - lambda1))));
	} else {
		tmp = R_m * Math.acos(Math.cos((phi1 - phi2)));
	}
	return R_s * tmp;
}

Python

R_m = math.fabs(R)
R_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 0.014:
		tmp = R_m * math.acos((math.cos(phi1) * math.cos((lambda2 - lambda1))))
	else:
		tmp = R_m * math.acos(math.cos((phi1 - phi2)))
	return R_s * tmp

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 0.014)
		tmp = Float64(R_m * acos(Float64(cos(phi1) * cos(Float64(lambda2 - lambda1)))));
	else
		tmp = Float64(R_m * acos(cos(Float64(phi1 - phi2))));
	end
	return Float64(R_s * tmp)
end

MATLAB

R_m = abs(R);
R_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 0.014)
		tmp = R_m * acos((cos(phi1) * cos((lambda2 - lambda1))));
	else
		tmp = R_m * acos(cos((phi1 - phi2)));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi2, 0.014], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.0140000000000000003

   1. Initial program 69.9%

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

   3. Simplified 69.9%

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

   5. Taylor expanded in phi2 around 0 49.8%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]

   if 0.0140000000000000003 < phi2

   1. Initial program 78.3%

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

   3. Simplified 78.3%

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

   5. Taylor expanded in lambda2 around 0 60.8%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \left(-\lambda_1\right)}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg 60.8%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]

   7. Simplified 60.8%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]

   8. Taylor expanded in lambda1 around 0 39.7%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 39.6%

         \[\leadsto R \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)} \]

      2. expm1-udef 39.6%

         \[\leadsto R \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)\right)} - 1\right)} \]

      3. cos-diff 29.3%

         \[\leadsto R \cdot \left(e^{\mathsf{log1p}\left(\cos^{-1} \color{blue}{\cos \left(\phi_1 - \phi_2\right)}\right)} - 1\right) \]

   10. Applied egg-rr 29.3%

       \[\leadsto R \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos^{-1} \cos \left(\phi_1 - \phi_2\right)\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 29.3%

          \[\leadsto R \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \cos \left(\phi_1 - \phi_2\right)\right)\right)} \]

       2. expm1-log1p 29.4%

          \[\leadsto R \cdot \color{blue}{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)} \]

   12. Simplified 29.4%

       \[\leadsto R \cdot \color{blue}{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.014:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 58.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ R_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.3 \cdot 10^{+27}:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (*
    R_s
    (if (<= phi1 -1.3e+27)
      (* R_m (acos (* (cos phi1) t_0)))
      (* R_m (acos (* (cos phi2) t_0)))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -1.3e+27) {
		tmp = R_m * acos((cos(phi1) * t_0));
	} else {
		tmp = R_m * acos((cos(phi2) * t_0));
	}
	return R_s * tmp;
}

Fortran

R_m = abs(R)
R_s = copysign(1.0d0, R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    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) :: tmp
    t_0 = cos((lambda2 - lambda1))
    if (phi1 <= (-1.3d+27)) then
        tmp = r_m * acos((cos(phi1) * t_0))
    else
        tmp = r_m * acos((cos(phi2) * t_0))
    end if
    code = r_s * tmp
end function

Java

R_m = Math.abs(R);
R_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -1.3e+27) {
		tmp = R_m * Math.acos((Math.cos(phi1) * t_0));
	} else {
		tmp = R_m * Math.acos((Math.cos(phi2) * t_0));
	}
	return R_s * tmp;
}

Python

R_m = math.fabs(R)
R_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	tmp = 0
	if phi1 <= -1.3e+27:
		tmp = R_m * math.acos((math.cos(phi1) * t_0))
	else:
		tmp = R_m * math.acos((math.cos(phi2) * t_0))
	return R_s * tmp

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi1 <= -1.3e+27)
		tmp = Float64(R_m * acos(Float64(cos(phi1) * t_0)));
	else
		tmp = Float64(R_m * acos(Float64(cos(phi2) * t_0)));
	end
	return Float64(R_s * tmp)
end

MATLAB

R_m = abs(R);
R_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	tmp = 0.0;
	if (phi1 <= -1.3e+27)
		tmp = R_m * acos((cos(phi1) * t_0));
	else
		tmp = R_m * acos((cos(phi2) * t_0));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi1, -1.3e+27], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.30000000000000004e27

   1. Initial program 72.2%

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

   3. Simplified 72.2%

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

   5. Taylor expanded in phi2 around 0 44.1%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]

   if -1.30000000000000004e27 < phi1

   1. Initial program 72.3%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in phi1 around 0 50.9%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.3 \cdot 10^{+27}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 34.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 0.07:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda2 0.07)
    (* R_m (acos (* (cos phi1) (cos lambda1))))
    (* R_m (acos (cos lambda2))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 0.07) {
		tmp = R_m * acos((cos(phi1) * cos(lambda1)));
	} else {
		tmp = R_m * acos(cos(lambda2));
	}
	return R_s * tmp;
}

Fortran

R_m = abs(R)
R_s = copysign(1.0d0, R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 0.07d0) then
        tmp = r_m * acos((cos(phi1) * cos(lambda1)))
    else
        tmp = r_m * acos(cos(lambda2))
    end if
    code = r_s * tmp
end function

Java

R_m = Math.abs(R);
R_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 0.07) {
		tmp = R_m * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R_m * Math.acos(Math.cos(lambda2));
	}
	return R_s * tmp;
}

Python

R_m = math.fabs(R)
R_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 0.07:
		tmp = R_m * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R_m * math.acos(math.cos(lambda2))
	return R_s * tmp

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 0.07)
		tmp = Float64(R_m * acos(Float64(cos(phi1) * cos(lambda1))));
	else
		tmp = Float64(R_m * acos(cos(lambda2)));
	end
	return Float64(R_s * tmp)
end

MATLAB

R_m = abs(R);
R_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 0.07)
		tmp = R_m * acos((cos(phi1) * cos(lambda1)));
	else
		tmp = R_m * acos(cos(lambda2));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda2, 0.07], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[Cos[lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 0.070000000000000007

   1. Initial program 77.0%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in phi2 around 0 42.1%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]

   6. Taylor expanded in lambda2 around 0 34.4%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(-\lambda_1\right)\right)} \]
   7. Step-by-step derivation

      1. cos-neg 34.4%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \lambda_1}\right) \]

   8. Simplified 34.4%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \lambda_1\right)} \]

   if 0.070000000000000007 < lambda2

   1. Initial program 57.4%

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

   3. Simplified 57.4%

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

   5. Taylor expanded in phi2 around 0 37.6%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]

   6. Taylor expanded in phi1 around 0 26.9%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \]

   7. Taylor expanded in lambda1 around 0 27.4%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \lambda_2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 0.07:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 37.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.0036:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda1 -0.0036)
    (* R_m (acos (* (cos phi1) (cos lambda1))))
    (* R_m (acos (* (cos phi1) (cos lambda2)))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -0.0036) {
		tmp = R_m * acos((cos(phi1) * cos(lambda1)));
	} else {
		tmp = R_m * acos((cos(phi1) * cos(lambda2)));
	}
	return R_s * tmp;
}

Fortran

R_m = abs(R)
R_s = copysign(1.0d0, R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-0.0036d0)) then
        tmp = r_m * acos((cos(phi1) * cos(lambda1)))
    else
        tmp = r_m * acos((cos(phi1) * cos(lambda2)))
    end if
    code = r_s * tmp
end function

Java

R_m = Math.abs(R);
R_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -0.0036) {
		tmp = R_m * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R_m * Math.acos((Math.cos(phi1) * Math.cos(lambda2)));
	}
	return R_s * tmp;
}

Python

R_m = math.fabs(R)
R_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -0.0036:
		tmp = R_m * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R_m * math.acos((math.cos(phi1) * math.cos(lambda2)))
	return R_s * tmp

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -0.0036)
		tmp = Float64(R_m * acos(Float64(cos(phi1) * cos(lambda1))));
	else
		tmp = Float64(R_m * acos(Float64(cos(phi1) * cos(lambda2))));
	end
	return Float64(R_s * tmp)
end

MATLAB

R_m = abs(R);
R_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -0.0036)
		tmp = R_m * acos((cos(phi1) * cos(lambda1)));
	else
		tmp = R_m * acos((cos(phi1) * cos(lambda2)));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda1, -0.0036], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -0.0035999999999999999

   1. Initial program 59.1%

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

   3. Simplified 59.1%

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

   5. Taylor expanded in phi2 around 0 37.7%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]

   6. Taylor expanded in lambda2 around 0 37.9%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(-\lambda_1\right)\right)} \]
   7. Step-by-step derivation

      1. cos-neg 37.9%

         \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \color{blue}{\cos \lambda_1}\right) \]

   8. Simplified 37.9%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \lambda_1\right)} \]

   if -0.0035999999999999999 < lambda1

   1. Initial program 76.0%

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

   3. Simplified 76.0%

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

   5. Taylor expanded in phi2 around 0 41.9%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]

   6. Taylor expanded in lambda1 around 0 32.3%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \lambda_2 \cdot \cos \phi_1\right)} \]
   7. Step-by-step derivation

      1. *-commutative 32.3%

         \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \lambda_2\right)} \]

   8. Simplified 32.3%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \lambda_2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.0036:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 33.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.00032:\\ \;\;\;\;R_m \cdot \left(\frac{\pi}{2} - \sin^{-1} \cos \left(\phi_1 - \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi1 -0.00032)
    (* R_m (- (/ PI 2.0) (asin (cos (- phi1 phi2)))))
    (* R_m (acos (cos (- lambda2 lambda1)))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.00032) {
		tmp = R_m * ((((double) M_PI) / 2.0) - asin(cos((phi1 - phi2))));
	} else {
		tmp = R_m * acos(cos((lambda2 - lambda1)));
	}
	return R_s * tmp;
}

Java

R_m = Math.abs(R);
R_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.00032) {
		tmp = R_m * ((Math.PI / 2.0) - Math.asin(Math.cos((phi1 - phi2))));
	} else {
		tmp = R_m * Math.acos(Math.cos((lambda2 - lambda1)));
	}
	return R_s * tmp;
}

Python

R_m = math.fabs(R)
R_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -0.00032:
		tmp = R_m * ((math.pi / 2.0) - math.asin(math.cos((phi1 - phi2))))
	else:
		tmp = R_m * math.acos(math.cos((lambda2 - lambda1)))
	return R_s * tmp

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -0.00032)
		tmp = Float64(R_m * Float64(Float64(pi / 2.0) - asin(cos(Float64(phi1 - phi2)))));
	else
		tmp = Float64(R_m * acos(cos(Float64(lambda2 - lambda1))));
	end
	return Float64(R_s * tmp)
end

MATLAB

R_m = abs(R);
R_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -0.00032)
		tmp = R_m * ((pi / 2.0) - asin(cos((phi1 - phi2))));
	else
		tmp = R_m * acos(cos((lambda2 - lambda1)));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi1, -0.00032], N[(R$95$m * N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcSin[N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.20000000000000026e-4

   1. Initial program 74.0%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in lambda2 around 0 55.0%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \left(-\lambda_1\right)}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg 55.0%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]

   7. Simplified 55.0%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]

   8. Taylor expanded in lambda1 around 0 35.4%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)} \]
   9. Step-by-step derivation

      1. acos-asin 35.5%

         \[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)\right)} \]

      2. cos-diff 26.6%

         \[\leadsto R \cdot \left(\frac{\pi}{2} - \sin^{-1} \color{blue}{\cos \left(\phi_1 - \phi_2\right)}\right) \]

   10. Applied egg-rr 26.6%

       \[\leadsto R \cdot \color{blue}{\left(\frac{\pi}{2} - \sin^{-1} \cos \left(\phi_1 - \phi_2\right)\right)} \]

   if -3.20000000000000026e-4 < phi1

   1. Initial program 71.5%

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

   3. Simplified 71.5%

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

   5. Taylor expanded in phi2 around 0 40.3%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]

   6. Taylor expanded in phi1 around 0 30.5%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.00032:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \cos \left(\phi_1 - \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 22.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 3.4 \cdot 10^{-227}:\\ \;\;\;\;R_m \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{elif}\;\lambda_2 \leq 8.5 \cdot 10^{-8}:\\ \;\;\;\;R_m \cdot \cos^{-1} \cos \phi_1\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda2 3.4e-227)
    (* R_m (acos (cos lambda1)))
    (if (<= lambda2 8.5e-8)
      (* R_m (acos (cos phi1)))
      (* R_m (acos (cos lambda2)))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 3.4e-227) {
		tmp = R_m * acos(cos(lambda1));
	} else if (lambda2 <= 8.5e-8) {
		tmp = R_m * acos(cos(phi1));
	} else {
		tmp = R_m * acos(cos(lambda2));
	}
	return R_s * tmp;
}

Fortran

R_m = abs(R)
R_s = copysign(1.0d0, R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 3.4d-227) then
        tmp = r_m * acos(cos(lambda1))
    else if (lambda2 <= 8.5d-8) then
        tmp = r_m * acos(cos(phi1))
    else
        tmp = r_m * acos(cos(lambda2))
    end if
    code = r_s * tmp
end function

Java

R_m = Math.abs(R);
R_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 3.4e-227) {
		tmp = R_m * Math.acos(Math.cos(lambda1));
	} else if (lambda2 <= 8.5e-8) {
		tmp = R_m * Math.acos(Math.cos(phi1));
	} else {
		tmp = R_m * Math.acos(Math.cos(lambda2));
	}
	return R_s * tmp;
}

Python

R_m = math.fabs(R)
R_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 3.4e-227:
		tmp = R_m * math.acos(math.cos(lambda1))
	elif lambda2 <= 8.5e-8:
		tmp = R_m * math.acos(math.cos(phi1))
	else:
		tmp = R_m * math.acos(math.cos(lambda2))
	return R_s * tmp

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 3.4e-227)
		tmp = Float64(R_m * acos(cos(lambda1)));
	elseif (lambda2 <= 8.5e-8)
		tmp = Float64(R_m * acos(cos(phi1)));
	else
		tmp = Float64(R_m * acos(cos(lambda2)));
	end
	return Float64(R_s * tmp)
end

MATLAB

R_m = abs(R);
R_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 3.4e-227)
		tmp = R_m * acos(cos(lambda1));
	elseif (lambda2 <= 8.5e-8)
		tmp = R_m * acos(cos(phi1));
	else
		tmp = R_m * acos(cos(lambda2));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda2, 3.4e-227], N[(R$95$m * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 8.5e-8], N[(R$95$m * N[ArcCos[N[Cos[phi1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[Cos[lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if lambda2 < 3.39999999999999979e-227

   1. Initial program 73.1%

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

   3. Simplified 73.2%

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

   5. Taylor expanded in phi2 around 0 42.3%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]

   6. Taylor expanded in phi1 around 0 29.0%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \]

   7. Taylor expanded in lambda2 around 0 21.0%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \left(-\lambda_1\right)} \]
   8. Step-by-step derivation

      1. cos-neg 54.9%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]

   9. Simplified 21.0%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \lambda_1} \]

   if 3.39999999999999979e-227 < lambda2 < 8.49999999999999935e-8

   1. Initial program 90.5%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in lambda2 around 0 90.5%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \left(-\lambda_1\right)}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg 90.5%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]

   7. Simplified 90.5%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]

   8. Taylor expanded in lambda1 around 0 51.1%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)} \]

   9. Taylor expanded in phi2 around 0 23.1%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \phi_1} \]

   if 8.49999999999999935e-8 < lambda2

   1. Initial program 57.1%

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

   3. Simplified 57.0%

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

   5. Taylor expanded in phi2 around 0 37.3%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]

   6. Taylor expanded in phi1 around 0 26.7%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \]

   7. Taylor expanded in lambda1 around 0 26.9%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \lambda_2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 22.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 3.4 \cdot 10^{-227}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{elif}\;\lambda_2 \leq 8.5 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 33.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1020000:\\ \;\;\;\;R_m \cdot \cos^{-1} \cos \phi_1\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi1 -1020000.0)
    (* R_m (acos (cos phi1)))
    (* R_m (acos (cos (- lambda2 lambda1)))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1020000.0) {
		tmp = R_m * acos(cos(phi1));
	} else {
		tmp = R_m * acos(cos((lambda2 - lambda1)));
	}
	return R_s * tmp;
}

Fortran

R_m = abs(R)
R_s = copysign(1.0d0, R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-1020000.0d0)) then
        tmp = r_m * acos(cos(phi1))
    else
        tmp = r_m * acos(cos((lambda2 - lambda1)))
    end if
    code = r_s * tmp
end function

Java

R_m = Math.abs(R);
R_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1020000.0) {
		tmp = R_m * Math.acos(Math.cos(phi1));
	} else {
		tmp = R_m * Math.acos(Math.cos((lambda2 - lambda1)));
	}
	return R_s * tmp;
}

Python

R_m = math.fabs(R)
R_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1020000.0:
		tmp = R_m * math.acos(math.cos(phi1))
	else:
		tmp = R_m * math.acos(math.cos((lambda2 - lambda1)))
	return R_s * tmp

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1020000.0)
		tmp = Float64(R_m * acos(cos(phi1)));
	else
		tmp = Float64(R_m * acos(cos(Float64(lambda2 - lambda1))));
	end
	return Float64(R_s * tmp)
end

MATLAB

R_m = abs(R);
R_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1020000.0)
		tmp = R_m * acos(cos(phi1));
	else
		tmp = R_m * acos(cos((lambda2 - lambda1)));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi1, -1020000.0], N[(R$95$m * N[ArcCos[N[Cos[phi1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.02e6

   1. Initial program 74.0%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in lambda2 around 0 55.0%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \left(-\lambda_1\right)}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg 55.0%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]

   7. Simplified 55.0%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]

   8. Taylor expanded in lambda1 around 0 35.4%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)} \]

   9. Taylor expanded in phi2 around 0 26.6%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \phi_1} \]

   if -1.02e6 < phi1

   1. Initial program 71.5%

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

   3. Simplified 71.5%

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

   5. Taylor expanded in phi2 around 0 40.3%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]

   6. Taylor expanded in phi1 around 0 30.5%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1020000:\\ \;\;\;\;R \cdot \cos^{-1} \cos \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 33.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.3 \cdot 10^{-5}:\\ \;\;\;\;R_m \cdot \cos^{-1} \cos \left(\phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi1 -2.3e-5)
    (* R_m (acos (cos (- phi1 phi2))))
    (* R_m (acos (cos (- lambda2 lambda1)))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.3e-5) {
		tmp = R_m * acos(cos((phi1 - phi2)));
	} else {
		tmp = R_m * acos(cos((lambda2 - lambda1)));
	}
	return R_s * tmp;
}

Fortran

R_m = abs(R)
R_s = copysign(1.0d0, R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-2.3d-5)) then
        tmp = r_m * acos(cos((phi1 - phi2)))
    else
        tmp = r_m * acos(cos((lambda2 - lambda1)))
    end if
    code = r_s * tmp
end function

Java

R_m = Math.abs(R);
R_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.3e-5) {
		tmp = R_m * Math.acos(Math.cos((phi1 - phi2)));
	} else {
		tmp = R_m * Math.acos(Math.cos((lambda2 - lambda1)));
	}
	return R_s * tmp;
}

Python

R_m = math.fabs(R)
R_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -2.3e-5:
		tmp = R_m * math.acos(math.cos((phi1 - phi2)))
	else:
		tmp = R_m * math.acos(math.cos((lambda2 - lambda1)))
	return R_s * tmp

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -2.3e-5)
		tmp = Float64(R_m * acos(cos(Float64(phi1 - phi2))));
	else
		tmp = Float64(R_m * acos(cos(Float64(lambda2 - lambda1))));
	end
	return Float64(R_s * tmp)
end

MATLAB

R_m = abs(R);
R_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -2.3e-5)
		tmp = R_m * acos(cos((phi1 - phi2)));
	else
		tmp = R_m * acos(cos((lambda2 - lambda1)));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi1, -2.3e-5], N[(R$95$m * N[ArcCos[N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.3e-5

   1. Initial program 74.0%

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

   3. Simplified 73.9%

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

   5. Taylor expanded in lambda2 around 0 55.0%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \left(-\lambda_1\right)}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg 55.0%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]

   7. Simplified 55.0%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]

   8. Taylor expanded in lambda1 around 0 35.4%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 35.4%

         \[\leadsto R \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)} \]

      2. expm1-udef 35.4%

         \[\leadsto R \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)\right)} - 1\right)} \]

      3. cos-diff 26.5%

         \[\leadsto R \cdot \left(e^{\mathsf{log1p}\left(\cos^{-1} \color{blue}{\cos \left(\phi_1 - \phi_2\right)}\right)} - 1\right) \]

   10. Applied egg-rr 26.5%

       \[\leadsto R \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\cos^{-1} \cos \left(\phi_1 - \phi_2\right)\right)} - 1\right)} \]
   11. Step-by-step derivation

       1. expm1-def 26.6%

          \[\leadsto R \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \cos \left(\phi_1 - \phi_2\right)\right)\right)} \]

       2. expm1-log1p 26.6%

          \[\leadsto R \cdot \color{blue}{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)} \]

   12. Simplified 26.6%

       \[\leadsto R \cdot \color{blue}{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)} \]

   if -2.3e-5 < phi1

   1. Initial program 71.5%

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

   3. Simplified 71.5%

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

   5. Taylor expanded in phi2 around 0 40.3%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]

   6. Taylor expanded in phi1 around 0 30.5%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.3 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 15.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.05 \cdot 10^{-8}:\\ \;\;\;\;R_m \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\left|R_m \cdot \left(\phi_1 - \phi_2\right)\right|\\ \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda1 -1.05e-8)
    (* R_m (acos (cos lambda1)))
    (fabs (* R_m (- phi1 phi2))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.05e-8) {
		tmp = R_m * acos(cos(lambda1));
	} else {
		tmp = fabs((R_m * (phi1 - phi2)));
	}
	return R_s * tmp;
}

Fortran

R_m = abs(R)
R_s = copysign(1.0d0, R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-1.05d-8)) then
        tmp = r_m * acos(cos(lambda1))
    else
        tmp = abs((r_m * (phi1 - phi2)))
    end if
    code = r_s * tmp
end function

Java

R_m = Math.abs(R);
R_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.05e-8) {
		tmp = R_m * Math.acos(Math.cos(lambda1));
	} else {
		tmp = Math.abs((R_m * (phi1 - phi2)));
	}
	return R_s * tmp;
}

Python

R_m = math.fabs(R)
R_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -1.05e-8:
		tmp = R_m * math.acos(math.cos(lambda1))
	else:
		tmp = math.fabs((R_m * (phi1 - phi2)))
	return R_s * tmp

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.05e-8)
		tmp = Float64(R_m * acos(cos(lambda1)));
	else
		tmp = abs(Float64(R_m * Float64(phi1 - phi2)));
	end
	return Float64(R_s * tmp)
end

MATLAB

R_m = abs(R);
R_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -1.05e-8)
		tmp = R_m * acos(cos(lambda1));
	else
		tmp = abs((R_m * (phi1 - phi2)));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda1, -1.05e-8], N[(R$95$m * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Abs[N[(R$95$m * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.04999999999999997e-8

   1. Initial program 59.7%

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

   3. Simplified 59.7%

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

   5. Taylor expanded in phi2 around 0 38.7%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]

   6. Taylor expanded in phi1 around 0 31.4%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \]

   7. Taylor expanded in lambda2 around 0 29.8%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \left(-\lambda_1\right)} \]
   8. Step-by-step derivation

      1. cos-neg 57.9%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]

   9. Simplified 29.8%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \lambda_1} \]

   if -1.04999999999999997e-8 < lambda1

   1. Initial program 75.9%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in lambda2 around 0 51.4%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \left(-\lambda_1\right)}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]
   6. Step-by-step derivation

      1. cos-neg 51.4%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]

   7. Simplified 51.4%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]

   8. Taylor expanded in lambda1 around 0 34.0%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)} \]
   9. Step-by-step derivation

      1. add-sqr-sqrt 33.9%

         \[\leadsto R \cdot \color{blue}{\left(\sqrt{\cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot \sqrt{\cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)}\right)} \]

      2. pow2 33.9%

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

      3. cos-diff 27.5%

         \[\leadsto R \cdot {\left(\sqrt{\cos^{-1} \color{blue}{\cos \left(\phi_1 - \phi_2\right)}}\right)}^{2} \]

   10. Applied egg-rr 27.5%

       \[\leadsto R \cdot \color{blue}{{\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}} \]
   11. Step-by-step derivation

       1. rem-square-sqrt 13.2%

          \[\leadsto \color{blue}{\sqrt{R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}} \cdot \sqrt{R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}}} \]

       2. sqrt-unprod 9.0%

          \[\leadsto \color{blue}{\sqrt{\left(R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}\right) \cdot \left(R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}\right)}} \]

       3. pow2 9.0%

          \[\leadsto \sqrt{\color{blue}{{\left(R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}\right)}^{2}}} \]

       4. unpow2 9.0%

          \[\leadsto \sqrt{{\left(R \cdot \color{blue}{\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)} \cdot \sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}\right)}^{2}} \]

       5. add-sqr-sqrt 9.0%

          \[\leadsto \sqrt{{\left(R \cdot \color{blue}{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}} \]

       6. acos-cos-s 4.1%

          \[\leadsto \sqrt{{\left(R \cdot \color{blue}{\left(\phi_1 - \phi_2\right)}\right)}^{2}} \]

   12. Applied egg-rr 4.1%

       \[\leadsto \color{blue}{\sqrt{{\left(R \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}} \]
   13. Step-by-step derivation

       1. unpow2 4.1%

          \[\leadsto \sqrt{\color{blue}{\left(R \cdot \left(\phi_1 - \phi_2\right)\right) \cdot \left(R \cdot \left(\phi_1 - \phi_2\right)\right)}} \]

       2. rem-sqrt-square 4.7%

          \[\leadsto \color{blue}{\left|R \cdot \left(\phi_1 - \phi_2\right)\right|} \]

   14. Simplified 4.7%

       \[\leadsto \color{blue}{\left|R \cdot \left(\phi_1 - \phi_2\right)\right|} \]
3. Recombined 2 regimes into one program.

4. Final simplification 10.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.05 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\left|R \cdot \left(\phi_1 - \phi_2\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 22.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 0.02:\\ \;\;\;\;R_m \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R_m \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda2 0.02)
    (* R_m (acos (cos lambda1)))
    (* R_m (acos (cos lambda2))))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 0.02) {
		tmp = R_m * acos(cos(lambda1));
	} else {
		tmp = R_m * acos(cos(lambda2));
	}
	return R_s * tmp;
}

Fortran

R_m = abs(R)
R_s = copysign(1.0d0, R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 0.02d0) then
        tmp = r_m * acos(cos(lambda1))
    else
        tmp = r_m * acos(cos(lambda2))
    end if
    code = r_s * tmp
end function

Java

R_m = Math.abs(R);
R_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 0.02) {
		tmp = R_m * Math.acos(Math.cos(lambda1));
	} else {
		tmp = R_m * Math.acos(Math.cos(lambda2));
	}
	return R_s * tmp;
}

Python

R_m = math.fabs(R)
R_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 0.02:
		tmp = R_m * math.acos(math.cos(lambda1))
	else:
		tmp = R_m * math.acos(math.cos(lambda2))
	return R_s * tmp

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 0.02)
		tmp = Float64(R_m * acos(cos(lambda1)));
	else
		tmp = Float64(R_m * acos(cos(lambda2)));
	end
	return Float64(R_s * tmp)
end

MATLAB

R_m = abs(R);
R_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 0.02)
		tmp = R_m * acos(cos(lambda1));
	else
		tmp = R_m * acos(cos(lambda2));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda2, 0.02], N[(R$95$m * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[Cos[lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 0.0200000000000000004

   1. Initial program 77.0%

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

   3. Simplified 77.0%

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

   5. Taylor expanded in phi2 around 0 42.1%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]

   6. Taylor expanded in phi1 around 0 26.5%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \]

   7. Taylor expanded in lambda2 around 0 20.5%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \left(-\lambda_1\right)} \]
   8. Step-by-step derivation

      1. cos-neg 63.3%

         \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]

   9. Simplified 20.5%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \lambda_1} \]

   if 0.0200000000000000004 < lambda2

   1. Initial program 57.4%

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

   3. Simplified 57.4%

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

   5. Taylor expanded in phi2 around 0 37.6%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]

   6. Taylor expanded in phi1 around 0 26.9%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \]

   7. Taylor expanded in lambda1 around 0 27.4%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \lambda_2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 0.02:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 8.9% accurate, 5.8× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R_s \cdot \left|R_m \cdot \left(\phi_1 - \phi_2\right)\right| \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R_s (fabs (* R_m (- phi1 phi2)))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * fabs((R_m * (phi1 - phi2)));
}

Fortran

R_m = abs(R)
R_s = copysign(1.0d0, R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r_s * abs((r_m * (phi1 - phi2)))
end function

Java

R_m = Math.abs(R);
R_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * Math.abs((R_m * (phi1 - phi2)));
}

Python

R_m = math.fabs(R)
R_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	return R_s * math.fabs((R_m * (phi1 - phi2)))

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	return Float64(R_s * abs(Float64(R_m * Float64(phi1 - phi2))))
end

MATLAB

R_m = abs(R);
R_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = R_s * abs((R_m * (phi1 - phi2)));
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * N[Abs[N[(R$95$m * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.2%

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

3. Simplified 72.2%

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

5. Taylor expanded in lambda2 around 0 52.8%

   \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \left(-\lambda_1\right)}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]
6. Step-by-step derivation

   1. cos-neg 52.8%

      \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]

7. Simplified 52.8%

   \[\leadsto R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \color{blue}{\cos \lambda_1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \]

8. Taylor expanded in lambda1 around 0 30.3%

   \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)} \]
9. Step-by-step derivation

   1. add-sqr-sqrt 30.3%

      \[\leadsto R \cdot \color{blue}{\left(\sqrt{\cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot \sqrt{\cos^{-1} \left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)}\right)} \]

   2. pow2 30.3%

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

   3. cos-diff 25.1%

      \[\leadsto R \cdot {\left(\sqrt{\cos^{-1} \color{blue}{\cos \left(\phi_1 - \phi_2\right)}}\right)}^{2} \]

10. Applied egg-rr 25.1%

    \[\leadsto R \cdot \color{blue}{{\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}} \]
11. Step-by-step derivation

    1. rem-square-sqrt 12.3%

       \[\leadsto \color{blue}{\sqrt{R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}} \cdot \sqrt{R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}}} \]

    2. sqrt-unprod 8.8%

       \[\leadsto \color{blue}{\sqrt{\left(R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}\right) \cdot \left(R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}\right)}} \]

    3. pow2 8.8%

       \[\leadsto \sqrt{\color{blue}{{\left(R \cdot {\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}\right)}^{2}}} \]

    4. unpow2 8.8%

       \[\leadsto \sqrt{{\left(R \cdot \color{blue}{\left(\sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)} \cdot \sqrt{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}\right)}^{2}} \]

    5. add-sqr-sqrt 8.8%

       \[\leadsto \sqrt{{\left(R \cdot \color{blue}{\cos^{-1} \cos \left(\phi_1 - \phi_2\right)}\right)}^{2}} \]

    6. acos-cos-s 4.1%

       \[\leadsto \sqrt{{\left(R \cdot \color{blue}{\left(\phi_1 - \phi_2\right)}\right)}^{2}} \]

12. Applied egg-rr 4.1%

    \[\leadsto \color{blue}{\sqrt{{\left(R \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}} \]
13. Step-by-step derivation

    1. unpow2 4.1%

       \[\leadsto \sqrt{\color{blue}{\left(R \cdot \left(\phi_1 - \phi_2\right)\right) \cdot \left(R \cdot \left(\phi_1 - \phi_2\right)\right)}} \]

    2. rem-sqrt-square 4.6%

       \[\leadsto \color{blue}{\left|R \cdot \left(\phi_1 - \phi_2\right)\right|} \]

14. Simplified 4.6%

    \[\leadsto \color{blue}{\left|R \cdot \left(\phi_1 - \phi_2\right)\right|} \]

15. Final simplification 4.6%

    \[\leadsto \left|R \cdot \left(\phi_1 - \phi_2\right)\right| \]
16. Add Preprocessing

Alternative 28: 6.3% accurate, 68.0× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.8 \cdot 10^{-87}:\\ \;\;\;\;\lambda_1 \cdot \left(-R_m\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R_m\\ \end{array} \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R_s (if (<= lambda2 2.8e-87) (* lambda1 (- R_m)) (* lambda2 R_m))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 2.8e-87) {
		tmp = lambda1 * -R_m;
	} else {
		tmp = lambda2 * R_m;
	}
	return R_s * tmp;
}

Fortran

R_m = abs(R)
R_s = copysign(1.0d0, R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 2.8d-87) then
        tmp = lambda1 * -r_m
    else
        tmp = lambda2 * r_m
    end if
    code = r_s * tmp
end function

Java

R_m = Math.abs(R);
R_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 2.8e-87) {
		tmp = lambda1 * -R_m;
	} else {
		tmp = lambda2 * R_m;
	}
	return R_s * tmp;
}

Python

R_m = math.fabs(R)
R_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 2.8e-87:
		tmp = lambda1 * -R_m
	else:
		tmp = lambda2 * R_m
	return R_s * tmp

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 2.8e-87)
		tmp = Float64(lambda1 * Float64(-R_m));
	else
		tmp = Float64(lambda2 * R_m);
	end
	return Float64(R_s * tmp)
end

MATLAB

R_m = abs(R);
R_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 2.8e-87)
		tmp = lambda1 * -R_m;
	else
		tmp = lambda2 * R_m;
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda2, 2.8e-87], N[(lambda1 * (-R$95$m)), $MachinePrecision], N[(lambda2 * R$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 2.8000000000000001e-87

   1. Initial program 75.9%

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

   3. Simplified 75.9%

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

   5. Taylor expanded in phi2 around 0 41.5%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]

   6. Taylor expanded in phi1 around 0 26.6%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \]

   7. Taylor expanded in lambda2 around 0 4.5%

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \lambda_1\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 4.5%

         \[\leadsto \color{blue}{-R \cdot \lambda_1} \]

      2. *-commutative 4.5%

         \[\leadsto -\color{blue}{\lambda_1 \cdot R} \]

      3. distribute-rgt-neg-in 4.5%

         \[\leadsto \color{blue}{\lambda_1 \cdot \left(-R\right)} \]

   9. Simplified 4.5%

      \[\leadsto \color{blue}{\lambda_1 \cdot \left(-R\right)} \]

   if 2.8000000000000001e-87 < lambda2

   1. Initial program 64.7%

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

   3. Simplified 64.7%

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

   5. Taylor expanded in phi2 around 0 39.9%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]

   6. Taylor expanded in phi1 around 0 26.4%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \]

   7. Taylor expanded in lambda2 around inf 8.7%

      \[\leadsto \color{blue}{R \cdot \lambda_2} \]
   8. Step-by-step derivation

      1. *-commutative 8.7%

         \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

   9. Simplified 8.7%

      \[\leadsto \color{blue}{\lambda_2 \cdot R} \]
3. Recombined 2 regimes into one program.

4. Final simplification 5.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.8 \cdot 10^{-87}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 5.6% accurate, 122.6× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R_s \cdot \left(R_m \cdot \left(\lambda_2 - \lambda_1\right)\right) \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R_s (* R_m (- lambda2 lambda1))))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (R_m * (lambda2 - lambda1));
}

Fortran

R_m = abs(R)
R_s = copysign(1.0d0, R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r_s * (r_m * (lambda2 - lambda1))
end function

Java

R_m = Math.abs(R);
R_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (R_m * (lambda2 - lambda1));
}

Python

R_m = math.fabs(R)
R_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	return R_s * (R_m * (lambda2 - lambda1))

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	return Float64(R_s * Float64(R_m * Float64(lambda2 - lambda1)))
end

MATLAB

R_m = abs(R);
R_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = R_s * (R_m * (lambda2 - lambda1));
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * N[(R$95$m * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.2%

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

3. Simplified 72.2%

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

5. Taylor expanded in phi2 around 0 41.0%

   \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]

6. Taylor expanded in phi1 around 0 26.6%

   \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \]

7. Taylor expanded in lambda2 around 0 4.8%

   \[\leadsto R \cdot \color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)} \]
8. Step-by-step derivation

   1. neg-mul-1 4.8%

      \[\leadsto R \cdot \left(\lambda_2 + \color{blue}{\left(-\lambda_1\right)}\right) \]

   2. sub-neg 4.8%

      \[\leadsto R \cdot \color{blue}{\left(\lambda_2 - \lambda_1\right)} \]

9. Simplified 4.8%

   \[\leadsto R \cdot \color{blue}{\left(\lambda_2 - \lambda_1\right)} \]

10. Final simplification 4.8%

    \[\leadsto R \cdot \left(\lambda_2 - \lambda_1\right) \]
11. Add Preprocessing

Alternative 30: 5.2% accurate, 204.3× speedup

Math

\[\begin{array}{l} R_m = \left|R\right| \\ R_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R_s \cdot \left(\lambda_2 \cdot R_m\right) \end{array} \]

FPCore

R_m = (fabs.f64 R)
R_s = (copysign.f64 1 R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R_s (* lambda2 R_m)))

C

R_m = fabs(R);
R_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (lambda2 * R_m);
}

Fortran

R_m = abs(R)
R_s = copysign(1.0d0, R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r_s * (lambda2 * r_m)
end function

Java

R_m = Math.abs(R);
R_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (lambda2 * R_m);
}

Python

R_m = math.fabs(R)
R_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	return R_s * (lambda2 * R_m)

Julia

R_m = abs(R)
R_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	return Float64(R_s * Float64(lambda2 * R_m))
end

MATLAB

R_m = abs(R);
R_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = R_s * (lambda2 * R_m);
end

Wolfram

R_m = N[Abs[R], $MachinePrecision]
R_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * N[(lambda2 * R$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.2%

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

3. Simplified 72.2%

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

5. Taylor expanded in phi2 around 0 41.0%

   \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]

6. Taylor expanded in phi1 around 0 26.6%

   \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \]

7. Taylor expanded in lambda2 around inf 4.9%

   \[\leadsto \color{blue}{R \cdot \lambda_2} \]
8. Step-by-step derivation

   1. *-commutative 4.9%

      \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

9. Simplified 4.9%

   \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

10. Final simplification 4.9%

    \[\leadsto \lambda_2 \cdot R \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024017 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Spherical law of cosines"
  :precision binary64
  (* (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))) R))