Spherical law of cosines

Percentage Accurate: 73.4% → 96.4%

Time: 35.1s

Alternatives: 29

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: 73.4% 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

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;t\_0 + t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right) \leq 0.99999:\\ \;\;\;\;\cos^{-1} \left(t\_0 + t\_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))) (t_1 (* (cos phi1) (cos phi2))))
   (if (<= (+ t_0 (* t_1 (cos (- lambda1 lambda2)))) 0.99999)
     (*
      (acos
       (+
        t_0
        (*
         t_1
         (fma
          (cos lambda2)
          (cos lambda1)
          (log1p (expm1 (* (sin lambda2) (sin lambda1))))))))
      R)
     (* R (- lambda2 lambda1)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double t_1 = cos(phi1) * cos(phi2);
	double tmp;
	if ((t_0 + (t_1 * cos((lambda1 - lambda2)))) <= 0.99999) {
		tmp = acos((t_0 + (t_1 * fma(cos(lambda2), cos(lambda1), log1p(expm1((sin(lambda2) * sin(lambda1)))))))) * R;
	} else {
		tmp = R * (lambda2 - lambda1);
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	t_1 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if (Float64(t_0 + Float64(t_1 * cos(Float64(lambda1 - lambda2)))) <= 0.99999)
		tmp = Float64(acos(Float64(t_0 + Float64(t_1 * fma(cos(lambda2), cos(lambda1), log1p(expm1(Float64(sin(lambda2) * sin(lambda1)))))))) * R);
	else
		tmp = Float64(R * Float64(lambda2 - lambda1));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 + N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.99999], N[(N[ArcCos[N[(t$95$0 + N[(t$95$1 * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[Log[1 + N[(Exp[N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))) < 0.999990000000000046

   1. Initial program 77.7%

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

      1. cos-diff 98.9%

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

      2. distribute-lft-in 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. distribute-lft-out 98.9%

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

      2. *-commutative 98.9%

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

      3. fma-undefine 98.9%

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

      4. *-commutative 98.9%

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

   6. Simplified 98.9%

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

      1. log1p-expm1-u 99.0%

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

   8. Applied egg-rr 99.0%

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

   if 0.999990000000000046 < (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))

   1. Initial program 7.5%

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

      1. *-commutative 7.5%

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

      2. *-commutative 7.5%

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

      3. *-commutative 7.5%

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

      4. *-commutative 7.5%

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

      5. associate-*l* 7.5%

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

      6. associate-*l* 7.5%

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

      7. *-commutative 7.5%

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

      8. cos-neg 7.5%

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

      9. sub-neg 7.5%

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

      10. +-commutative 7.5%

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

      11. distribute-neg-out 7.5%

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

      12. remove-double-neg 7.5%

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

      13. sub-neg 7.5%

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

   3. Simplified 7.5%

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

   5. Taylor expanded in phi2 around 0 7.5%

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

   6. Taylor expanded in phi1 around 0 7.5%

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

   7. Taylor expanded in lambda2 around 0 61.1%

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

      1. neg-mul-1 61.1%

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

      2. sub-neg 61.1%

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

   9. Simplified 61.1%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) \leq 0.99999:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\cos^{-1} \left(t\_0 + t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 10^{-7}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(t\_1, \sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1, t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))) (t_1 (* (cos phi1) (cos phi2))))
   (if (<= (acos (+ t_0 (* t_1 (cos (- lambda1 lambda2))))) 1e-7)
     (* R (- lambda2 lambda1))
     (*
      R
      (acos
       (fma
        t_1
        (+ (* (sin lambda2) (sin lambda1)) (* (cos lambda2) (cos lambda1)))
        t_0))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double t_1 = cos(phi1) * cos(phi2);
	double tmp;
	if (acos((t_0 + (t_1 * cos((lambda1 - lambda2))))) <= 1e-7) {
		tmp = R * (lambda2 - lambda1);
	} else {
		tmp = R * acos(fma(t_1, ((sin(lambda2) * sin(lambda1)) + (cos(lambda2) * cos(lambda1))), t_0));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	t_1 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if (acos(Float64(t_0 + Float64(t_1 * cos(Float64(lambda1 - lambda2))))) <= 1e-7)
		tmp = Float64(R * Float64(lambda2 - lambda1));
	else
		tmp = Float64(R * acos(fma(t_1, Float64(Float64(sin(lambda2) * sin(lambda1)) + Float64(cos(lambda2) * cos(lambda1))), t_0)));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[ArcCos[N[(t$95$0 + N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1e-7], N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$1 * N[(N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))) < 9.9999999999999995e-8

   1. Initial program 7.5%

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

      1. *-commutative 7.5%

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

      2. *-commutative 7.5%

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

      3. *-commutative 7.5%

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

      4. *-commutative 7.5%

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

      5. associate-*l* 7.5%

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

      6. associate-*l* 7.5%

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

      7. *-commutative 7.5%

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

      8. cos-neg 7.5%

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

      9. sub-neg 7.5%

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

      10. +-commutative 7.5%

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

      11. distribute-neg-out 7.5%

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

      12. remove-double-neg 7.5%

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

      13. sub-neg 7.5%

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

   3. Simplified 7.5%

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

   5. Taylor expanded in phi2 around 0 7.5%

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

   6. Taylor expanded in phi1 around 0 7.5%

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

   7. Taylor expanded in lambda2 around 0 61.1%

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

      1. neg-mul-1 61.1%

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

      2. sub-neg 61.1%

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

   9. Simplified 61.1%

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

   if 9.9999999999999995e-8 < (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))))

   1. Initial program 77.7%

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

      1. *-commutative 77.7%

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

      2. *-commutative 77.7%

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

      3. *-commutative 77.7%

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

      4. *-commutative 77.7%

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

      5. associate-*l* 77.7%

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

      6. associate-*l* 77.7%

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

      7. *-commutative 77.7%

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

      8. cos-neg 77.7%

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

      9. sub-neg 77.7%

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

      10. +-commutative 77.7%

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

      11. distribute-neg-out 77.7%

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

      12. remove-double-neg 77.7%

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

      13. sub-neg 77.7%

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

   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
   5. Step-by-step derivation

      1. cos-diff 55.1%

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

      2. +-commutative 55.1%

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

      3. *-commutative 55.1%

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

      4. *-commutative 55.1%

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

   6. Applied egg-rr 98.9%

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

4. Final simplification 97.3%

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

Alternative 3: 96.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\cos^{-1} \left(t\_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 10^{-7}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<=
        (acos (+ t_0 (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
        1e-7)
     (* R (- lambda2 lambda1))
     (*
      R
      (acos
       (+
        t_0
        (*
         (cos phi1)
         (*
          (cos phi2)
          (+
           (* (sin lambda2) (sin lambda1))
           (* (cos lambda2) (cos lambda1)))))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, 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))))) <= 1e-7) {
		tmp = R * (lambda2 - lambda1);
	} else {
		tmp = R * acos((t_0 + (cos(phi1) * (cos(phi2) * ((sin(lambda2) * sin(lambda1)) + (cos(lambda2) * cos(lambda1)))))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(phi1) * sin(phi2)
    if (acos((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) <= 1d-7) then
        tmp = r * (lambda2 - lambda1)
    else
        tmp = r * acos((t_0 + (cos(phi1) * (cos(phi2) * ((sin(lambda2) * sin(lambda1)) + (cos(lambda2) * cos(lambda1)))))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, 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))))) <= 1e-7) {
		tmp = R * (lambda2 - lambda1);
	} else {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * (Math.cos(phi2) * ((Math.sin(lambda2) * Math.sin(lambda1)) + (Math.cos(lambda2) * Math.cos(lambda1)))))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, 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))))) <= 1e-7:
		tmp = R * (lambda2 - lambda1)
	else:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * (math.cos(phi2) * ((math.sin(lambda2) * math.sin(lambda1)) + (math.cos(lambda2) * math.cos(lambda1)))))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) <= 1e-7)
		tmp = Float64(R * Float64(lambda2 - lambda1));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * Float64(cos(phi2) * Float64(Float64(sin(lambda2) * sin(lambda1)) + Float64(cos(lambda2) * cos(lambda1))))))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if (acos((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) <= 1e-7)
		tmp = R * (lambda2 - lambda1);
	else
		tmp = R * acos((t_0 + (cos(phi1) * (cos(phi2) * ((sin(lambda2) * sin(lambda1)) + (cos(lambda2) * cos(lambda1)))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[ArcCos[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1e-7], N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))) < 9.9999999999999995e-8

   1. Initial program 7.5%

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

      1. *-commutative 7.5%

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

      2. *-commutative 7.5%

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

      3. *-commutative 7.5%

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

      4. *-commutative 7.5%

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

      5. associate-*l* 7.5%

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

      6. associate-*l* 7.5%

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

      7. *-commutative 7.5%

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

      8. cos-neg 7.5%

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

      9. sub-neg 7.5%

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

      10. +-commutative 7.5%

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

      11. distribute-neg-out 7.5%

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

      12. remove-double-neg 7.5%

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

      13. sub-neg 7.5%

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

   3. Simplified 7.5%

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

   5. Taylor expanded in phi2 around 0 7.5%

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

   6. Taylor expanded in phi1 around 0 7.5%

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

   7. Taylor expanded in lambda2 around 0 61.1%

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

      1. neg-mul-1 61.1%

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

      2. sub-neg 61.1%

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

   9. Simplified 61.1%

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

   if 9.9999999999999995e-8 < (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))))

   1. Initial program 77.7%

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

      1. cos-diff 98.9%

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

      2. distribute-lft-in 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. distribute-lft-out 98.9%

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

      2. *-commutative 98.9%

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

      3. fma-undefine 98.9%

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

      4. *-commutative 98.9%

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

   6. Simplified 98.9%

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

   7. Taylor expanded in phi1 around inf 98.9%

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

4. Final simplification 97.3%

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

Alternative 4: 96.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;t\_0 + t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right) \leq 0.99999:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + t\_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))) (t_1 (* (cos phi1) (cos phi2))))
   (if (<= (+ t_0 (* t_1 (cos (- lambda1 lambda2)))) 0.99999)
     (*
      R
      (acos
       (+
        t_0
        (*
         t_1
         (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1)))))))
     (* R (- lambda2 lambda1)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double t_1 = cos(phi1) * cos(phi2);
	double tmp;
	if ((t_0 + (t_1 * cos((lambda1 - lambda2)))) <= 0.99999) {
		tmp = R * acos((t_0 + (t_1 * fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1))))));
	} else {
		tmp = R * (lambda2 - lambda1);
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	t_1 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if (Float64(t_0 + Float64(t_1 * cos(Float64(lambda1 - lambda2)))) <= 0.99999)
		tmp = Float64(R * acos(Float64(t_0 + Float64(t_1 * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1)))))));
	else
		tmp = Float64(R * Float64(lambda2 - lambda1));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 + N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.99999], N[(R * N[ArcCos[N[(t$95$0 + N[(t$95$1 * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))) < 0.999990000000000046

   1. Initial program 77.7%

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

      1. cos-diff 98.9%

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

      2. distribute-lft-in 98.9%

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

   4. Applied egg-rr 98.9%

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

      1. distribute-lft-out 98.9%

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

      2. *-commutative 98.9%

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

      3. fma-undefine 98.9%

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

      4. *-commutative 98.9%

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

   6. Simplified 98.9%

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

   if 0.999990000000000046 < (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))

   1. Initial program 7.5%

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

      1. *-commutative 7.5%

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

      2. *-commutative 7.5%

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

      3. *-commutative 7.5%

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

      4. *-commutative 7.5%

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

      5. associate-*l* 7.5%

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

      6. associate-*l* 7.5%

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

      7. *-commutative 7.5%

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

      8. cos-neg 7.5%

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

      9. sub-neg 7.5%

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

      10. +-commutative 7.5%

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

      11. distribute-neg-out 7.5%

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

      12. remove-double-neg 7.5%

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

      13. sub-neg 7.5%

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

   3. Simplified 7.5%

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

   5. Taylor expanded in phi2 around 0 7.5%

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

   6. Taylor expanded in phi1 around 0 7.5%

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

   7. Taylor expanded in lambda2 around 0 61.1%

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

      1. neg-mul-1 61.1%

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

      2. sub-neg 61.1%

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

   9. Simplified 61.1%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) \leq 0.99999:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -2.4 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 4.6 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right), t\_0\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= phi2 -2.4e-6)
     (*
      R
      (acos
       (fma
        (sin phi1)
        (sin phi2)
        (* (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))
     (if (<= phi2 4.6e-7)
       (*
        R
        (acos
         (+
          t_0
          (*
           (cos phi1)
           (+
            (* (sin lambda2) (sin lambda1))
            (* (cos lambda2) (cos lambda1)))))))
       (*
        R
        (log
         (exp
          (acos
           (fma
            (cos phi1)
            (* (cos phi2) (cos (- lambda2 lambda1)))
            t_0)))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi2 <= -2.4e-6) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * (cos(phi2) * cos((lambda1 - lambda2))))));
	} else if (phi2 <= 4.6e-7) {
		tmp = R * acos((t_0 + (cos(phi1) * ((sin(lambda2) * sin(lambda1)) + (cos(lambda2) * cos(lambda1))))));
	} else {
		tmp = R * log(exp(acos(fma(cos(phi1), (cos(phi2) * cos((lambda2 - lambda1))), t_0))));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi2 <= -2.4e-6)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))))));
	elseif (phi2 <= 4.6e-7)
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * Float64(Float64(sin(lambda2) * sin(lambda1)) + Float64(cos(lambda2) * cos(lambda1)))))));
	else
		tmp = Float64(R * log(exp(acos(fma(cos(phi1), Float64(cos(phi2) * cos(Float64(lambda2 - lambda1))), t_0)))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -2.4e-6], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 4.6e-7], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[Log[N[Exp[N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -2.3999999999999999e-6

   1. Initial program 79.6%

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

   3. Simplified 79.7%

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

   if -2.3999999999999999e-6 < phi2 < 4.5999999999999999e-7

   1. Initial program 68.9%

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

      1. cos-diff 90.8%

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

      2. distribute-lft-in 90.8%

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

   4. Applied egg-rr 90.8%

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

      1. distribute-lft-out 90.8%

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

      2. *-commutative 90.8%

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

      3. fma-undefine 90.8%

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

      4. *-commutative 90.8%

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

   6. Simplified 90.8%

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

   7. Taylor expanded in phi2 around 0 90.8%

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

   if 4.5999999999999999e-7 < phi2

   1. Initial program 80.4%

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

      1. *-commutative 80.4%

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

      2. *-commutative 80.4%

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

      3. *-commutative 80.4%

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

      4. *-commutative 80.4%

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

      5. associate-*l* 80.4%

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

      6. associate-*l* 80.4%

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

      7. *-commutative 80.4%

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

      8. cos-neg 80.4%

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

      9. sub-neg 80.4%

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

      10. +-commutative 80.4%

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

      11. distribute-neg-out 80.4%

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

      12. remove-double-neg 80.4%

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

      13. sub-neg 80.4%

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

   3. Simplified 80.5%

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

      1. add-log-exp 80.6%

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

      2. cos-diff 99.4%

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

      3. *-commutative 99.4%

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

      4. *-commutative 99.4%

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

      5. cos-diff 80.6%

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

      6. fma-define 80.4%

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

      7. associate-*r* 80.5%

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

      8. fma-undefine 80.6%

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

   6. Applied egg-rr 80.6%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.4 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 4.6 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.85 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.55 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -1.85e-5)
   (*
    R
    (acos
     (fma
      (sin phi1)
      (sin phi2)
      (* (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))
   (if (<= phi2 1.55e-6)
     (*
      R
      (acos
       (+
        (* (sin phi1) (sin phi2))
        (*
         (cos phi1)
         (+
          (* (sin lambda2) (sin lambda1))
          (* (cos lambda2) (cos lambda1)))))))
     (*
      R
      (log
       (exp
        (acos
         (fma
          (sin phi1)
          (sin phi2)
          (* (cos phi1) (* (cos phi2) (cos (- lambda2 lambda1))))))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -1.85e-5) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * (cos(phi2) * cos((lambda1 - lambda2))))));
	} else if (phi2 <= 1.55e-6) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * ((sin(lambda2) * sin(lambda1)) + (cos(lambda2) * cos(lambda1))))));
	} else {
		tmp = R * log(exp(acos(fma(sin(phi1), sin(phi2), (cos(phi1) * (cos(phi2) * cos((lambda2 - lambda1))))))));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -1.85e-5)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))))));
	elseif (phi2 <= 1.55e-6)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * Float64(Float64(sin(lambda2) * sin(lambda1)) + Float64(cos(lambda2) * cos(lambda1)))))));
	else
		tmp = Float64(R * log(exp(acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * Float64(cos(phi2) * cos(Float64(lambda2 - lambda1)))))))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -1.85e-5], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.55e-6], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[Log[N[Exp[N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.84999999999999991e-5

   1. Initial program 79.6%

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

   3. Simplified 79.7%

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

   if -1.84999999999999991e-5 < phi2 < 1.55e-6

   1. Initial program 68.9%

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

      1. cos-diff 90.8%

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

      2. distribute-lft-in 90.8%

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

   4. Applied egg-rr 90.8%

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

      1. distribute-lft-out 90.8%

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

      2. *-commutative 90.8%

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

      3. fma-undefine 90.8%

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

      4. *-commutative 90.8%

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

   6. Simplified 90.8%

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

   7. Taylor expanded in phi2 around 0 90.8%

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

   if 1.55e-6 < phi2

   1. Initial program 80.4%

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

      1. cos-diff 99.3%

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

      2. distribute-lft-in 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. distribute-lft-out 99.3%

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

      2. *-commutative 99.3%

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

      3. fma-undefine 99.3%

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

      4. *-commutative 99.3%

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

   6. Simplified 99.3%

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

      1. log1p-expm1-u 99.3%

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

   8. Applied egg-rr 99.3%

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

      1. add-log-exp 99.3%

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

      2. fma-define 99.5%

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

      3. fma-undefine 99.5%

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

      4. log1p-expm1-u 99.5%

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

      5. cos-diff 80.6%

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

      6. associate-*l* 80.6%

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

   10. Applied egg-rr 80.6%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.85 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.55 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -1.95 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot t\_0\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.4 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_1 + \cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, t\_0, t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos (- lambda1 lambda2))))
        (t_1 (* (sin phi1) (sin phi2))))
   (if (<= phi2 -1.95e-7)
     (* R (acos (fma (sin phi1) (sin phi2) (* (cos phi1) t_0))))
     (if (<= phi2 1.4e-10)
       (*
        R
        (acos
         (+
          t_1
          (*
           (cos phi1)
           (+
            (* (sin lambda2) (sin lambda1))
            (* (cos lambda2) (cos lambda1)))))))
       (* R (acos (fma (cos phi1) t_0 t_1)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos((lambda1 - lambda2));
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi2 <= -1.95e-7) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * t_0)));
	} else if (phi2 <= 1.4e-10) {
		tmp = R * acos((t_1 + (cos(phi1) * ((sin(lambda2) * sin(lambda1)) + (cos(lambda2) * cos(lambda1))))));
	} else {
		tmp = R * acos(fma(cos(phi1), t_0, t_1));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi2 <= -1.95e-7)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * t_0))));
	elseif (phi2 <= 1.4e-10)
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(phi1) * Float64(Float64(sin(lambda2) * sin(lambda1)) + Float64(cos(lambda2) * cos(lambda1)))))));
	else
		tmp = Float64(R * acos(fma(cos(phi1), t_0, t_1)));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.95e-7], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.4e-10], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.95000000000000012e-7

   1. Initial program 79.6%

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

   3. Simplified 79.7%

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

   if -1.95000000000000012e-7 < phi2 < 1.40000000000000008e-10

   1. Initial program 68.9%

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

      1. cos-diff 90.8%

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

      2. distribute-lft-in 90.8%

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

   4. Applied egg-rr 90.8%

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

      1. distribute-lft-out 90.8%

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

      2. *-commutative 90.8%

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

      3. fma-undefine 90.8%

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

      4. *-commutative 90.8%

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

   6. Simplified 90.8%

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

   7. Taylor expanded in phi2 around 0 90.8%

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

   if 1.40000000000000008e-10 < phi2

   1. Initial program 80.4%

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

   3. Simplified 80.5%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.95 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.4 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -6.5 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot t\_0\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.4 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, t\_0, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos (- lambda1 lambda2)))))
   (if (<= phi2 -6.5e-6)
     (* R (acos (fma (sin phi1) (sin phi2) (* (cos phi1) t_0))))
     (if (<= phi2 1.4e-10)
       (*
        R
        (acos
         (+
          (*
           (cos phi1)
           (+ (* (sin lambda2) (sin lambda1)) (* (cos lambda2) (cos lambda1))))
          (* (sin phi1) phi2))))
       (* R (acos (fma (cos phi1) t_0 (* (sin phi1) (sin phi2)))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -6.5e-6) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * t_0)));
	} else if (phi2 <= 1.4e-10) {
		tmp = R * acos(((cos(phi1) * ((sin(lambda2) * sin(lambda1)) + (cos(lambda2) * cos(lambda1)))) + (sin(phi1) * phi2)));
	} else {
		tmp = R * acos(fma(cos(phi1), t_0, (sin(phi1) * sin(phi2))));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi2 <= -6.5e-6)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * t_0))));
	elseif (phi2 <= 1.4e-10)
		tmp = Float64(R * acos(Float64(Float64(cos(phi1) * Float64(Float64(sin(lambda2) * sin(lambda1)) + Float64(cos(lambda2) * cos(lambda1)))) + Float64(sin(phi1) * phi2))));
	else
		tmp = Float64(R * acos(fma(cos(phi1), t_0, Float64(sin(phi1) * sin(phi2)))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -6.5e-6], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.4e-10], N[(R * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -6.4999999999999996e-6

   1. Initial program 79.6%

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

   3. Simplified 79.7%

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

   if -6.4999999999999996e-6 < phi2 < 1.40000000000000008e-10

   1. Initial program 68.9%

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

      1. *-commutative 68.9%

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

      2. *-commutative 68.9%

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

      3. *-commutative 68.9%

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

      4. *-commutative 68.9%

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

      5. associate-*l* 68.9%

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

      6. associate-*l* 68.9%

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

      7. *-commutative 68.9%

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

      8. cos-neg 68.9%

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

      9. sub-neg 68.9%

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

      10. +-commutative 68.9%

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

      11. distribute-neg-out 68.9%

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

      12. remove-double-neg 68.9%

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

      13. sub-neg 68.9%

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

   3. Simplified 68.9%

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

      1. cos-diff 90.3%

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

      2. +-commutative 90.3%

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

      3. *-commutative 90.3%

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

      4. *-commutative 90.3%

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

   6. Applied egg-rr 90.8%

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

   7. Taylor expanded in phi2 around 0 90.8%

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

   if 1.40000000000000008e-10 < phi2

   1. Initial program 80.4%

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

   3. Simplified 80.5%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -6.5 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.4 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.55 \cdot 10^{-9} \lor \neg \left(\phi_2 \leq 9 \cdot 10^{-12}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi2 -1.55e-9) (not (<= phi2 9e-12)))
   (*
    R
    (acos
     (fma
      (* (cos phi1) (cos phi2))
      (cos (- lambda2 lambda1))
      (* (sin phi1) (sin phi2)))))
   (*
    R
    (acos
     (*
      (cos phi1)
      (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -1.55e-9) || !(phi2 <= 9e-12)) {
		tmp = R * acos(fma((cos(phi1) * cos(phi2)), cos((lambda2 - lambda1)), (sin(phi1) * sin(phi2))));
	} else {
		tmp = R * acos((cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1)))));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi2 <= -1.55e-9) || !(phi2 <= 9e-12))
		tmp = Float64(R * acos(fma(Float64(cos(phi1) * cos(phi2)), cos(Float64(lambda2 - lambda1)), Float64(sin(phi1) * sin(phi2)))));
	else
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1))))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi2, -1.55e-9], N[Not[LessEqual[phi2, 9e-12]], $MachinePrecision]], N[(R * 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 * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.55000000000000002e-9 or 8.99999999999999962e-12 < phi2

   1. Initial program 79.7%

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

      1. *-commutative 79.7%

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

      2. *-commutative 79.7%

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

      3. *-commutative 79.7%

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

      4. *-commutative 79.7%

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

      5. associate-*l* 79.7%

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

      6. associate-*l* 79.7%

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

      7. *-commutative 79.7%

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

      8. cos-neg 79.7%

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

      9. sub-neg 79.7%

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

      10. +-commutative 79.7%

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

      11. distribute-neg-out 79.7%

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

      12. remove-double-neg 79.7%

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

      13. sub-neg 79.7%

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

   3. Simplified 79.7%

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

   if -1.55000000000000002e-9 < phi2 < 8.99999999999999962e-12

   1. Initial program 69.1%

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

      1. *-commutative 69.1%

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

      2. *-commutative 69.1%

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

      3. *-commutative 69.1%

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

      4. *-commutative 69.1%

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

      5. associate-*l* 69.1%

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

      6. associate-*l* 69.1%

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

      7. *-commutative 69.1%

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

      8. cos-neg 69.1%

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

      9. sub-neg 69.1%

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

      10. +-commutative 69.1%

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

      11. distribute-neg-out 69.1%

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

      12. remove-double-neg 69.1%

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

      13. sub-neg 69.1%

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

   3. Simplified 69.1%

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

   5. Taylor expanded in phi2 around 0 68.9%

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

      1. cos-diff 90.4%

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

      2. *-commutative 90.4%

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

      3. *-commutative 90.4%

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

   7. Applied egg-rr 90.4%

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

      1. *-commutative 90.4%

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

      2. fma-undefine 90.4%

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

      3. *-commutative 90.4%

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

   9. Simplified 90.4%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.55 \cdot 10^{-9} \lor \neg \left(\phi_2 \leq 9 \cdot 10^{-12}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -6.4 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), t\_0\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 8.2 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= phi2 -6.4e-9)
     (* R (acos (fma (* (cos phi1) (cos phi2)) (cos (- lambda2 lambda1)) t_0)))
     (if (<= phi2 8.2e-11)
       (*
        R
        (acos
         (*
          (cos phi1)
          (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1))))))
       (*
        R
        (acos
         (fma (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))) t_0)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi2 <= -6.4e-9) {
		tmp = R * acos(fma((cos(phi1) * cos(phi2)), cos((lambda2 - lambda1)), t_0));
	} else if (phi2 <= 8.2e-11) {
		tmp = R * acos((cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1)))));
	} else {
		tmp = R * acos(fma(cos(phi1), (cos(phi2) * cos((lambda1 - lambda2))), t_0));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi2 <= -6.4e-9)
		tmp = Float64(R * acos(fma(Float64(cos(phi1) * cos(phi2)), cos(Float64(lambda2 - lambda1)), t_0)));
	elseif (phi2 <= 8.2e-11)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1))))));
	else
		tmp = Float64(R * acos(fma(cos(phi1), Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))), t_0)));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -6.4e-9], N[(R * 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], If[LessEqual[phi2, 8.2e-11], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * 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]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -6.40000000000000023e-9

   1. Initial program 79.9%

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

      1. *-commutative 79.9%

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

      2. *-commutative 79.9%

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

      3. *-commutative 79.9%

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

      4. *-commutative 79.9%

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

      5. associate-*l* 79.9%

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

      6. associate-*l* 79.9%

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

      7. *-commutative 79.9%

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

      8. cos-neg 79.9%

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

      9. sub-neg 79.9%

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

      10. +-commutative 79.9%

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

      11. distribute-neg-out 79.9%

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

      12. remove-double-neg 79.9%

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

      13. sub-neg 79.9%

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

   3. Simplified 79.9%

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

   if -6.40000000000000023e-9 < phi2 < 8.2000000000000001e-11

   1. Initial program 68.6%

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

      1. *-commutative 68.6%

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

      2. *-commutative 68.6%

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

      3. *-commutative 68.6%

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

      4. *-commutative 68.6%

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

      5. associate-*l* 68.6%

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

      6. associate-*l* 68.6%

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

      7. *-commutative 68.6%

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

      8. cos-neg 68.6%

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

      9. sub-neg 68.6%

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

      10. +-commutative 68.6%

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

      11. distribute-neg-out 68.6%

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

      12. remove-double-neg 68.6%

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

      13. sub-neg 68.6%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in phi2 around 0 68.5%

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

      1. cos-diff 90.2%

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

      2. *-commutative 90.2%

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

      3. *-commutative 90.2%

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

   7. Applied egg-rr 90.2%

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

      1. *-commutative 90.2%

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

      2. fma-undefine 90.3%

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

      3. *-commutative 90.3%

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

   9. Simplified 90.3%

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

   if 8.2000000000000001e-11 < phi2

   1. Initial program 80.4%

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

   3. Simplified 80.5%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -6.4 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 8.2 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -4.8 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot t\_0\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 3.4 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, t\_0, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos (- lambda1 lambda2)))))
   (if (<= phi2 -4.8e-8)
     (* R (acos (fma (sin phi1) (sin phi2) (* (cos phi1) t_0))))
     (if (<= phi2 3.4e-11)
       (*
        R
        (acos
         (*
          (cos phi1)
          (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1))))))
       (* R (acos (fma (cos phi1) t_0 (* (sin phi1) (sin phi2)))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -4.8e-8) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * t_0)));
	} else if (phi2 <= 3.4e-11) {
		tmp = R * acos((cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1)))));
	} else {
		tmp = R * acos(fma(cos(phi1), t_0, (sin(phi1) * sin(phi2))));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi2 <= -4.8e-8)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * t_0))));
	elseif (phi2 <= 3.4e-11)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1))))));
	else
		tmp = Float64(R * acos(fma(cos(phi1), t_0, Float64(sin(phi1) * sin(phi2)))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -4.8e-8], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 3.4e-11], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -4.79999999999999997e-8

   1. Initial program 79.6%

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

   3. Simplified 79.7%

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

   if -4.79999999999999997e-8 < phi2 < 3.3999999999999999e-11

   1. Initial program 68.9%

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

      1. *-commutative 68.9%

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

      2. *-commutative 68.9%

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

      3. *-commutative 68.9%

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

      4. *-commutative 68.9%

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

      5. associate-*l* 68.9%

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

      6. associate-*l* 68.9%

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

      7. *-commutative 68.9%

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

      8. cos-neg 68.9%

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

      9. sub-neg 68.9%

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

      10. +-commutative 68.9%

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

      11. distribute-neg-out 68.9%

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

      12. remove-double-neg 68.9%

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

      13. sub-neg 68.9%

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

   3. Simplified 68.9%

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

   5. Taylor expanded in phi2 around 0 68.8%

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

      1. cos-diff 90.3%

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

      2. *-commutative 90.3%

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

      3. *-commutative 90.3%

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

   7. Applied egg-rr 90.3%

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

      1. *-commutative 90.3%

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

      2. fma-undefine 90.4%

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

      3. *-commutative 90.4%

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

   9. Simplified 90.4%

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

   if 3.3999999999999999e-11 < phi2

   1. Initial program 80.4%

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

   3. Simplified 80.5%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -4.8 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 3.4 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 83.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -7 \cdot 10^{-10} \lor \neg \left(\phi_2 \leq 4.6 \cdot 10^{-11}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi2 -7e-10) (not (<= phi2 4.6e-11)))
   (*
    R
    (acos
     (+
      (* (sin phi1) (sin phi2))
      (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))))
   (*
    R
    (acos
     (*
      (cos phi1)
      (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -7e-10) || !(phi2 <= 4.6e-11)) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	} else {
		tmp = R * acos((cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1)))));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi2 <= -7e-10) || !(phi2 <= 4.6e-11))
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1))))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi2, -7e-10], N[Not[LessEqual[phi2, 4.6e-11]], $MachinePrecision]], N[(R * 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]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -6.99999999999999961e-10 or 4.60000000000000027e-11 < phi2

   1. Initial program 80.1%

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

   if -6.99999999999999961e-10 < phi2 < 4.60000000000000027e-11

   1. Initial program 68.6%

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

      1. *-commutative 68.6%

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

      2. *-commutative 68.6%

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

      3. *-commutative 68.6%

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

      4. *-commutative 68.6%

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

      5. associate-*l* 68.6%

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

      6. associate-*l* 68.6%

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

      7. *-commutative 68.6%

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

      8. cos-neg 68.6%

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

      9. sub-neg 68.6%

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

      10. +-commutative 68.6%

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

      11. distribute-neg-out 68.6%

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

      12. remove-double-neg 68.6%

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

      13. sub-neg 68.6%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in phi2 around 0 68.5%

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

      1. cos-diff 90.2%

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

      2. *-commutative 90.2%

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

      3. *-commutative 90.2%

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

   7. Applied egg-rr 90.2%

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

      1. *-commutative 90.2%

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

      2. fma-undefine 90.3%

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

      3. *-commutative 90.3%

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

   9. Simplified 90.3%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -7 \cdot 10^{-10} \lor \neg \left(\phi_2 \leq 4.6 \cdot 10^{-11}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ t_1 := \sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\\ \mathbf{if}\;\phi_2 \leq -2.3 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 5.2:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_1\right)\\ \mathbf{elif}\;\phi_2 \leq 2.6 \cdot 10^{+156} \lor \neg \left(\phi_2 \leq 1.95 \cdot 10^{+262}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          R
          (acos
           (+
            (* (sin phi1) (sin phi2))
            (* (* (cos phi1) (cos phi2)) (cos lambda2))))))
        (t_1
         (+ (* (sin lambda2) (sin lambda1)) (* (cos lambda2) (cos lambda1)))))
   (if (<= phi2 -2.3e-7)
     t_0
     (if (<= phi2 5.2)
       (* R (acos (* (cos phi1) t_1)))
       (if (or (<= phi2 2.6e+156) (not (<= phi2 1.95e+262)))
         (* R (acos (* (cos phi2) t_1)))
         t_0)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos(lambda2))));
	double t_1 = (sin(lambda2) * sin(lambda1)) + (cos(lambda2) * cos(lambda1));
	double tmp;
	if (phi2 <= -2.3e-7) {
		tmp = t_0;
	} else if (phi2 <= 5.2) {
		tmp = R * acos((cos(phi1) * t_1));
	} else if ((phi2 <= 2.6e+156) || !(phi2 <= 1.95e+262)) {
		tmp = R * acos((cos(phi2) * t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = r * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos(lambda2))))
    t_1 = (sin(lambda2) * sin(lambda1)) + (cos(lambda2) * cos(lambda1))
    if (phi2 <= (-2.3d-7)) then
        tmp = t_0
    else if (phi2 <= 5.2d0) then
        tmp = r * acos((cos(phi1) * t_1))
    else if ((phi2 <= 2.6d+156) .or. (.not. (phi2 <= 1.95d+262))) then
        tmp = r * acos((cos(phi2) * t_1))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos(lambda2))));
	double t_1 = (Math.sin(lambda2) * Math.sin(lambda1)) + (Math.cos(lambda2) * Math.cos(lambda1));
	double tmp;
	if (phi2 <= -2.3e-7) {
		tmp = t_0;
	} else if (phi2 <= 5.2) {
		tmp = R * Math.acos((Math.cos(phi1) * t_1));
	} else if ((phi2 <= 2.6e+156) || !(phi2 <= 1.95e+262)) {
		tmp = R * Math.acos((Math.cos(phi2) * t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + ((math.cos(phi1) * math.cos(phi2)) * math.cos(lambda2))))
	t_1 = (math.sin(lambda2) * math.sin(lambda1)) + (math.cos(lambda2) * math.cos(lambda1))
	tmp = 0
	if phi2 <= -2.3e-7:
		tmp = t_0
	elif phi2 <= 5.2:
		tmp = R * math.acos((math.cos(phi1) * t_1))
	elif (phi2 <= 2.6e+156) or not (phi2 <= 1.95e+262):
		tmp = R * math.acos((math.cos(phi2) * t_1))
	else:
		tmp = t_0
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(lambda2)))))
	t_1 = Float64(Float64(sin(lambda2) * sin(lambda1)) + Float64(cos(lambda2) * cos(lambda1)))
	tmp = 0.0
	if (phi2 <= -2.3e-7)
		tmp = t_0;
	elseif (phi2 <= 5.2)
		tmp = Float64(R * acos(Float64(cos(phi1) * t_1)));
	elseif ((phi2 <= 2.6e+156) || !(phi2 <= 1.95e+262))
		tmp = Float64(R * acos(Float64(cos(phi2) * t_1)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos(lambda2))));
	t_1 = (sin(lambda2) * sin(lambda1)) + (cos(lambda2) * cos(lambda1));
	tmp = 0.0;
	if (phi2 <= -2.3e-7)
		tmp = t_0;
	elseif (phi2 <= 5.2)
		tmp = R * acos((cos(phi1) * t_1));
	elseif ((phi2 <= 2.6e+156) || ~((phi2 <= 1.95e+262)))
		tmp = R * acos((cos(phi2) * t_1));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(R * 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]}, Block[{t$95$1 = N[(N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -2.3e-7], t$95$0, If[LessEqual[phi2, 5.2], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[phi2, 2.6e+156], N[Not[LessEqual[phi2, 1.95e+262]], $MachinePrecision]], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -2.29999999999999995e-7 or 2.60000000000000019e156 < phi2 < 1.94999999999999993e262

   1. Initial program 82.8%

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

   3. Taylor expanded in lambda1 around 0 54.0%

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

      1. cos-neg 54.0%

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

      2. associate-*r* 54.0%

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

      3. *-commutative 54.0%

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

   5. Simplified 54.0%

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

   if -2.29999999999999995e-7 < phi2 < 5.20000000000000018

   1. Initial program 68.5%

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

      1. *-commutative 68.5%

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

      2. *-commutative 68.5%

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

      3. *-commutative 68.5%

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

      4. *-commutative 68.5%

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

      5. associate-*l* 68.5%

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

      6. associate-*l* 68.5%

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

      7. *-commutative 68.5%

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

      8. cos-neg 68.5%

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

      9. sub-neg 68.5%

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

      10. +-commutative 68.5%

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

      11. distribute-neg-out 68.5%

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

      12. remove-double-neg 68.5%

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

      13. sub-neg 68.5%

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

   3. Simplified 68.5%

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

   5. Taylor expanded in phi2 around 0 68.4%

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

      1. cos-diff 89.8%

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

      2. +-commutative 89.8%

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

      3. *-commutative 89.8%

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

      4. *-commutative 89.8%

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

   7. Applied egg-rr 89.8%

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

   if 5.20000000000000018 < phi2 < 2.60000000000000019e156 or 1.94999999999999993e262 < phi2

   1. Initial program 74.2%

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

      1. *-commutative 74.2%

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

      2. *-commutative 74.2%

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

      3. *-commutative 74.2%

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

      4. *-commutative 74.2%

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

      5. associate-*l* 74.2%

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

      6. associate-*l* 74.2%

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

      7. *-commutative 74.2%

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

      8. cos-neg 74.2%

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

      9. sub-neg 74.2%

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

      10. +-commutative 74.2%

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

      11. distribute-neg-out 74.2%

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

      12. remove-double-neg 74.2%

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

      13. sub-neg 74.2%

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

   3. Simplified 74.3%

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

      1. cos-diff 19.9%

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

      2. +-commutative 19.9%

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

      3. *-commutative 19.9%

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

      4. *-commutative 19.9%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in phi1 around 0 59.5%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.3 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 5.2:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.6 \cdot 10^{+156} \lor \neg \left(\phi_2 \leq 1.95 \cdot 10^{+262}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\\ \mathbf{if}\;\phi_2 \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 5.2:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_1\right)\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{+102}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2)))
        (t_1
         (+ (* (sin lambda2) (sin lambda1)) (* (cos lambda2) (cos lambda1)))))
   (if (<= phi2 -3.2e-7)
     (* R (acos (+ t_0 (* (* (cos phi1) (cos phi2)) (cos lambda2)))))
     (if (<= phi2 5.2)
       (* R (acos (* (cos phi1) t_1)))
       (if (<= phi2 1.05e+102)
         (* R (acos (* (cos phi2) t_1)))
         (* R (acos (+ t_0 (* (cos phi1) (* (cos phi2) (cos lambda1)))))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double t_1 = (sin(lambda2) * sin(lambda1)) + (cos(lambda2) * cos(lambda1));
	double tmp;
	if (phi2 <= -3.2e-7) {
		tmp = R * acos((t_0 + ((cos(phi1) * cos(phi2)) * cos(lambda2))));
	} else if (phi2 <= 5.2) {
		tmp = R * acos((cos(phi1) * t_1));
	} else if (phi2 <= 1.05e+102) {
		tmp = R * acos((cos(phi2) * t_1));
	} else {
		tmp = R * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda1)))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(phi1) * sin(phi2)
    t_1 = (sin(lambda2) * sin(lambda1)) + (cos(lambda2) * cos(lambda1))
    if (phi2 <= (-3.2d-7)) then
        tmp = r * acos((t_0 + ((cos(phi1) * cos(phi2)) * cos(lambda2))))
    else if (phi2 <= 5.2d0) then
        tmp = r * acos((cos(phi1) * t_1))
    else if (phi2 <= 1.05d+102) then
        tmp = r * acos((cos(phi2) * t_1))
    else
        tmp = r * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda1)))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(phi1) * Math.sin(phi2);
	double t_1 = (Math.sin(lambda2) * Math.sin(lambda1)) + (Math.cos(lambda2) * Math.cos(lambda1));
	double tmp;
	if (phi2 <= -3.2e-7) {
		tmp = R * Math.acos((t_0 + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos(lambda2))));
	} else if (phi2 <= 5.2) {
		tmp = R * Math.acos((Math.cos(phi1) * t_1));
	} else if (phi2 <= 1.05e+102) {
		tmp = R * Math.acos((Math.cos(phi2) * t_1));
	} else {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * (Math.cos(phi2) * Math.cos(lambda1)))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * math.sin(phi2)
	t_1 = (math.sin(lambda2) * math.sin(lambda1)) + (math.cos(lambda2) * math.cos(lambda1))
	tmp = 0
	if phi2 <= -3.2e-7:
		tmp = R * math.acos((t_0 + ((math.cos(phi1) * math.cos(phi2)) * math.cos(lambda2))))
	elif phi2 <= 5.2:
		tmp = R * math.acos((math.cos(phi1) * t_1))
	elif phi2 <= 1.05e+102:
		tmp = R * math.acos((math.cos(phi2) * t_1))
	else:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * (math.cos(phi2) * math.cos(lambda1)))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	t_1 = Float64(Float64(sin(lambda2) * sin(lambda1)) + Float64(cos(lambda2) * cos(lambda1)))
	tmp = 0.0
	if (phi2 <= -3.2e-7)
		tmp = Float64(R * acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(lambda2)))));
	elseif (phi2 <= 5.2)
		tmp = Float64(R * acos(Float64(cos(phi1) * t_1)));
	elseif (phi2 <= 1.05e+102)
		tmp = Float64(R * acos(Float64(cos(phi2) * t_1)));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * Float64(cos(phi2) * cos(lambda1))))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(phi1) * sin(phi2);
	t_1 = (sin(lambda2) * sin(lambda1)) + (cos(lambda2) * cos(lambda1));
	tmp = 0.0;
	if (phi2 <= -3.2e-7)
		tmp = R * acos((t_0 + ((cos(phi1) * cos(phi2)) * cos(lambda2))));
	elseif (phi2 <= 5.2)
		tmp = R * acos((cos(phi1) * t_1));
	elseif (phi2 <= 1.05e+102)
		tmp = R * acos((cos(phi2) * t_1));
	else
		tmp = R * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda1)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, 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[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -3.2e-7], N[(R * N[ArcCos[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 5.2], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.05e+102], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if phi2 < -3.2000000000000001e-7

   1. Initial program 79.6%

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

   3. Taylor expanded in lambda1 around 0 50.3%

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

      1. cos-neg 50.3%

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

      2. associate-*r* 50.3%

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

      3. *-commutative 50.3%

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

   5. Simplified 50.3%

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

   if -3.2000000000000001e-7 < phi2 < 5.20000000000000018

   1. Initial program 68.5%

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

      1. *-commutative 68.5%

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

      2. *-commutative 68.5%

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

      3. *-commutative 68.5%

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

      4. *-commutative 68.5%

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

      5. associate-*l* 68.5%

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

      6. associate-*l* 68.5%

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

      7. *-commutative 68.5%

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

      8. cos-neg 68.5%

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

      9. sub-neg 68.5%

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

      10. +-commutative 68.5%

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

      11. distribute-neg-out 68.5%

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

      12. remove-double-neg 68.5%

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

      13. sub-neg 68.5%

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

   3. Simplified 68.5%

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

   5. Taylor expanded in phi2 around 0 68.4%

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

      1. cos-diff 89.8%

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

      2. +-commutative 89.8%

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

      3. *-commutative 89.8%

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

      4. *-commutative 89.8%

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

   7. Applied egg-rr 89.8%

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

   if 5.20000000000000018 < phi2 < 1.05000000000000001e102

   1. Initial program 71.3%

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

      1. *-commutative 71.3%

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

      2. *-commutative 71.3%

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

      3. *-commutative 71.3%

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

      4. *-commutative 71.3%

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

      5. associate-*l* 71.3%

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

      6. associate-*l* 71.3%

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

      7. *-commutative 71.3%

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

      8. cos-neg 71.3%

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

      9. sub-neg 71.3%

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

      10. +-commutative 71.3%

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

      11. distribute-neg-out 71.3%

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

      12. remove-double-neg 71.3%

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

      13. sub-neg 71.3%

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

   3. Simplified 71.3%

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

      1. cos-diff 20.8%

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

      2. +-commutative 20.8%

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

      3. *-commutative 20.8%

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

      4. *-commutative 20.8%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in phi1 around 0 63.5%

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

   if 1.05000000000000001e102 < phi2

   1. Initial program 86.1%

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

   3. Taylor expanded in lambda2 around 0 65.3%

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

      1. *-commutative 65.3%

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

      2. associate-*r* 65.3%

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

   5. Simplified 65.3%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.2 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 5.2:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{+102}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.4 \cdot 10^{-11} \lor \neg \left(\phi_2 \leq 6.4 \cdot 10^{-12}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi2 -1.4e-11) (not (<= phi2 6.4e-12)))
   (*
    R
    (acos
     (+
      (* (sin phi1) (sin phi2))
      (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))))
   (*
    R
    (acos
     (*
      (cos phi1)
      (+ (* (sin lambda2) (sin lambda1)) (* (cos lambda2) (cos lambda1))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -1.4e-11) || !(phi2 <= 6.4e-12)) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	} else {
		tmp = R * acos((cos(phi1) * ((sin(lambda2) * sin(lambda1)) + (cos(lambda2) * cos(lambda1)))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if ((phi2 <= (-1.4d-11)) .or. (.not. (phi2 <= 6.4d-12))) then
        tmp = r * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))))
    else
        tmp = r * acos((cos(phi1) * ((sin(lambda2) * sin(lambda1)) + (cos(lambda2) * cos(lambda1)))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -1.4e-11) || !(phi2 <= 6.4e-12)) {
		tmp = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2)))));
	} else {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.sin(lambda2) * Math.sin(lambda1)) + (Math.cos(lambda2) * Math.cos(lambda1)))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (phi2 <= -1.4e-11) or not (phi2 <= 6.4e-12):
		tmp = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + ((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2)))))
	else:
		tmp = R * math.acos((math.cos(phi1) * ((math.sin(lambda2) * math.sin(lambda1)) + (math.cos(lambda2) * math.cos(lambda1)))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi2 <= -1.4e-11) || !(phi2 <= 6.4e-12))
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(sin(lambda2) * sin(lambda1)) + Float64(cos(lambda2) * cos(lambda1))))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((phi2 <= -1.4e-11) || ~((phi2 <= 6.4e-12)))
		tmp = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	else
		tmp = R * acos((cos(phi1) * ((sin(lambda2) * sin(lambda1)) + (cos(lambda2) * cos(lambda1)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi2, -1.4e-11], N[Not[LessEqual[phi2, 6.4e-12]], $MachinePrecision]], N[(R * 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]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.4e-11 or 6.4000000000000002e-12 < phi2

   1. Initial program 79.7%

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

   if -1.4e-11 < phi2 < 6.4000000000000002e-12

   1. Initial program 69.1%

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

      1. *-commutative 69.1%

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

      2. *-commutative 69.1%

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

      3. *-commutative 69.1%

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

      4. *-commutative 69.1%

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

      5. associate-*l* 69.1%

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

      6. associate-*l* 69.1%

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

      7. *-commutative 69.1%

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

      8. cos-neg 69.1%

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

      9. sub-neg 69.1%

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

      10. +-commutative 69.1%

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

      11. distribute-neg-out 69.1%

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

      12. remove-double-neg 69.1%

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

      13. sub-neg 69.1%

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

   3. Simplified 69.1%

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

   5. Taylor expanded in phi2 around 0 68.9%

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

      1. cos-diff 90.4%

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

      2. +-commutative 90.4%

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

      3. *-commutative 90.4%

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

      4. *-commutative 90.4%

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

   7. Applied egg-rr 90.4%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.4 \cdot 10^{-11} \lor \neg \left(\phi_2 \leq 6.4 \cdot 10^{-12}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 60.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5.2:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 5.2)
   (*
    R
    (acos
     (*
      (cos phi1)
      (+ (* (sin lambda2) (sin lambda1)) (* (cos lambda2) (cos lambda1))))))
   (* R (acos (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5.2) {
		tmp = R * acos((cos(phi1) * ((sin(lambda2) * sin(lambda1)) + (cos(lambda2) * cos(lambda1)))));
	} else {
		tmp = R * acos(((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 5.2d0) then
        tmp = r * acos((cos(phi1) * ((sin(lambda2) * sin(lambda1)) + (cos(lambda2) * cos(lambda1)))))
    else
        tmp = r * acos(((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5.2) {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.sin(lambda2) * Math.sin(lambda1)) + (Math.cos(lambda2) * Math.cos(lambda1)))));
	} else {
		tmp = R * Math.acos(((Math.cos(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 5.2:
		tmp = R * math.acos((math.cos(phi1) * ((math.sin(lambda2) * math.sin(lambda1)) + (math.cos(lambda2) * math.cos(lambda1)))))
	else:
		tmp = R * math.acos(((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 5.2)
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(sin(lambda2) * sin(lambda1)) + Float64(cos(lambda2) * cos(lambda1))))));
	else
		tmp = Float64(R * acos(Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 5.2)
		tmp = R * acos((cos(phi1) * ((sin(lambda2) * sin(lambda1)) + (cos(lambda2) * cos(lambda1)))));
	else
		tmp = R * acos(((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 5.2], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 5.20000000000000018

   1. Initial program 72.6%

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

      1. *-commutative 72.6%

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

      2. *-commutative 72.6%

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

      3. *-commutative 72.6%

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

      4. *-commutative 72.6%

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

      5. associate-*l* 72.6%

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

      6. associate-*l* 72.6%

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

      7. *-commutative 72.6%

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

      8. cos-neg 72.6%

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

      9. sub-neg 72.6%

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

      10. +-commutative 72.6%

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

      11. distribute-neg-out 72.6%

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

      12. remove-double-neg 72.6%

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

      13. sub-neg 72.6%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in phi2 around 0 49.6%

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

      1. cos-diff 63.4%

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

      2. +-commutative 63.4%

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

      3. *-commutative 63.4%

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

      4. *-commutative 63.4%

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

   7. Applied egg-rr 63.4%

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

   if 5.20000000000000018 < phi2

   1. Initial program 81.4%

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

      1. *-commutative 81.4%

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

      2. sin-mult 45.6%

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

      3. +-commutative 45.6%

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

   4. Applied egg-rr 45.6%

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

   5. Taylor expanded in phi2 around 0 45.6%

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

      1. cos-neg 45.6%

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

      2. +-inverses 45.6%

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

   7. Simplified 45.6%

      \[\leadsto \cos^{-1} \left(\frac{\color{blue}{0}}{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.2%

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

Alternative 17: 62.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \lambda_2 \cdot \sin \lambda_1 + \cos \lambda_2 \cdot \cos \lambda_1\\ \mathbf{if}\;\phi_2 \leq 5.2:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (+ (* (sin lambda2) (sin lambda1)) (* (cos lambda2) (cos lambda1)))))
   (if (<= phi2 5.2)
     (* R (acos (* (cos phi1) t_0)))
     (* R (acos (* (cos phi2) t_0))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (sin(lambda2) * sin(lambda1)) + (cos(lambda2) * cos(lambda1));
	double tmp;
	if (phi2 <= 5.2) {
		tmp = R * acos((cos(phi1) * t_0));
	} else {
		tmp = R * acos((cos(phi2) * t_0));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sin(lambda2) * sin(lambda1)) + (cos(lambda2) * cos(lambda1))
    if (phi2 <= 5.2d0) then
        tmp = r * acos((cos(phi1) * t_0))
    else
        tmp = r * acos((cos(phi2) * t_0))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (Math.sin(lambda2) * Math.sin(lambda1)) + (Math.cos(lambda2) * Math.cos(lambda1));
	double tmp;
	if (phi2 <= 5.2) {
		tmp = R * Math.acos((Math.cos(phi1) * t_0));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * t_0));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (math.sin(lambda2) * math.sin(lambda1)) + (math.cos(lambda2) * math.cos(lambda1))
	tmp = 0
	if phi2 <= 5.2:
		tmp = R * math.acos((math.cos(phi1) * t_0))
	else:
		tmp = R * math.acos((math.cos(phi2) * t_0))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(sin(lambda2) * sin(lambda1)) + Float64(cos(lambda2) * cos(lambda1)))
	tmp = 0.0
	if (phi2 <= 5.2)
		tmp = Float64(R * acos(Float64(cos(phi1) * t_0)));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * t_0)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (sin(lambda2) * sin(lambda1)) + (cos(lambda2) * cos(lambda1));
	tmp = 0.0;
	if (phi2 <= 5.2)
		tmp = R * acos((cos(phi1) * t_0));
	else
		tmp = R * acos((cos(phi2) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, 5.2], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 5.20000000000000018

   1. Initial program 72.6%

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

      1. *-commutative 72.6%

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

      2. *-commutative 72.6%

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

      3. *-commutative 72.6%

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

      4. *-commutative 72.6%

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

      5. associate-*l* 72.6%

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

      6. associate-*l* 72.6%

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

      7. *-commutative 72.6%

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

      8. cos-neg 72.6%

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

      9. sub-neg 72.6%

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

      10. +-commutative 72.6%

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

      11. distribute-neg-out 72.6%

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

      12. remove-double-neg 72.6%

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

      13. sub-neg 72.6%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in phi2 around 0 49.6%

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

      1. cos-diff 63.4%

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

      2. +-commutative 63.4%

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

      3. *-commutative 63.4%

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

      4. *-commutative 63.4%

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

   7. Applied egg-rr 63.4%

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

   if 5.20000000000000018 < phi2

   1. Initial program 81.4%

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

      1. *-commutative 81.4%

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

      2. *-commutative 81.4%

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

      3. *-commutative 81.4%

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

      4. *-commutative 81.4%

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

      5. associate-*l* 81.4%

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

      6. associate-*l* 81.4%

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

      7. *-commutative 81.4%

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

      8. cos-neg 81.4%

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

      9. sub-neg 81.4%

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

      10. +-commutative 81.4%

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

      11. distribute-neg-out 81.4%

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

      12. remove-double-neg 81.4%

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

      13. sub-neg 81.4%

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

   3. Simplified 81.5%

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

      1. cos-diff 19.3%

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

      2. +-commutative 19.3%

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

      3. *-commutative 19.3%

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

      4. *-commutative 19.3%

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

   6. Applied egg-rr 99.4%

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

   7. Taylor expanded in phi1 around 0 53.2%

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

4. Final simplification 61.0%

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

Alternative 18: 63.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2))))
   (if (<= (- lambda1 lambda2) -1e-8)
     (* R (acos (* t_0 (cos (- lambda1 lambda2)))))
     (* R (acos (fma (sin phi1) (sin phi2) t_0))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * cos(phi2);
	double tmp;
	if ((lambda1 - lambda2) <= -1e-8) {
		tmp = R * acos((t_0 * cos((lambda1 - lambda2))));
	} else {
		tmp = R * acos(fma(sin(phi1), sin(phi2), t_0));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -1e-8)
		tmp = Float64(R * acos(Float64(t_0 * cos(Float64(lambda1 - lambda2)))));
	else
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), t_0)));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -1e-8], N[(R * N[ArcCos[N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -1e-8

   1. Initial program 77.1%

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

      1. *-commutative 77.1%

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

      2. sin-mult 57.4%

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

      3. +-commutative 57.4%

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

   4. Applied egg-rr 57.4%

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

   5. Taylor expanded in phi2 around 0 57.4%

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

      1. cos-neg 57.4%

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

      2. +-inverses 57.4%

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

   7. Simplified 57.4%

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

   if -1e-8 < (-.f64 lambda1 lambda2)

   1. Initial program 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in lambda2 around 0 55.2%

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

   6. Taylor expanded in lambda1 around 0 40.8%

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

4. Final simplification 47.3%

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

Alternative 19: 58.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R (acos (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * acos(((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))));
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 = r * acos(((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))
end function

Java

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

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.acos(((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2))))

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * acos(Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2)))))
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * acos(((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))));
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.7%

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

   1. *-commutative 74.7%

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

   2. sin-mult 59.2%

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

   3. +-commutative 59.2%

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

4. Applied egg-rr 59.2%

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

5. Taylor expanded in phi2 around 0 59.2%

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

   1. cos-neg 59.2%

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

   2. +-inverses 59.2%

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

7. Simplified 59.2%

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

8. Final simplification 59.2%

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

Alternative 20: 49.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 5.2:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi2 5.2)
     (* R (acos (* (cos phi1) t_0)))
     (* R (acos (* (cos phi2) t_0))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 5.2) {
		tmp = R * acos((cos(phi1) * t_0));
	} else {
		tmp = R * acos((cos(phi2) * t_0));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((lambda2 - lambda1))
    if (phi2 <= 5.2d0) then
        tmp = r * acos((cos(phi1) * t_0))
    else
        tmp = r * acos((cos(phi2) * t_0))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 5.2) {
		tmp = R * Math.acos((Math.cos(phi1) * t_0));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * t_0));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	tmp = 0
	if phi2 <= 5.2:
		tmp = R * math.acos((math.cos(phi1) * t_0))
	else:
		tmp = R * math.acos((math.cos(phi2) * t_0))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi2 <= 5.2)
		tmp = Float64(R * acos(Float64(cos(phi1) * t_0)));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * t_0)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	tmp = 0.0;
	if (phi2 <= 5.2)
		tmp = R * acos((cos(phi1) * t_0));
	else
		tmp = R * acos((cos(phi2) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 5.2], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 5.20000000000000018

   1. Initial program 72.6%

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

      1. *-commutative 72.6%

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

      2. *-commutative 72.6%

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

      3. *-commutative 72.6%

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

      4. *-commutative 72.6%

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

      5. associate-*l* 72.6%

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

      6. associate-*l* 72.6%

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

      7. *-commutative 72.6%

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

      8. cos-neg 72.6%

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

      9. sub-neg 72.6%

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

      10. +-commutative 72.6%

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

      11. distribute-neg-out 72.6%

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

      12. remove-double-neg 72.6%

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

      13. sub-neg 72.6%

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

   3. Simplified 72.6%

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

   5. Taylor expanded in phi2 around 0 49.6%

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

   if 5.20000000000000018 < phi2

   1. Initial program 81.4%

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

      1. *-commutative 81.4%

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

      2. *-commutative 81.4%

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

      3. *-commutative 81.4%

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

      4. *-commutative 81.4%

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

      5. associate-*l* 81.4%

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

      6. associate-*l* 81.4%

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

      7. *-commutative 81.4%

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

      8. cos-neg 81.4%

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

      9. sub-neg 81.4%

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

      10. +-commutative 81.4%

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

      11. distribute-neg-out 81.4%

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

      12. remove-double-neg 81.4%

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

      13. sub-neg 81.4%

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

   3. Simplified 81.5%

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

   5. Taylor expanded in phi1 around 0 45.2%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5.2:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 37.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 1.35e-6)
   (* R (acos (* (cos phi1) (cos lambda1))))
   (* R (acos (cos (- lambda2 lambda1))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.35e-6) {
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	} else {
		tmp = R * acos(cos((lambda2 - lambda1)));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 1.35d-6) then
        tmp = r * acos((cos(phi1) * cos(lambda1)))
    else
        tmp = r * acos(cos((lambda2 - lambda1)))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.35e-6) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R * Math.acos(Math.cos((lambda2 - lambda1)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 1.35e-6:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R * math.acos(math.cos((lambda2 - lambda1)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1.35e-6)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda1))));
	else
		tmp = Float64(R * acos(cos(Float64(lambda2 - lambda1))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 1.35e-6)
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	else
		tmp = R * acos(cos((lambda2 - lambda1)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 1.35e-6], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.34999999999999999e-6

   1. Initial program 78.8%

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

      1. *-commutative 78.8%

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

      2. *-commutative 78.8%

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

      3. *-commutative 78.8%

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

      4. *-commutative 78.8%

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

      5. associate-*l* 78.8%

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

      6. associate-*l* 78.8%

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

      7. *-commutative 78.8%

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

      8. cos-neg 78.8%

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

      9. sub-neg 78.8%

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

      10. +-commutative 78.8%

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

      11. distribute-neg-out 78.8%

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

      12. remove-double-neg 78.8%

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

      13. sub-neg 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in phi2 around 0 43.1%

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

   6. Taylor expanded in lambda2 around 0 37.4%

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

      1. cos-neg 37.4%

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

   8. Simplified 37.4%

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

   if 1.34999999999999999e-6 < lambda2

   1. Initial program 62.3%

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

      1. *-commutative 62.3%

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

      2. *-commutative 62.3%

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

      3. *-commutative 62.3%

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

      4. *-commutative 62.3%

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

      5. associate-*l* 62.3%

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

      6. associate-*l* 62.3%

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

      7. *-commutative 62.3%

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

      8. cos-neg 62.3%

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

      9. sub-neg 62.3%

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

      10. +-commutative 62.3%

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

      11. distribute-neg-out 62.3%

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

      12. remove-double-neg 62.3%

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

      13. sub-neg 62.3%

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

   3. Simplified 62.3%

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

   5. Taylor expanded in phi2 around 0 40.3%

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

   6. Taylor expanded in phi1 around 0 27.6%

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

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 42.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 2.4e-5)
   (* R (acos (* (cos phi1) (cos lambda1))))
   (* R (acos (* (cos phi1) (cos lambda2))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 2.4e-5) {
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	} else {
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 2.4d-5) then
        tmp = r * acos((cos(phi1) * cos(lambda1)))
    else
        tmp = r * acos((cos(phi1) * cos(lambda2)))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 2.4e-5) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda2)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 2.4e-5:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda2)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 2.4e-5)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda1))));
	else
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda2))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 2.4e-5)
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	else
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 2.4e-5], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 2.4000000000000001e-5

   1. Initial program 78.4%

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

      1. *-commutative 78.4%

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

      2. *-commutative 78.4%

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

      3. *-commutative 78.4%

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

      4. *-commutative 78.4%

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

      5. associate-*l* 78.4%

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

      6. associate-*l* 78.4%

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

      7. *-commutative 78.4%

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

      8. cos-neg 78.4%

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

      9. sub-neg 78.4%

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

      10. +-commutative 78.4%

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

      11. distribute-neg-out 78.4%

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

      12. remove-double-neg 78.4%

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

      13. sub-neg 78.4%

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

   3. Simplified 78.5%

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

   5. Taylor expanded in phi2 around 0 43.0%

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

   6. Taylor expanded in lambda2 around 0 37.3%

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

      1. cos-neg 37.3%

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

   8. Simplified 37.3%

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

   if 2.4000000000000001e-5 < lambda2

   1. Initial program 62.9%

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

      1. *-commutative 62.9%

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

      2. *-commutative 62.9%

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

      3. *-commutative 62.9%

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

      4. *-commutative 62.9%

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

      5. associate-*l* 62.9%

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

      6. associate-*l* 62.9%

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

      7. *-commutative 62.9%

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

      8. cos-neg 62.9%

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

      9. sub-neg 62.9%

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

      10. +-commutative 62.9%

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

      11. distribute-neg-out 62.9%

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

      12. remove-double-neg 62.9%

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

      13. sub-neg 62.9%

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

   3. Simplified 62.8%

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

   5. Taylor expanded in phi2 around 0 40.7%

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

   6. Taylor expanded in lambda1 around 0 41.0%

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

      1. *-commutative 41.0%

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

   8. Simplified 41.0%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 42.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R (acos (* (cos phi1) (cos (- lambda2 lambda1))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * acos((cos(phi1) * cos((lambda2 - lambda1))));
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 = r * acos((cos(phi1) * cos((lambda2 - lambda1))))
end function

Java

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

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.acos((math.cos(phi1) * math.cos((lambda2 - lambda1))))

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * acos(Float64(cos(phi1) * cos(Float64(lambda2 - lambda1)))))
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * acos((cos(phi1) * cos((lambda2 - lambda1))));
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.7%

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

   1. *-commutative 74.7%

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

   2. *-commutative 74.7%

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

   3. *-commutative 74.7%

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

   4. *-commutative 74.7%

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

   5. associate-*l* 74.7%

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

   6. associate-*l* 74.7%

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

   7. *-commutative 74.7%

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

   8. cos-neg 74.7%

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

   9. sub-neg 74.7%

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

   10. +-commutative 74.7%

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

   11. distribute-neg-out 74.7%

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

   12. remove-double-neg 74.7%

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

   13. sub-neg 74.7%

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

3. Simplified 74.7%

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

5. Taylor expanded in phi2 around 0 42.4%

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

6. Final simplification 42.4%

   \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \]
7. Add Preprocessing

Alternative 24: 26.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 5.3 \cdot 10^{-95}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{elif}\;\lambda_2 \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 5.3e-95)
   (* R (acos (cos lambda1)))
   (if (<= lambda2 7.2e-6)
     (* R (- lambda2 lambda1))
     (* R (acos (cos lambda2))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 5.3e-95) {
		tmp = R * acos(cos(lambda1));
	} else if (lambda2 <= 7.2e-6) {
		tmp = R * (lambda2 - lambda1);
	} else {
		tmp = R * acos(cos(lambda2));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 5.3d-95) then
        tmp = r * acos(cos(lambda1))
    else if (lambda2 <= 7.2d-6) then
        tmp = r * (lambda2 - lambda1)
    else
        tmp = r * acos(cos(lambda2))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 5.3e-95) {
		tmp = R * Math.acos(Math.cos(lambda1));
	} else if (lambda2 <= 7.2e-6) {
		tmp = R * (lambda2 - lambda1);
	} else {
		tmp = R * Math.acos(Math.cos(lambda2));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 5.3e-95:
		tmp = R * math.acos(math.cos(lambda1))
	elif lambda2 <= 7.2e-6:
		tmp = R * (lambda2 - lambda1)
	else:
		tmp = R * math.acos(math.cos(lambda2))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 5.3e-95)
		tmp = Float64(R * acos(cos(lambda1)));
	elseif (lambda2 <= 7.2e-6)
		tmp = Float64(R * Float64(lambda2 - lambda1));
	else
		tmp = Float64(R * acos(cos(lambda2)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 5.3e-95)
		tmp = R * acos(cos(lambda1));
	elseif (lambda2 <= 7.2e-6)
		tmp = R * (lambda2 - lambda1);
	else
		tmp = R * acos(cos(lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 5.3e-95], N[(R * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 7.2e-6], N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[Cos[lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < 5.2999999999999998e-95

   1. Initial program 79.1%

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

      1. *-commutative 79.1%

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

      2. *-commutative 79.1%

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

      3. *-commutative 79.1%

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

      4. *-commutative 79.1%

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

      5. associate-*l* 79.1%

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

      6. associate-*l* 79.1%

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

      7. *-commutative 79.1%

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

      8. cos-neg 79.1%

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

      9. sub-neg 79.1%

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

      10. +-commutative 79.1%

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

      11. distribute-neg-out 79.1%

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

      12. remove-double-neg 79.1%

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

      13. sub-neg 79.1%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in phi2 around 0 42.8%

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

   6. Taylor expanded in phi1 around 0 23.0%

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

   7. Taylor expanded in lambda2 around 0 19.3%

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

      1. cos-neg 19.3%

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

   9. Simplified 19.3%

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

   if 5.2999999999999998e-95 < lambda2 < 7.19999999999999967e-6

   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

      1. *-commutative 76.2%

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

      2. *-commutative 76.2%

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

      3. *-commutative 76.2%

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

      4. *-commutative 76.2%

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

      5. associate-*l* 76.2%

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

      6. associate-*l* 76.2%

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

      7. *-commutative 76.2%

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

      8. cos-neg 76.2%

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

      9. sub-neg 76.2%

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

      10. +-commutative 76.2%

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

      11. distribute-neg-out 76.2%

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

      12. remove-double-neg 76.2%

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

      13. sub-neg 76.2%

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

   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. Taylor expanded in phi2 around 0 45.1%

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

   6. Taylor expanded in phi1 around 0 29.7%

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

   7. Taylor expanded in lambda2 around 0 25.9%

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

      1. neg-mul-1 25.9%

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

      2. sub-neg 25.9%

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

   9. Simplified 25.9%

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

   if 7.19999999999999967e-6 < lambda2

   1. Initial program 62.3%

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

      1. *-commutative 62.3%

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

      2. *-commutative 62.3%

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

      3. *-commutative 62.3%

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

      4. *-commutative 62.3%

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

      5. associate-*l* 62.3%

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

      6. associate-*l* 62.3%

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

      7. *-commutative 62.3%

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

      8. cos-neg 62.3%

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

      9. sub-neg 62.3%

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

      10. +-commutative 62.3%

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

      11. distribute-neg-out 62.3%

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

      12. remove-double-neg 62.3%

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

      13. sub-neg 62.3%

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

   3. Simplified 62.3%

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

   5. Taylor expanded in phi2 around 0 40.3%

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

   6. Taylor expanded in phi1 around 0 27.6%

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

   7. Taylor expanded in lambda1 around 0 27.2%

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

4. Final simplification 21.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 5.3 \cdot 10^{-95}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{elif}\;\lambda_2 \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 20.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.5 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -1.5e-7)
   (* R (acos (cos lambda1)))
   (* R (- lambda2 lambda1))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.5e-7) {
		tmp = R * acos(cos(lambda1));
	} else {
		tmp = R * (lambda2 - lambda1);
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-1.5d-7)) then
        tmp = r * acos(cos(lambda1))
    else
        tmp = r * (lambda2 - lambda1)
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.5e-7) {
		tmp = R * Math.acos(Math.cos(lambda1));
	} else {
		tmp = R * (lambda2 - lambda1);
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -1.5e-7:
		tmp = R * math.acos(math.cos(lambda1))
	else:
		tmp = R * (lambda2 - lambda1)
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.5e-7)
		tmp = Float64(R * acos(cos(lambda1)));
	else
		tmp = Float64(R * Float64(lambda2 - lambda1));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -1.5e-7)
		tmp = R * acos(cos(lambda1));
	else
		tmp = R * (lambda2 - lambda1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -1.5e-7], N[(R * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.4999999999999999e-7

   1. Initial program 67.2%

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

      1. *-commutative 67.2%

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

      2. *-commutative 67.2%

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

      3. *-commutative 67.2%

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

      4. *-commutative 67.2%

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

      5. associate-*l* 67.2%

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

      6. associate-*l* 67.2%

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

      7. *-commutative 67.2%

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

      8. cos-neg 67.2%

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

      9. sub-neg 67.2%

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

      10. +-commutative 67.2%

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

      11. distribute-neg-out 67.2%

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

      12. remove-double-neg 67.2%

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

      13. sub-neg 67.2%

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

   3. Simplified 67.3%

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

   5. Taylor expanded in phi2 around 0 34.3%

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

   6. Taylor expanded in phi1 around 0 25.9%

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

   7. Taylor expanded in lambda2 around 0 24.7%

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

      1. cos-neg 24.7%

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

   9. Simplified 24.7%

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

   if -1.4999999999999999e-7 < lambda1

   1. Initial program 77.3%

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

      1. *-commutative 77.3%

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

      2. *-commutative 77.3%

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

      3. *-commutative 77.3%

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

      4. *-commutative 77.3%

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

      5. associate-*l* 77.3%

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

      6. associate-*l* 77.3%

         \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right) \]

      7. *-commutative 77.3%

         \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

      8. cos-neg 77.3%

         \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(-\left(\lambda_1 - \lambda_2\right)\right)}\right) \]

      9. sub-neg 77.3%

         \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(-\color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right)\right) \]

      10. +-commutative 77.3%

          \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(-\color{blue}{\left(\left(-\lambda_2\right) + \lambda_1\right)}\right)\right) \]

      11. distribute-neg-out 77.3%

          \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\left(-\left(-\lambda_2\right)\right) + \left(-\lambda_1\right)\right)}\right) \]

      12. remove-double-neg 77.3%

          \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\color{blue}{\lambda_2} + \left(-\lambda_1\right)\right)\right) \]

      13. sub-neg 77.3%

          \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \]

   3. Simplified 77.3%

      \[\leadsto \color{blue}{R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi2 around 0 45.3%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]

   6. Taylor expanded in phi1 around 0 24.4%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \]

   7. Taylor expanded in lambda2 around 0 6.6%

      \[\leadsto R \cdot \color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)} \]
   8. Step-by-step derivation

      1. neg-mul-1 6.6%

         \[\leadsto R \cdot \left(\lambda_2 + \color{blue}{\left(-\lambda_1\right)}\right) \]

      2. sub-neg 6.6%

         \[\leadsto R \cdot \color{blue}{\left(\lambda_2 - \lambda_1\right)} \]

   9. Simplified 6.6%

      \[\leadsto R \cdot \color{blue}{\left(\lambda_2 - \lambda_1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 11.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.5 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 26.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R (acos (cos (- lambda2 lambda1)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * acos(cos((lambda2 - lambda1)));
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 = r * acos(cos((lambda2 - lambda1)))
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.acos(Math.cos((lambda2 - lambda1)));
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.acos(math.cos((lambda2 - lambda1)))

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * acos(cos(Float64(lambda2 - lambda1))))
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * acos(cos((lambda2 - lambda1)));
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[ArcCos[N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.7%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
2. Step-by-step derivation

   1. *-commutative 74.7%

      \[\leadsto \color{blue}{R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   2. *-commutative 74.7%

      \[\leadsto R \cdot \cos^{-1} \left(\color{blue}{\sin \phi_2 \cdot \sin \phi_1} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

   3. *-commutative 74.7%

      \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_2 \cdot \sin \phi_1 + \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

   4. *-commutative 74.7%

      \[\leadsto R \cdot \cos^{-1} \left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

   5. associate-*l* 74.7%

      \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right) \]

   6. associate-*l* 74.7%

      \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right) \]

   7. *-commutative 74.7%

      \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

   8. cos-neg 74.7%

      \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(-\left(\lambda_1 - \lambda_2\right)\right)}\right) \]

   9. sub-neg 74.7%

      \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(-\color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right)\right) \]

   10. +-commutative 74.7%

       \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(-\color{blue}{\left(\left(-\lambda_2\right) + \lambda_1\right)}\right)\right) \]

   11. distribute-neg-out 74.7%

       \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\left(-\left(-\lambda_2\right)\right) + \left(-\lambda_1\right)\right)}\right) \]

   12. remove-double-neg 74.7%

       \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\color{blue}{\lambda_2} + \left(-\lambda_1\right)\right)\right) \]

   13. sub-neg 74.7%

       \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \]

3. Simplified 74.7%

   \[\leadsto \color{blue}{R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in phi2 around 0 42.4%

   \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]

6. Taylor expanded in phi1 around 0 24.8%

   \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \]

7. Final simplification 24.8%

   \[\leadsto R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right) \]
8. Add Preprocessing

Alternative 27: 8.6% accurate, 68.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.25 \cdot 10^{-85}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -2.25e-85) (* lambda1 (- R)) (* lambda2 R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -2.25e-85) {
		tmp = lambda1 * -R;
	} else {
		tmp = lambda2 * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-2.25d-85)) then
        tmp = lambda1 * -r
    else
        tmp = lambda2 * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -2.25e-85) {
		tmp = lambda1 * -R;
	} else {
		tmp = lambda2 * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -2.25e-85:
		tmp = lambda1 * -R
	else:
		tmp = lambda2 * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -2.25e-85)
		tmp = Float64(lambda1 * Float64(-R));
	else
		tmp = Float64(lambda2 * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -2.25e-85)
		tmp = lambda1 * -R;
	else
		tmp = lambda2 * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -2.25e-85], N[(lambda1 * (-R)), $MachinePrecision], N[(lambda2 * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2.25000000000000002e-85

   1. Initial program 69.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Step-by-step derivation

      1. *-commutative 69.1%

         \[\leadsto \color{blue}{R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      2. *-commutative 69.1%

         \[\leadsto R \cdot \cos^{-1} \left(\color{blue}{\sin \phi_2 \cdot \sin \phi_1} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

      3. *-commutative 69.1%

         \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_2 \cdot \sin \phi_1 + \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

      4. *-commutative 69.1%

         \[\leadsto R \cdot \cos^{-1} \left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

      5. associate-*l* 69.1%

         \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right) \]

      6. associate-*l* 69.1%

         \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right) \]

      7. *-commutative 69.1%

         \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

      8. cos-neg 69.1%

         \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(-\left(\lambda_1 - \lambda_2\right)\right)}\right) \]

      9. sub-neg 69.1%

         \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(-\color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right)\right) \]

      10. +-commutative 69.1%

          \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(-\color{blue}{\left(\left(-\lambda_2\right) + \lambda_1\right)}\right)\right) \]

      11. distribute-neg-out 69.1%

          \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\left(-\left(-\lambda_2\right)\right) + \left(-\lambda_1\right)\right)}\right) \]

      12. remove-double-neg 69.1%

          \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\color{blue}{\lambda_2} + \left(-\lambda_1\right)\right)\right) \]

      13. sub-neg 69.1%

          \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \]

   3. Simplified 69.2%

      \[\leadsto \color{blue}{R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi2 around 0 37.8%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]

   6. Taylor expanded in phi1 around 0 24.7%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \]

   7. Taylor expanded in lambda2 around 0 9.1%

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \lambda_1\right)} \]
   8. Step-by-step derivation

      1. mul-1-neg 9.1%

         \[\leadsto \color{blue}{-R \cdot \lambda_1} \]

      2. *-commutative 9.1%

         \[\leadsto -\color{blue}{\lambda_1 \cdot R} \]

      3. distribute-rgt-neg-in 9.1%

         \[\leadsto \color{blue}{\lambda_1 \cdot \left(-R\right)} \]

   9. Simplified 9.1%

      \[\leadsto \color{blue}{\lambda_1 \cdot \left(-R\right)} \]

   if -2.25000000000000002e-85 < lambda1

   1. Initial program 77.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Step-by-step derivation

      1. *-commutative 77.3%

         \[\leadsto \color{blue}{R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

      2. *-commutative 77.3%

         \[\leadsto R \cdot \cos^{-1} \left(\color{blue}{\sin \phi_2 \cdot \sin \phi_1} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

      3. *-commutative 77.3%

         \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_2 \cdot \sin \phi_1 + \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

      4. *-commutative 77.3%

         \[\leadsto R \cdot \cos^{-1} \left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

      5. associate-*l* 77.3%

         \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right) \]

      6. associate-*l* 77.3%

         \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right) \]

      7. *-commutative 77.3%

         \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

      8. cos-neg 77.3%

         \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(-\left(\lambda_1 - \lambda_2\right)\right)}\right) \]

      9. sub-neg 77.3%

         \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(-\color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right)\right) \]

      10. +-commutative 77.3%

          \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(-\color{blue}{\left(\left(-\lambda_2\right) + \lambda_1\right)}\right)\right) \]

      11. distribute-neg-out 77.3%

          \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\left(-\left(-\lambda_2\right)\right) + \left(-\lambda_1\right)\right)}\right) \]

      12. remove-double-neg 77.3%

          \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\color{blue}{\lambda_2} + \left(-\lambda_1\right)\right)\right) \]

      13. sub-neg 77.3%

          \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \]

   3. Simplified 77.3%

      \[\leadsto \color{blue}{R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi2 around 0 44.6%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]

   6. Taylor expanded in phi1 around 0 24.8%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \]

   7. Taylor expanded in lambda2 around inf 6.3%

      \[\leadsto \color{blue}{R \cdot \lambda_2} \]
   8. Step-by-step derivation

      1. *-commutative 6.3%

         \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

   9. Simplified 6.3%

      \[\leadsto \color{blue}{\lambda_2 \cdot R} \]
3. Recombined 2 regimes into one program.

4. Final simplification 7.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.25 \cdot 10^{-85}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 9.0% accurate, 122.6× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ R \cdot \left(\lambda_2 - \lambda_1\right) \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R (- lambda2 lambda1)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * (lambda2 - lambda1);
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 = r * (lambda2 - lambda1)
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * (lambda2 - lambda1);
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return R * (lambda2 - lambda1)

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * Float64(lambda2 - lambda1))
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * (lambda2 - lambda1);
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.7%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
2. Step-by-step derivation

   1. *-commutative 74.7%

      \[\leadsto \color{blue}{R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   2. *-commutative 74.7%

      \[\leadsto R \cdot \cos^{-1} \left(\color{blue}{\sin \phi_2 \cdot \sin \phi_1} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

   3. *-commutative 74.7%

      \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_2 \cdot \sin \phi_1 + \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

   4. *-commutative 74.7%

      \[\leadsto R \cdot \cos^{-1} \left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

   5. associate-*l* 74.7%

      \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right) \]

   6. associate-*l* 74.7%

      \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right) \]

   7. *-commutative 74.7%

      \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

   8. cos-neg 74.7%

      \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(-\left(\lambda_1 - \lambda_2\right)\right)}\right) \]

   9. sub-neg 74.7%

      \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(-\color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right)\right) \]

   10. +-commutative 74.7%

       \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(-\color{blue}{\left(\left(-\lambda_2\right) + \lambda_1\right)}\right)\right) \]

   11. distribute-neg-out 74.7%

       \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\left(-\left(-\lambda_2\right)\right) + \left(-\lambda_1\right)\right)}\right) \]

   12. remove-double-neg 74.7%

       \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\color{blue}{\lambda_2} + \left(-\lambda_1\right)\right)\right) \]

   13. sub-neg 74.7%

       \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \]

3. Simplified 74.7%

   \[\leadsto \color{blue}{R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in phi2 around 0 42.4%

   \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]

6. Taylor expanded in phi1 around 0 24.8%

   \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \]

7. Taylor expanded in lambda2 around 0 6.2%

   \[\leadsto R \cdot \color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)} \]
8. Step-by-step derivation

   1. neg-mul-1 6.2%

      \[\leadsto R \cdot \left(\lambda_2 + \color{blue}{\left(-\lambda_1\right)}\right) \]

   2. sub-neg 6.2%

      \[\leadsto R \cdot \color{blue}{\left(\lambda_2 - \lambda_1\right)} \]

9. Simplified 6.2%

   \[\leadsto R \cdot \color{blue}{\left(\lambda_2 - \lambda_1\right)} \]

10. Final simplification 6.2%

    \[\leadsto R \cdot \left(\lambda_2 - \lambda_1\right) \]
11. Add Preprocessing

Alternative 29: 6.9% accurate, 204.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \lambda_2 \cdot R \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2) :precision binary64 (* lambda2 R))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return lambda2 * R;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 = lambda2 * r
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return lambda2 * R;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return lambda2 * R

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(lambda2 * R)
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = lambda2 * R;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(lambda2 * R), $MachinePrecision]

Derivation

1. Initial program 74.7%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
2. Step-by-step derivation

   1. *-commutative 74.7%

      \[\leadsto \color{blue}{R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \]

   2. *-commutative 74.7%

      \[\leadsto R \cdot \cos^{-1} \left(\color{blue}{\sin \phi_2 \cdot \sin \phi_1} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

   3. *-commutative 74.7%

      \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_2 \cdot \sin \phi_1 + \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

   4. *-commutative 74.7%

      \[\leadsto R \cdot \cos^{-1} \left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2} + \left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

   5. associate-*l* 74.7%

      \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right) \]

   6. associate-*l* 74.7%

      \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_2 \cdot \cos \phi_1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right) \]

   7. *-commutative 74.7%

      \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \]

   8. cos-neg 74.7%

      \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(-\left(\lambda_1 - \lambda_2\right)\right)}\right) \]

   9. sub-neg 74.7%

      \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(-\color{blue}{\left(\lambda_1 + \left(-\lambda_2\right)\right)}\right)\right) \]

   10. +-commutative 74.7%

       \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(-\color{blue}{\left(\left(-\lambda_2\right) + \lambda_1\right)}\right)\right) \]

   11. distribute-neg-out 74.7%

       \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\left(-\left(-\lambda_2\right)\right) + \left(-\lambda_1\right)\right)}\right) \]

   12. remove-double-neg 74.7%

       \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\color{blue}{\lambda_2} + \left(-\lambda_1\right)\right)\right) \]

   13. sub-neg 74.7%

       \[\leadsto R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}\right) \]

3. Simplified 74.7%

   \[\leadsto \color{blue}{R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in phi2 around 0 42.4%

   \[\leadsto R \cdot \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \]

6. Taylor expanded in phi1 around 0 24.8%

   \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \]

7. Taylor expanded in lambda2 around inf 5.5%

   \[\leadsto \color{blue}{R \cdot \lambda_2} \]
8. Step-by-step derivation

   1. *-commutative 5.5%

      \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

9. Simplified 5.5%

   \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

10. Final simplification 5.5%

    \[\leadsto \lambda_2 \cdot R \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024034 
(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))