Spherical law of cosines

Percentage Accurate: 73.7% → 96.4%

Time: 23.5s

Alternatives: 27

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: 73.7% 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: 96.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(t\_1 + t\_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\_m \leq 0:\\ \;\;\;\;R\_m \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(t\_0, \cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1, t\_1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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))))) R_m) 0.0)
      (* R_m (- lambda2 lambda1))
      (*
       R_m
       (acos
        (fma
         t_0
         (+ (* (cos lambda2) (cos lambda1)) (* (sin lambda2) (sin lambda1)))
         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(phi1) * cos(phi2);
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if ((acos((t_1 + (t_0 * cos((lambda1 - lambda2))))) * R_m) <= 0.0) {
		tmp = R_m * (lambda2 - lambda1);
	} else {
		tmp = R_m * acos(fma(t_0, ((cos(lambda2) * cos(lambda1)) + (sin(lambda2) * sin(lambda1))), 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 = Float64(cos(phi1) * cos(phi2))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (Float64(acos(Float64(t_1 + Float64(t_0 * cos(Float64(lambda1 - lambda2))))) * R_m) <= 0.0)
		tmp = Float64(R_m * Float64(lambda2 - lambda1));
	else
		tmp = Float64(R_m * acos(fma(t_0, Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda2) * sin(lambda1))), 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[(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[(N[ArcCos[N[(t$95$1 + N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R$95$m), $MachinePrecision], 0.0], N[(R$95$m * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(t$95$0 * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 69.7%

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

   3. Simplified 69.8%

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

   5. Taylor expanded in phi2 around 0 44.0%

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

   6. Taylor expanded in phi1 around 0 26.4%

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

   7. Taylor expanded in R around 0 6.1%

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

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

   1. Initial program 74.3%

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

   3. Simplified 74.4%

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

      1. cos-diff 99.1%

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

      2. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

4. Final simplification 51.5%

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

Alternative 2: 96.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(t\_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\_m \leq 0:\\ \;\;\;\;R\_m \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(t\_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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)))))
          R_m)
         0.0)
      (* R_m (- lambda2 lambda1))
      (*
       R_m
       (acos
        (+
         t_0
         (*
          (cos phi1)
          (*
           (cos phi2)
           (+
            (* (cos lambda2) (cos lambda1))
            (* (sin lambda2) (sin 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 = sin(phi1) * sin(phi2);
	double tmp;
	if ((acos((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R_m) <= 0.0) {
		tmp = R_m * (lambda2 - lambda1);
	} else {
		tmp = R_m * acos((t_0 + (cos(phi1) * (cos(phi2) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda2) * sin(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) :: t_0
    real(8) :: tmp
    t_0 = sin(phi1) * sin(phi2)
    if ((acos((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * r_m) <= 0.0d0) then
        tmp = r_m * (lambda2 - lambda1)
    else
        tmp = r_m * acos((t_0 + (cos(phi1) * (cos(phi2) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda2) * sin(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 t_0 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if ((Math.acos((t_0 + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2))))) * R_m) <= 0.0) {
		tmp = R_m * (lambda2 - lambda1);
	} else {
		tmp = R_m * Math.acos((t_0 + (Math.cos(phi1) * (Math.cos(phi2) * ((Math.cos(lambda2) * Math.cos(lambda1)) + (Math.sin(lambda2) * Math.sin(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):
	t_0 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if (math.acos((t_0 + ((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2))))) * R_m) <= 0.0:
		tmp = R_m * (lambda2 - lambda1)
	else:
		tmp = R_m * math.acos((t_0 + (math.cos(phi1) * (math.cos(phi2) * ((math.cos(lambda2) * math.cos(lambda1)) + (math.sin(lambda2) * math.sin(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(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (Float64(acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) * R_m) <= 0.0)
		tmp = Float64(R_m * Float64(lambda2 - lambda1));
	else
		tmp = Float64(R_m * acos(Float64(t_0 + Float64(cos(phi1) * Float64(cos(phi2) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda2) * sin(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)
	t_0 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if ((acos((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R_m) <= 0.0)
		tmp = R_m * (lambda2 - lambda1);
	else
		tmp = R_m * acos((t_0 + (cos(phi1) * (cos(phi2) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda2) * sin(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_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[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], 0.0], N[(R$95$m * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 69.7%

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

   3. Simplified 69.8%

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

   5. Taylor expanded in phi2 around 0 44.0%

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

   6. Taylor expanded in phi1 around 0 26.4%

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

   7. Taylor expanded in R around 0 6.1%

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

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

   1. Initial program 74.3%

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

   3. Simplified 74.4%

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

      1. cos-diff 99.1%

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

      2. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

   7. Taylor expanded in phi1 around 0 99.0%

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

4. Final simplification 51.5%

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

Alternative 3: 83.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.7 \cdot 10^{-5}:\\ \;\;\;\;R\_m \cdot \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t\_0, t\_1\right)\right)}\right)\\ \mathbf{elif}\;\phi_1 \leq 4 \cdot 10^{-31}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1, t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, t\_0, t\_1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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))) (t_1 (* (sin phi1) (sin phi2))))
   (*
    R_s
    (if (<= phi1 -2.7e-5)
      (* R_m (log (exp (acos (fma (cos phi1) (* (cos phi2) t_0) t_1)))))
      (if (<= phi1 4e-31)
        (*
         R_m
         (acos
          (fma
           (cos phi2)
           (+ (* (cos lambda2) (cos lambda1)) (* (sin lambda2) (sin lambda1)))
           t_1)))
        (* 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((lambda2 - lambda1));
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi1 <= -2.7e-5) {
		tmp = R_m * log(exp(acos(fma(cos(phi1), (cos(phi2) * t_0), t_1))));
	} else if (phi1 <= 4e-31) {
		tmp = R_m * acos(fma(cos(phi2), ((cos(lambda2) * cos(lambda1)) + (sin(lambda2) * sin(lambda1))), t_1));
	} 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(lambda2 - lambda1))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi1 <= -2.7e-5)
		tmp = Float64(R_m * log(exp(acos(fma(cos(phi1), Float64(cos(phi2) * t_0), t_1)))));
	elseif (phi1 <= 4e-31)
		tmp = Float64(R_m * acos(fma(cos(phi2), Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda2) * sin(lambda1))), t_1)));
	else
		tmp = Float64(R_m * acos(fma(Float64(cos(phi1) * 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[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi1, -2.7e-5], N[(R$95$m * N[Log[N[Exp[N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 4e-31], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.6999999999999999e-5

   1. Initial program 78.0%

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

   3. Simplified 78.0%

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

      1. add-log-exp 78.0%

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

      2. cos-diff 99.3%

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

      3. *-commutative 99.3%

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

      4. *-commutative 99.3%

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

      5. cos-diff 78.0%

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

      6. fma-define 77.9%

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

      7. associate-*r* 77.9%

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

      8. fma-undefine 78.0%

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

      9. cos-diff 99.3%

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

      10. *-commutative 99.3%

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

      11. *-commutative 99.3%

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

      12. cos-diff 78.0%

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

   6. Applied egg-rr 78.0%

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

   if -2.6999999999999999e-5 < phi1 < 4e-31

   1. Initial program 63.4%

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

   3. Simplified 63.4%

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

      1. cos-diff 88.7%

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

      2. +-commutative 88.7%

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

   6. Applied egg-rr 88.7%

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

   7. Taylor expanded in phi1 around 0 88.3%

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

   if 4e-31 < phi1

   1. Initial program 79.3%

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

   3. Simplified 79.4%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.7 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\\ \mathbf{elif}\;\phi_1 \leq 4 \cdot 10^{-31}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, \cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.1 \cdot 10^{-14}:\\ \;\;\;\;R\_m \cdot \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t\_0, t\_1\right)\right)}\right)\\ \mathbf{elif}\;\phi_1 \leq 4 \cdot 10^{-31}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, t\_0, t\_1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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))) (t_1 (* (sin phi1) (sin phi2))))
   (*
    R_s
    (if (<= phi1 -3.1e-14)
      (* R_m (log (exp (acos (fma (cos phi1) (* (cos phi2) t_0) t_1)))))
      (if (<= phi1 4e-31)
        (*
         R_m
         (acos
          (*
           (cos phi2)
           (+
            (* (cos lambda2) (cos lambda1))
            (* (sin lambda2) (sin 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((lambda2 - lambda1));
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi1 <= -3.1e-14) {
		tmp = R_m * log(exp(acos(fma(cos(phi1), (cos(phi2) * t_0), t_1))));
	} else if (phi1 <= 4e-31) {
		tmp = R_m * acos((cos(phi2) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda2) * sin(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(lambda2 - lambda1))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi1 <= -3.1e-14)
		tmp = Float64(R_m * log(exp(acos(fma(cos(phi1), Float64(cos(phi2) * t_0), t_1)))));
	elseif (phi1 <= 4e-31)
		tmp = Float64(R_m * acos(Float64(cos(phi2) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda2) * sin(lambda1))))));
	else
		tmp = Float64(R_m * acos(fma(Float64(cos(phi1) * 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[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi1, -3.1e-14], N[(R$95$m * N[Log[N[Exp[N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 4e-31], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.10000000000000004e-14

   1. Initial program 76.2%

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

   3. Simplified 76.2%

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

      1. add-log-exp 76.2%

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

      2. cos-diff 98.1%

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

      3. *-commutative 98.1%

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

      4. *-commutative 98.1%

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

      5. cos-diff 76.2%

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

      6. fma-define 76.2%

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

      7. associate-*r* 76.2%

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

      8. fma-undefine 76.3%

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

      9. cos-diff 98.1%

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

      10. *-commutative 98.1%

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

      11. *-commutative 98.1%

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

      12. cos-diff 76.3%

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

   6. Applied egg-rr 76.3%

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

   if -3.10000000000000004e-14 < phi1 < 4e-31

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

      1. cos-diff 89.1%

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

      2. +-commutative 89.1%

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

   6. Applied egg-rr 89.1%

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

      1. expm1-log1p-u 89.1%

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

      2. expm1-undefine 89.1%

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

   8. Applied egg-rr 89.1%

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

   9. Taylor expanded in phi1 around 0 89.1%

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

   if 4e-31 < phi1

   1. Initial program 79.3%

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

   3. Simplified 79.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.1 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\\ \mathbf{elif}\;\phi_1 \leq 4 \cdot 10^{-31}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.8 \cdot 10^{-19}:\\ \;\;\;\;R\_m \cdot e^{\log \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t\_0, t\_1\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 4 \cdot 10^{-31}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, t\_0, t\_1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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))) (t_1 (* (sin phi1) (sin phi2))))
   (*
    R_s
    (if (<= phi1 -2.8e-19)
      (* R_m (exp (log (acos (fma (cos phi1) (* (cos phi2) t_0) t_1)))))
      (if (<= phi1 4e-31)
        (*
         R_m
         (acos
          (*
           (cos phi2)
           (+
            (* (cos lambda2) (cos lambda1))
            (* (sin lambda2) (sin 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((lambda2 - lambda1));
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi1 <= -2.8e-19) {
		tmp = R_m * exp(log(acos(fma(cos(phi1), (cos(phi2) * t_0), t_1))));
	} else if (phi1 <= 4e-31) {
		tmp = R_m * acos((cos(phi2) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda2) * sin(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(lambda2 - lambda1))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi1 <= -2.8e-19)
		tmp = Float64(R_m * exp(log(acos(fma(cos(phi1), Float64(cos(phi2) * t_0), t_1)))));
	elseif (phi1 <= 4e-31)
		tmp = Float64(R_m * acos(Float64(cos(phi2) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda2) * sin(lambda1))))));
	else
		tmp = Float64(R_m * acos(fma(Float64(cos(phi1) * 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[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi1, -2.8e-19], N[(R$95$m * N[Exp[N[Log[N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 4e-31], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.80000000000000003e-19

   1. Initial program 76.2%

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

   3. Simplified 76.2%

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

      1. add-exp-log 76.2%

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

      2. cos-diff 98.1%

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

      3. *-commutative 98.1%

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

      4. *-commutative 98.1%

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

      5. cos-diff 76.2%

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

      6. fma-define 76.2%

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

      7. associate-*r* 76.2%

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

      8. fma-undefine 76.3%

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

      9. cos-diff 98.1%

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

      10. *-commutative 98.1%

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

      11. *-commutative 98.1%

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

      12. cos-diff 76.3%

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

   6. Applied egg-rr 76.3%

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

   if -2.80000000000000003e-19 < phi1 < 4e-31

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

      1. cos-diff 89.1%

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

      2. +-commutative 89.1%

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

   6. Applied egg-rr 89.1%

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

      1. expm1-log1p-u 89.1%

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

      2. expm1-undefine 89.1%

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

   8. Applied egg-rr 89.1%

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

   9. Taylor expanded in phi1 around 0 89.1%

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

   if 4e-31 < phi1

   1. Initial program 79.3%

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

   3. Simplified 79.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.8 \cdot 10^{-19}:\\ \;\;\;\;R \cdot e^{\log \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)}\\ \mathbf{elif}\;\phi_1 \leq 4 \cdot 10^{-31}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.1 \cdot 10^{-14} \lor \neg \left(\phi_1 \leq 4 \cdot 10^{-31}\right):\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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 (<= phi1 -3.1e-14) (not (<= phi1 4e-31)))
    (*
     R_m
     (acos
      (fma
       (* (cos phi1) (cos phi2))
       (cos (- lambda2 lambda1))
       (* (sin phi1) (sin phi2)))))
    (*
     R_m
     (acos
      (*
       (cos phi2)
       (+
        (* (cos lambda2) (cos lambda1))
        (* (sin lambda2) (sin 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 <= -3.1e-14) || !(phi1 <= 4e-31)) {
		tmp = R_m * acos(fma((cos(phi1) * cos(phi2)), cos((lambda2 - lambda1)), (sin(phi1) * sin(phi2))));
	} else {
		tmp = R_m * acos((cos(phi2) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda2) * sin(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 <= -3.1e-14) || !(phi1 <= 4e-31))
		tmp = Float64(R_m * acos(fma(Float64(cos(phi1) * cos(phi2)), cos(Float64(lambda2 - lambda1)), Float64(sin(phi1) * sin(phi2)))));
	else
		tmp = Float64(R_m * acos(Float64(cos(phi2) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda2) * sin(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[phi1, -3.1e-14], N[Not[LessEqual[phi1, 4e-31]], $MachinePrecision]], N[(R$95$m * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.10000000000000004e-14 or 4e-31 < phi1

   1. Initial program 77.6%

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

   3. Simplified 77.7%

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

   if -3.10000000000000004e-14 < phi1 < 4e-31

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

      1. cos-diff 89.1%

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

      2. +-commutative 89.1%

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

   6. Applied egg-rr 89.1%

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

      1. expm1-log1p-u 89.1%

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

      2. expm1-undefine 89.1%

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

   8. Applied egg-rr 89.1%

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

   9. Taylor expanded in phi1 around 0 89.1%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.1 \cdot 10^{-14} \lor \neg \left(\phi_1 \leq 4 \cdot 10^{-31}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.1 \cdot 10^{-14}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), t\_0\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 4 \cdot 10^{-31}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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 (<= phi1 -3.1e-14)
      (*
       R_m
       (acos (fma (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))) t_0)))
      (if (<= phi1 4e-31)
        (*
         R_m
         (acos
          (*
           (cos phi2)
           (+
            (* (cos lambda2) (cos lambda1))
            (* (sin lambda2) (sin lambda1))))))
        (*
         R_m
         (acos
          (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 (phi1 <= -3.1e-14) {
		tmp = R_m * acos(fma(cos(phi1), (cos(phi2) * cos((lambda1 - lambda2))), t_0));
	} else if (phi1 <= 4e-31) {
		tmp = R_m * acos((cos(phi2) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda2) * sin(lambda1)))));
	} else {
		tmp = R_m * acos(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 (phi1 <= -3.1e-14)
		tmp = Float64(R_m * acos(fma(cos(phi1), Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))), t_0)));
	elseif (phi1 <= 4e-31)
		tmp = Float64(R_m * acos(Float64(cos(phi2) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda2) * sin(lambda1))))));
	else
		tmp = Float64(R_m * acos(fma(Float64(cos(phi1) * 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[phi1, -3.1e-14], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 4e-31], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.10000000000000004e-14

   1. Initial program 76.2%

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

   3. Simplified 76.3%

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

   if -3.10000000000000004e-14 < phi1 < 4e-31

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

      1. cos-diff 89.1%

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

      2. +-commutative 89.1%

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

   6. Applied egg-rr 89.1%

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

      1. expm1-log1p-u 89.1%

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

      2. expm1-undefine 89.1%

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

   8. Applied egg-rr 89.1%

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

   9. Taylor expanded in phi1 around 0 89.1%

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

   if 4e-31 < phi1

   1. Initial program 79.3%

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

   3. Simplified 79.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.1 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 4 \cdot 10^{-31}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -3.6 \cdot 10^{-13}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 0.044:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(t\_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 4.8 \cdot 10^{+201}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(t\_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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)))
        (t_1
         (+ (* (cos lambda2) (cos lambda1)) (* (sin lambda2) (sin lambda1)))))
   (*
    R_s
    (if (<= lambda2 -3.6e-13)
      (* R_m (acos (* (cos phi2) t_1)))
      (if (<= lambda2 0.044)
        (* R_m (acos (+ t_0 (* (* (cos phi1) (cos phi2)) (cos lambda1)))))
        (if (<= lambda2 4.8e+201)
          (* R_m (acos (+ t_0 (* (cos phi2) (* (cos phi1) (cos lambda2))))))
          (* R_m (acos (* (cos phi1) 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 = sin(phi1) * sin(phi2);
	double t_1 = (cos(lambda2) * cos(lambda1)) + (sin(lambda2) * sin(lambda1));
	double tmp;
	if (lambda2 <= -3.6e-13) {
		tmp = R_m * acos((cos(phi2) * t_1));
	} else if (lambda2 <= 0.044) {
		tmp = R_m * acos((t_0 + ((cos(phi1) * cos(phi2)) * cos(lambda1))));
	} else if (lambda2 <= 4.8e+201) {
		tmp = R_m * acos((t_0 + (cos(phi2) * (cos(phi1) * cos(lambda2)))));
	} else {
		tmp = R_m * acos((cos(phi1) * t_1));
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = sin(phi1) * sin(phi2)
    t_1 = (cos(lambda2) * cos(lambda1)) + (sin(lambda2) * sin(lambda1))
    if (lambda2 <= (-3.6d-13)) then
        tmp = r_m * acos((cos(phi2) * t_1))
    else if (lambda2 <= 0.044d0) then
        tmp = r_m * acos((t_0 + ((cos(phi1) * cos(phi2)) * cos(lambda1))))
    else if (lambda2 <= 4.8d+201) then
        tmp = r_m * acos((t_0 + (cos(phi2) * (cos(phi1) * cos(lambda2)))))
    else
        tmp = r_m * acos((cos(phi1) * t_1))
    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 t_1 = (Math.cos(lambda2) * Math.cos(lambda1)) + (Math.sin(lambda2) * Math.sin(lambda1));
	double tmp;
	if (lambda2 <= -3.6e-13) {
		tmp = R_m * Math.acos((Math.cos(phi2) * t_1));
	} else if (lambda2 <= 0.044) {
		tmp = R_m * Math.acos((t_0 + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos(lambda1))));
	} else if (lambda2 <= 4.8e+201) {
		tmp = R_m * Math.acos((t_0 + (Math.cos(phi2) * (Math.cos(phi1) * Math.cos(lambda2)))));
	} else {
		tmp = R_m * Math.acos((Math.cos(phi1) * t_1));
	}
	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)
	t_1 = (math.cos(lambda2) * math.cos(lambda1)) + (math.sin(lambda2) * math.sin(lambda1))
	tmp = 0
	if lambda2 <= -3.6e-13:
		tmp = R_m * math.acos((math.cos(phi2) * t_1))
	elif lambda2 <= 0.044:
		tmp = R_m * math.acos((t_0 + ((math.cos(phi1) * math.cos(phi2)) * math.cos(lambda1))))
	elif lambda2 <= 4.8e+201:
		tmp = R_m * math.acos((t_0 + (math.cos(phi2) * (math.cos(phi1) * math.cos(lambda2)))))
	else:
		tmp = R_m * math.acos((math.cos(phi1) * 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 = Float64(sin(phi1) * sin(phi2))
	t_1 = Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda2) * sin(lambda1)))
	tmp = 0.0
	if (lambda2 <= -3.6e-13)
		tmp = Float64(R_m * acos(Float64(cos(phi2) * t_1)));
	elseif (lambda2 <= 0.044)
		tmp = Float64(R_m * acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(lambda1)))));
	elseif (lambda2 <= 4.8e+201)
		tmp = Float64(R_m * acos(Float64(t_0 + Float64(cos(phi2) * Float64(cos(phi1) * cos(lambda2))))));
	else
		tmp = Float64(R_m * acos(Float64(cos(phi1) * t_1)));
	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);
	t_1 = (cos(lambda2) * cos(lambda1)) + (sin(lambda2) * sin(lambda1));
	tmp = 0.0;
	if (lambda2 <= -3.6e-13)
		tmp = R_m * acos((cos(phi2) * t_1));
	elseif (lambda2 <= 0.044)
		tmp = R_m * acos((t_0 + ((cos(phi1) * cos(phi2)) * cos(lambda1))));
	elseif (lambda2 <= 4.8e+201)
		tmp = R_m * acos((t_0 + (cos(phi2) * (cos(phi1) * cos(lambda2)))));
	else
		tmp = R_m * acos((cos(phi1) * t_1));
	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]}, Block[{t$95$1 = N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[lambda2, -3.6e-13], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 0.044], N[(R$95$m * N[ArcCos[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 4.8e+201], N[(R$95$m * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if lambda2 < -3.5999999999999998e-13

   1. Initial program 61.3%

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

   3. Simplified 61.3%

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

      1. cos-diff 99.1%

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

      2. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. expm1-log1p-u 99.1%

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

      2. expm1-undefine 99.0%

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

   8. Applied egg-rr 99.0%

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

   9. Taylor expanded in phi1 around 0 56.2%

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

   if -3.5999999999999998e-13 < lambda2 < 0.043999999999999997

   1. Initial program 88.5%

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

   3. Taylor expanded in lambda2 around 0 88.5%

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

   if 0.043999999999999997 < lambda2 < 4.79999999999999985e201

   1. Initial program 57.9%

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

      1. add-cube-cbrt 57.9%

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\color{blue}{\left(\left(\sqrt[3]{\cos \phi_1} \cdot \sqrt[3]{\cos \phi_1}\right) \cdot \sqrt[3]{\cos \phi_1}\right)} \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. pow3 57.9%

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\color{blue}{{\left(\sqrt[3]{\cos \phi_1}\right)}^{3}} \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   4. Applied egg-rr 57.9%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\color{blue}{{\left(\sqrt[3]{\cos \phi_1}\right)}^{3}} \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

   5. Taylor expanded in lambda1 around 0 57.7%

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

      1. cos-neg 57.7%

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

      2. associate-*r* 57.7%

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

      3. *-commutative 57.7%

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

      4. associate-*r* 57.7%

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

      5. *-commutative 57.7%

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

   7. Simplified 57.7%

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

   if 4.79999999999999985e201 < lambda2

   1. Initial program 50.8%

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

   3. Simplified 50.9%

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

   5. Taylor expanded in phi2 around 0 40.8%

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

      1. cos-diff 98.6%

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

      2. +-commutative 98.6%

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

   7. Applied egg-rr 70.7%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -3.6 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 0.044:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 4.8 \cdot 10^{+201}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -3.6 \cdot 10^{-13}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 0.044:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(t\_0 + t\_2 \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 7 \cdot 10^{+198}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(t\_0 + t\_2 \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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)))
        (t_1
         (+ (* (cos lambda2) (cos lambda1)) (* (sin lambda2) (sin lambda1))))
        (t_2 (* (cos phi1) (cos phi2))))
   (*
    R_s
    (if (<= lambda2 -3.6e-13)
      (* R_m (acos (* (cos phi2) t_1)))
      (if (<= lambda2 0.044)
        (* R_m (acos (+ t_0 (* t_2 (cos lambda1)))))
        (if (<= lambda2 7e+198)
          (* R_m (acos (+ t_0 (* t_2 (cos lambda2)))))
          (* R_m (acos (* (cos phi1) 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 = sin(phi1) * sin(phi2);
	double t_1 = (cos(lambda2) * cos(lambda1)) + (sin(lambda2) * sin(lambda1));
	double t_2 = cos(phi1) * cos(phi2);
	double tmp;
	if (lambda2 <= -3.6e-13) {
		tmp = R_m * acos((cos(phi2) * t_1));
	} else if (lambda2 <= 0.044) {
		tmp = R_m * acos((t_0 + (t_2 * cos(lambda1))));
	} else if (lambda2 <= 7e+198) {
		tmp = R_m * acos((t_0 + (t_2 * cos(lambda2))));
	} else {
		tmp = R_m * acos((cos(phi1) * t_1));
	}
	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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sin(phi1) * sin(phi2)
    t_1 = (cos(lambda2) * cos(lambda1)) + (sin(lambda2) * sin(lambda1))
    t_2 = cos(phi1) * cos(phi2)
    if (lambda2 <= (-3.6d-13)) then
        tmp = r_m * acos((cos(phi2) * t_1))
    else if (lambda2 <= 0.044d0) then
        tmp = r_m * acos((t_0 + (t_2 * cos(lambda1))))
    else if (lambda2 <= 7d+198) then
        tmp = r_m * acos((t_0 + (t_2 * cos(lambda2))))
    else
        tmp = r_m * acos((cos(phi1) * t_1))
    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 t_1 = (Math.cos(lambda2) * Math.cos(lambda1)) + (Math.sin(lambda2) * Math.sin(lambda1));
	double t_2 = Math.cos(phi1) * Math.cos(phi2);
	double tmp;
	if (lambda2 <= -3.6e-13) {
		tmp = R_m * Math.acos((Math.cos(phi2) * t_1));
	} else if (lambda2 <= 0.044) {
		tmp = R_m * Math.acos((t_0 + (t_2 * Math.cos(lambda1))));
	} else if (lambda2 <= 7e+198) {
		tmp = R_m * Math.acos((t_0 + (t_2 * Math.cos(lambda2))));
	} else {
		tmp = R_m * Math.acos((Math.cos(phi1) * t_1));
	}
	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)
	t_1 = (math.cos(lambda2) * math.cos(lambda1)) + (math.sin(lambda2) * math.sin(lambda1))
	t_2 = math.cos(phi1) * math.cos(phi2)
	tmp = 0
	if lambda2 <= -3.6e-13:
		tmp = R_m * math.acos((math.cos(phi2) * t_1))
	elif lambda2 <= 0.044:
		tmp = R_m * math.acos((t_0 + (t_2 * math.cos(lambda1))))
	elif lambda2 <= 7e+198:
		tmp = R_m * math.acos((t_0 + (t_2 * math.cos(lambda2))))
	else:
		tmp = R_m * math.acos((math.cos(phi1) * 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 = Float64(sin(phi1) * sin(phi2))
	t_1 = Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda2) * sin(lambda1)))
	t_2 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if (lambda2 <= -3.6e-13)
		tmp = Float64(R_m * acos(Float64(cos(phi2) * t_1)));
	elseif (lambda2 <= 0.044)
		tmp = Float64(R_m * acos(Float64(t_0 + Float64(t_2 * cos(lambda1)))));
	elseif (lambda2 <= 7e+198)
		tmp = Float64(R_m * acos(Float64(t_0 + Float64(t_2 * cos(lambda2)))));
	else
		tmp = Float64(R_m * acos(Float64(cos(phi1) * t_1)));
	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);
	t_1 = (cos(lambda2) * cos(lambda1)) + (sin(lambda2) * sin(lambda1));
	t_2 = cos(phi1) * cos(phi2);
	tmp = 0.0;
	if (lambda2 <= -3.6e-13)
		tmp = R_m * acos((cos(phi2) * t_1));
	elseif (lambda2 <= 0.044)
		tmp = R_m * acos((t_0 + (t_2 * cos(lambda1))));
	elseif (lambda2 <= 7e+198)
		tmp = R_m * acos((t_0 + (t_2 * cos(lambda2))));
	else
		tmp = R_m * acos((cos(phi1) * t_1));
	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]}, Block[{t$95$1 = N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[lambda2, -3.6e-13], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 0.044], N[(R$95$m * N[ArcCos[N[(t$95$0 + N[(t$95$2 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 7e+198], N[(R$95$m * N[ArcCos[N[(t$95$0 + N[(t$95$2 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if lambda2 < -3.5999999999999998e-13

   1. Initial program 61.3%

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

   3. Simplified 61.3%

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

      1. cos-diff 99.1%

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

      2. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. expm1-log1p-u 99.1%

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

      2. expm1-undefine 99.0%

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

   8. Applied egg-rr 99.0%

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

   9. Taylor expanded in phi1 around 0 56.2%

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

   if -3.5999999999999998e-13 < lambda2 < 0.043999999999999997

   1. Initial program 88.5%

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

   3. Taylor expanded in lambda2 around 0 88.5%

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

   if 0.043999999999999997 < lambda2 < 7.00000000000000026e198

   1. Initial program 57.9%

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

   3. Simplified 57.9%

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

   5. Taylor expanded in lambda1 around 0 57.7%

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

   if 7.00000000000000026e198 < lambda2

   1. Initial program 50.8%

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

   3. Simplified 50.9%

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

   5. Taylor expanded in phi2 around 0 40.8%

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

      1. cos-diff 98.6%

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

      2. +-commutative 98.6%

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

   7. Applied egg-rr 70.7%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -3.6 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 0.044:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 7 \cdot 10^{+198}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.1 \cdot 10^{-14} \lor \neg \left(\phi_1 \leq 4 \cdot 10^{-31}\right):\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\_m\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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 (<= phi1 -3.1e-14) (not (<= phi1 4e-31)))
    (*
     (acos
      (+
       (* (sin phi1) (sin phi2))
       (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
     R_m)
    (*
     R_m
     (acos
      (*
       (cos phi2)
       (+
        (* (cos lambda2) (cos lambda1))
        (* (sin lambda2) (sin 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 <= -3.1e-14) || !(phi1 <= 4e-31)) {
		tmp = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R_m;
	} else {
		tmp = R_m * acos((cos(phi2) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda2) * sin(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 <= (-3.1d-14)) .or. (.not. (phi1 <= 4d-31))) then
        tmp = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * r_m
    else
        tmp = r_m * acos((cos(phi2) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda2) * sin(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 <= -3.1e-14) || !(phi1 <= 4e-31)) {
		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(phi2) * ((Math.cos(lambda2) * Math.cos(lambda1)) + (Math.sin(lambda2) * Math.sin(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 <= -3.1e-14) or not (phi1 <= 4e-31):
		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(phi2) * ((math.cos(lambda2) * math.cos(lambda1)) + (math.sin(lambda2) * math.sin(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 <= -3.1e-14) || !(phi1 <= 4e-31))
		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(phi2) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda2) * sin(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 <= -3.1e-14) || ~((phi1 <= 4e-31)))
		tmp = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R_m;
	else
		tmp = R_m * acos((cos(phi2) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda2) * sin(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[Or[LessEqual[phi1, -3.1e-14], N[Not[LessEqual[phi1, 4e-31]], $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[phi2], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.10000000000000004e-14 or 4e-31 < phi1

   1. Initial program 77.6%

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

   if -3.10000000000000004e-14 < phi1 < 4e-31

   1. Initial program 64.0%

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

   3. Simplified 64.0%

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

      1. cos-diff 89.1%

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

      2. +-commutative 89.1%

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

   6. Applied egg-rr 89.1%

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

      1. expm1-log1p-u 89.1%

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

      2. expm1-undefine 89.1%

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

   8. Applied egg-rr 89.1%

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

   9. Taylor expanded in phi1 around 0 89.1%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.1 \cdot 10^{-14} \lor \neg \left(\phi_1 \leq 4 \cdot 10^{-31}\right):\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.0142:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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) (cos lambda1)) (* (sin lambda2) (sin lambda1)))))
   (*
    R_s
    (if (<= phi2 -0.0142)
      (*
       R_m
       (acos
        (+
         (* (sin phi1) (sin phi2))
         (* (* (cos phi1) (cos phi2)) (cos lambda2)))))
      (if (<= phi2 6.5e-5)
        (* 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) * cos(lambda1)) + (sin(lambda2) * sin(lambda1));
	double tmp;
	if (phi2 <= -0.0142) {
		tmp = R_m * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos(lambda2))));
	} else if (phi2 <= 6.5e-5) {
		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) * cos(lambda1)) + (sin(lambda2) * sin(lambda1))
    if (phi2 <= (-0.0142d0)) then
        tmp = r_m * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos(lambda2))))
    else if (phi2 <= 6.5d-5) 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) * Math.cos(lambda1)) + (Math.sin(lambda2) * Math.sin(lambda1));
	double tmp;
	if (phi2 <= -0.0142) {
		tmp = R_m * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos(lambda2))));
	} else if (phi2 <= 6.5e-5) {
		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) * math.cos(lambda1)) + (math.sin(lambda2) * math.sin(lambda1))
	tmp = 0
	if phi2 <= -0.0142:
		tmp = R_m * math.acos(((math.sin(phi1) * math.sin(phi2)) + ((math.cos(phi1) * math.cos(phi2)) * math.cos(lambda2))))
	elif phi2 <= 6.5e-5:
		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 = Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda2) * sin(lambda1)))
	tmp = 0.0
	if (phi2 <= -0.0142)
		tmp = Float64(R_m * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(lambda2)))));
	elseif (phi2 <= 6.5e-5)
		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) * cos(lambda1)) + (sin(lambda2) * sin(lambda1));
	tmp = 0.0;
	if (phi2 <= -0.0142)
		tmp = R_m * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos(lambda2))));
	elseif (phi2 <= 6.5e-5)
		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[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi2, -0.0142], 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], If[LessEqual[phi2, 6.5e-5], 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 3 regimes

2. if phi2 < -0.014200000000000001

   1. Initial program 79.0%

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

   3. Simplified 79.0%

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

   5. Taylor expanded in lambda1 around 0 62.0%

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

   if -0.014200000000000001 < phi2 < 6.49999999999999943e-5

   1. Initial program 68.4%

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

   3. Simplified 68.4%

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

   5. Taylor expanded in phi2 around 0 67.5%

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

      1. cos-diff 90.4%

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

      2. +-commutative 90.4%

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

   7. Applied egg-rr 89.4%

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

   if 6.49999999999999943e-5 < phi2

   1. Initial program 71.9%

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

   3. Simplified 72.0%

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

      1. cos-diff 99.1%

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

      2. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. expm1-log1p-u 99.1%

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

      2. expm1-undefine 99.1%

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

   8. Applied egg-rr 99.1%

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

   9. Taylor expanded in phi1 around 0 59.6%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.0142:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 62.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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) (cos lambda1)) (* (sin lambda2) (sin lambda1)))))
   (*
    R_s
    (if (<= phi2 2.1e-5)
      (* 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) * cos(lambda1)) + (sin(lambda2) * sin(lambda1));
	double tmp;
	if (phi2 <= 2.1e-5) {
		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) * cos(lambda1)) + (sin(lambda2) * sin(lambda1))
    if (phi2 <= 2.1d-5) 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) * Math.cos(lambda1)) + (Math.sin(lambda2) * Math.sin(lambda1));
	double tmp;
	if (phi2 <= 2.1e-5) {
		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) * math.cos(lambda1)) + (math.sin(lambda2) * math.sin(lambda1))
	tmp = 0
	if phi2 <= 2.1e-5:
		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 = Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda2) * sin(lambda1)))
	tmp = 0.0
	if (phi2 <= 2.1e-5)
		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) * cos(lambda1)) + (sin(lambda2) * sin(lambda1));
	tmp = 0.0;
	if (phi2 <= 2.1e-5)
		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[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi2, 2.1e-5], 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 phi2 < 2.09999999999999988e-5

   1. Initial program 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in phi2 around 0 51.0%

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

      1. cos-diff 93.4%

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

      2. +-commutative 93.4%

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

   7. Applied egg-rr 65.3%

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

   if 2.09999999999999988e-5 < phi2

   1. Initial program 71.9%

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

   3. Simplified 72.0%

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

      1. cos-diff 99.1%

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

      2. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. expm1-log1p-u 99.1%

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

      2. expm1-undefine 99.1%

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

   8. Applied egg-rr 99.1%

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

   9. Taylor expanded in phi1 around 0 59.6%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.1 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.0% accurate, 1.0× speedup

Math

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

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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 0.00032)
    (*
     R_m
     (acos
      (*
       (cos phi1)
       (+ (* (cos lambda2) (cos lambda1)) (* (sin lambda2) (sin lambda1))))))
    (*
     R_m
     (acos
      (fma (cos phi2) (cos (- lambda2 lambda1)) (* (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 tmp;
	if (phi2 <= 0.00032) {
		tmp = R_m * acos((cos(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda2) * sin(lambda1)))));
	} else {
		tmp = R_m * acos(fma(cos(phi2), cos((lambda2 - lambda1)), (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)
	tmp = 0.0
	if (phi2 <= 0.00032)
		tmp = Float64(R_m * acos(Float64(cos(phi1) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda2) * sin(lambda1))))));
	else
		tmp = Float64(R_m * acos(fma(cos(phi2), cos(Float64(lambda2 - lambda1)), 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_] := N[(R$95$s * If[LessEqual[phi2, 0.00032], 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[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.20000000000000026e-4

   1. Initial program 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in phi2 around 0 51.0%

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

      1. cos-diff 93.4%

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

      2. +-commutative 93.4%

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

   7. Applied egg-rr 65.3%

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

   if 3.20000000000000026e-4 < phi2

   1. Initial program 71.9%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in phi1 around 0 42.0%

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

4. Final simplification 59.9%

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

Alternative 14: 49.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;R\_m \cdot \log \left(e^{\cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2, t\_0, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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 4.2e-5)
      (* R_m (log (exp (acos (* (cos phi1) t_0)))))
      (* R_m (acos (fma (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 <= 4.2e-5) {
		tmp = R_m * log(exp(acos((cos(phi1) * t_0))));
	} else {
		tmp = R_m * acos(fma(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 <= 4.2e-5)
		tmp = Float64(R_m * log(exp(acos(Float64(cos(phi1) * t_0)))));
	else
		tmp = Float64(R_m * acos(fma(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, 4.2e-5], N[(R$95$m * N[Log[N[Exp[N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.19999999999999977e-5

   1. Initial program 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in phi2 around 0 51.0%

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

      1. add-log-exp 51.0%

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

   7. Applied egg-rr 51.0%

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

   if 4.19999999999999977e-5 < phi2

   1. Initial program 71.9%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in phi1 around 0 42.0%

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

Alternative 15: 49.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;R\_m \cdot \log \left(e^{\cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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 1.3e-5)
    (* R_m (log (exp (acos (* (cos phi1) (cos (- lambda2 lambda1)))))))
    (*
     R_m
     (acos
      (+
       (* (sin phi1) (sin phi2))
       (* (cos phi2) (cos (- lambda1 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 <= 1.3e-5) {
		tmp = R_m * log(exp(acos((cos(phi1) * cos((lambda2 - lambda1))))));
	} else {
		tmp = R_m * acos(((sin(phi1) * sin(phi2)) + (cos(phi2) * cos((lambda1 - 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 <= 1.3d-5) then
        tmp = r_m * log(exp(acos((cos(phi1) * cos((lambda2 - lambda1))))))
    else
        tmp = r_m * acos(((sin(phi1) * sin(phi2)) + (cos(phi2) * cos((lambda1 - 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 <= 1.3e-5) {
		tmp = R_m * Math.log(Math.exp(Math.acos((Math.cos(phi1) * Math.cos((lambda2 - lambda1))))));
	} else {
		tmp = R_m * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + (Math.cos(phi2) * Math.cos((lambda1 - 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 <= 1.3e-5:
		tmp = R_m * math.log(math.exp(math.acos((math.cos(phi1) * math.cos((lambda2 - lambda1))))))
	else:
		tmp = R_m * math.acos(((math.sin(phi1) * math.sin(phi2)) + (math.cos(phi2) * math.cos((lambda1 - 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 <= 1.3e-5)
		tmp = Float64(R_m * log(exp(acos(Float64(cos(phi1) * cos(Float64(lambda2 - lambda1)))))));
	else
		tmp = Float64(R_m * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi2) * cos(Float64(lambda1 - 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 <= 1.3e-5)
		tmp = R_m * log(exp(acos((cos(phi1) * cos((lambda2 - lambda1))))));
	else
		tmp = R_m * acos(((sin(phi1) * sin(phi2)) + (cos(phi2) * cos((lambda1 - 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[phi2, 1.3e-5], N[(R$95$m * N[Log[N[Exp[N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.29999999999999992e-5

   1. Initial program 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in phi2 around 0 51.0%

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

      1. add-log-exp 51.0%

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

   7. Applied egg-rr 51.0%

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

   if 1.29999999999999992e-5 < phi2

   1. Initial program 71.9%

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

   3. Taylor expanded in phi1 around 0 42.0%

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

4. Final simplification 48.9%

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

Alternative 16: 50.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.5 \cdot 10^{-5}:\\ \;\;\;\;R\_m \cdot \log \left(e^{\cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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 2.5e-5)
      (* R_m (log (exp (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 (phi2 <= 2.5e-5) {
		tmp = R_m * log(exp(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 (phi2 <= 2.5d-5) then
        tmp = r_m * log(exp(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 (phi2 <= 2.5e-5) {
		tmp = R_m * Math.log(Math.exp(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 phi2 <= 2.5e-5:
		tmp = R_m * math.log(math.exp(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 (phi2 <= 2.5e-5)
		tmp = Float64(R_m * log(exp(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 (phi2 <= 2.5e-5)
		tmp = R_m * log(exp(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[phi2, 2.5e-5], N[(R$95$m * N[Log[N[Exp[N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $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 phi2 < 2.50000000000000012e-5

   1. Initial program 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in phi2 around 0 51.0%

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

      1. add-log-exp 51.0%

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

   7. Applied egg-rr 51.0%

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

   if 2.50000000000000012e-5 < phi2

   1. Initial program 71.9%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in phi1 around 0 41.0%

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

Alternative 17: 50.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.85 \cdot 10^{-5}:\\ \;\;\;\;R\_m \cdot e^{\log \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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 1.85e-5)
      (* R_m (exp (log (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 (phi2 <= 1.85e-5) {
		tmp = R_m * exp(log(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 (phi2 <= 1.85d-5) then
        tmp = r_m * exp(log(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 (phi2 <= 1.85e-5) {
		tmp = R_m * Math.exp(Math.log(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 phi2 <= 1.85e-5:
		tmp = R_m * math.exp(math.log(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 (phi2 <= 1.85e-5)
		tmp = Float64(R_m * exp(log(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 (phi2 <= 1.85e-5)
		tmp = R_m * exp(log(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[phi2, 1.85e-5], N[(R$95$m * N[Exp[N[Log[N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $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 phi2 < 1.84999999999999991e-5

   1. Initial program 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in phi2 around 0 51.0%

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

      1. add-exp-log 51.0%

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

   7. Applied egg-rr 51.0%

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

   if 1.84999999999999991e-5 < phi2

   1. Initial program 71.9%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in phi1 around 0 41.0%

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

Alternative 18: 50.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5.4 \cdot 10^{-5}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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 5.4e-5)
      (* 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 (phi2 <= 5.4e-5) {
		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 (phi2 <= 5.4d-5) 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 (phi2 <= 5.4e-5) {
		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 phi2 <= 5.4e-5:
		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 (phi2 <= 5.4e-5)
		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 (phi2 <= 5.4e-5)
		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[phi2, 5.4e-5], 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 phi2 < 5.3999999999999998e-5

   1. Initial program 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in phi2 around 0 51.0%

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

   if 5.3999999999999998e-5 < phi2

   1. Initial program 71.9%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in phi1 around 0 41.0%

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

Alternative 19: 47.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.012:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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 0.012)
    (* R_m (acos (* (cos phi1) (cos (- lambda2 lambda1)))))
    (* R_m (acos (* (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 tmp;
	if (phi2 <= 0.012) {
		tmp = R_m * acos((cos(phi1) * cos((lambda2 - lambda1))));
	} else {
		tmp = R_m * acos((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) :: tmp
    if (phi2 <= 0.012d0) then
        tmp = r_m * acos((cos(phi1) * cos((lambda2 - lambda1))))
    else
        tmp = r_m * acos((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 tmp;
	if (phi2 <= 0.012) {
		tmp = R_m * Math.acos((Math.cos(phi1) * Math.cos((lambda2 - lambda1))));
	} else {
		tmp = R_m * Math.acos((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):
	tmp = 0
	if phi2 <= 0.012:
		tmp = R_m * math.acos((math.cos(phi1) * math.cos((lambda2 - lambda1))))
	else:
		tmp = R_m * math.acos((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)
	tmp = 0.0
	if (phi2 <= 0.012)
		tmp = Float64(R_m * acos(Float64(cos(phi1) * cos(Float64(lambda2 - lambda1)))));
	else
		tmp = Float64(R_m * acos(Float64(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)
	tmp = 0.0;
	if (phi2 <= 0.012)
		tmp = R_m * acos((cos(phi1) * cos((lambda2 - lambda1))));
	else
		tmp = R_m * acos((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_] := N[(R$95$s * If[LessEqual[phi2, 0.012], 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[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.012

   1. Initial program 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in phi2 around 0 51.0%

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

   if 0.012 < phi2

   1. Initial program 71.9%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in lambda1 around 0 55.9%

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

      1. *-commutative 55.9%

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

      2. associate-*r* 55.9%

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

      3. cos-neg 55.9%

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

      4. fma-define 56.0%

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

      5. cos-neg 56.0%

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

   7. Simplified 56.0%

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

   8. Taylor expanded in phi1 around 0 35.6%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.012:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 36.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.000105:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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 0.000105)
    (* R_m (acos (* (cos phi1) (cos lambda1))))
    (* R_m (acos (* (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 tmp;
	if (phi2 <= 0.000105) {
		tmp = R_m * acos((cos(phi1) * cos(lambda1)));
	} else {
		tmp = R_m * acos((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) :: tmp
    if (phi2 <= 0.000105d0) then
        tmp = r_m * acos((cos(phi1) * cos(lambda1)))
    else
        tmp = r_m * acos((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 tmp;
	if (phi2 <= 0.000105) {
		tmp = R_m * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R_m * Math.acos((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):
	tmp = 0
	if phi2 <= 0.000105:
		tmp = R_m * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R_m * math.acos((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)
	tmp = 0.0
	if (phi2 <= 0.000105)
		tmp = Float64(R_m * acos(Float64(cos(phi1) * cos(lambda1))));
	else
		tmp = Float64(R_m * acos(Float64(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)
	tmp = 0.0;
	if (phi2 <= 0.000105)
		tmp = R_m * acos((cos(phi1) * cos(lambda1)));
	else
		tmp = R_m * acos((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_] := N[(R$95$s * If[LessEqual[phi2, 0.000105], 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[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.05e-4

   1. Initial program 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in phi2 around 0 51.0%

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

   6. Taylor expanded in lambda2 around 0 36.5%

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

      1. cos-neg 36.5%

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

   8. Simplified 36.5%

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

   if 1.05e-4 < phi2

   1. Initial program 71.9%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in lambda1 around 0 55.9%

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

      1. *-commutative 55.9%

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

      2. associate-*r* 55.9%

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

      3. cos-neg 55.9%

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

      4. fma-define 56.0%

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

      5. cos-neg 56.0%

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

   7. Simplified 56.0%

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

   8. Taylor expanded in phi1 around 0 35.6%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.000105:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 37.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.1 \cdot 10^{-6}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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 -3.1e-6)
    (* R_m (acos (cos (- lambda2 lambda1))))
    (* R_m (acos (* (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 tmp;
	if (lambda1 <= -3.1e-6) {
		tmp = R_m * acos(cos((lambda2 - lambda1)));
	} else {
		tmp = R_m * acos((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) :: tmp
    if (lambda1 <= (-3.1d-6)) then
        tmp = r_m * acos(cos((lambda2 - lambda1)))
    else
        tmp = r_m * acos((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 tmp;
	if (lambda1 <= -3.1e-6) {
		tmp = R_m * Math.acos(Math.cos((lambda2 - lambda1)));
	} else {
		tmp = R_m * Math.acos((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):
	tmp = 0
	if lambda1 <= -3.1e-6:
		tmp = R_m * math.acos(math.cos((lambda2 - lambda1)))
	else:
		tmp = R_m * math.acos((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)
	tmp = 0.0
	if (lambda1 <= -3.1e-6)
		tmp = Float64(R_m * acos(cos(Float64(lambda2 - lambda1))));
	else
		tmp = Float64(R_m * acos(Float64(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)
	tmp = 0.0;
	if (lambda1 <= -3.1e-6)
		tmp = R_m * acos(cos((lambda2 - lambda1)));
	else
		tmp = R_m * acos((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_] := N[(R$95$s * If[LessEqual[lambda1, -3.1e-6], N[(R$95$m * N[ArcCos[N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -3.1e-6

   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 38.2%

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

   6. Taylor expanded in phi1 around 0 27.4%

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

   if -3.1e-6 < lambda1

   1. Initial program 76.9%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in lambda1 around 0 65.9%

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

      1. *-commutative 65.9%

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

      2. associate-*r* 65.9%

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

      3. cos-neg 65.9%

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

      4. fma-define 66.0%

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

      5. cos-neg 66.0%

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

   7. Simplified 66.0%

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

   8. Taylor expanded in phi1 around 0 33.7%

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

4. Final simplification 31.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.1 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 26.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -5 \cdot 10^{-8}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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 -5e-8)
    (* 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 (lambda1 <= -5e-8) {
		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 (lambda1 <= (-5d-8)) 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 (lambda1 <= -5e-8) {
		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 lambda1 <= -5e-8:
		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 (lambda1 <= -5e-8)
		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 (lambda1 <= -5e-8)
		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[lambda1, -5e-8], 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 lambda1 < -4.9999999999999998e-8

   1. Initial program 60.3%

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

   3. Simplified 60.3%

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

   5. Taylor expanded in phi2 around 0 37.9%

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

   6. Taylor expanded in phi1 around 0 27.2%

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

   7. Taylor expanded in lambda2 around 0 26.7%

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

      1. cos-neg 26.7%

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

   9. Simplified 26.7%

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

   if -4.9999999999999998e-8 < lambda1

   1. Initial program 76.7%

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

   3. Simplified 76.8%

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

   5. Taylor expanded in phi2 around 0 45.5%

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

   6. Taylor expanded in phi1 around 0 26.2%

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

   7. Taylor expanded in lambda1 around 0 20.3%

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

Alternative 23: 26.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \left(R\_m \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\right) \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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 (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) {
	return R_s * (R_m * acos(cos((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 * acos(cos((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 * Math.acos(Math.cos((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 * math.acos(math.cos((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 * acos(cos(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 * acos(cos((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[ArcCos[N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.0%

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

3. Simplified 72.0%

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

5. Taylor expanded in phi2 around 0 43.3%

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

6. Taylor expanded in phi1 around 0 26.5%

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

Alternative 24: 17.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \left(R\_m \cdot \cos^{-1} \cos \lambda_1\right) \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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 (acos (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) {
	return R_s * (R_m * acos(cos(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 * acos(cos(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 * Math.acos(Math.cos(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 * math.acos(math.cos(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 * acos(cos(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 * acos(cos(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[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.0%

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

3. Simplified 72.0%

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

5. Taylor expanded in phi2 around 0 43.3%

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

6. Taylor expanded in phi1 around 0 26.5%

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

7. Taylor expanded in lambda2 around 0 17.5%

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

   1. cos-neg 17.5%

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

9. Simplified 17.5%

   \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \lambda_1} \]
10. Add Preprocessing

Alternative 25: 8.8% 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 #s(literal 1 binary64) 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 72.0%

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

3. Simplified 72.0%

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

5. Taylor expanded in phi2 around 0 43.3%

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

6. Taylor expanded in phi1 around 0 26.5%

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

7. Taylor expanded in R around 0 5.0%

   \[\leadsto \color{blue}{R \cdot \left(\lambda_2 - \lambda_1\right)} \]
8. Add Preprocessing

Alternative 26: 6.8% accurate, 153.3× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \left(\lambda_1 \cdot \left(-R\_m\right)\right) \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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 (* lambda1 (- 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 * (lambda1 * -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 * (lambda1 * -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 * (lambda1 * -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 * (lambda1 * -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(lambda1 * Float64(-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 * (lambda1 * -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[(lambda1 * (-R$95$m)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.0%

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

3. Simplified 72.0%

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

5. Taylor expanded in phi2 around 0 43.3%

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

6. Taylor expanded in phi1 around 0 26.5%

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

7. Taylor expanded in lambda2 around 0 5.2%

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

   1. associate-*r* 5.2%

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

   2. mul-1-neg 5.2%

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

9. Simplified 5.2%

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

10. Final simplification 5.2%

    \[\leadsto \lambda_1 \cdot \left(-R\right) \]
11. Add Preprocessing

Alternative 27: 6.7% 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 #s(literal 1 binary64) 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 72.0%

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

3. Simplified 72.0%

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

5. Taylor expanded in phi2 around 0 43.3%

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

6. Taylor expanded in phi1 around 0 26.5%

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

7. Taylor expanded in lambda2 around inf 4.7%

   \[\leadsto \color{blue}{R \cdot \lambda_2} \]
8. Step-by-step derivation

   1. *-commutative 4.7%

      \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

9. Simplified 4.7%

   \[\leadsto \color{blue}{\lambda_2 \cdot R} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024182 
(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))