Spherical law of cosines

Percentage Accurate: 74.0% → 94.4%

Time: 29.5s

Alternatives: 22

Speedup: 0.7×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * r
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 74.0% 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: 94.4% accurate, 0.5× speedup

Math

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

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
 (*
  (acos
   (fma
    (sin phi1)
    (sin phi2)
    (*
     (cos phi1)
     (fma
      (cos lambda1)
      (* (cos lambda2) (cos phi2))
      (* (cos phi2) (* (sin lambda1) (sin 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 acos(fma(sin(phi1), sin(phi2), (cos(phi1) * fma(cos(lambda1), (cos(lambda2) * cos(phi2)), (cos(phi2) * (sin(lambda1) * sin(lambda2))))))) * R;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * fma(cos(lambda1), Float64(cos(lambda2) * cos(phi2)), Float64(cos(phi2) * Float64(sin(lambda1) * sin(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[(N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. cos-diff 94.3%

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

   2. +-commutative 94.3%

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

6. Applied egg-rr 94.3%

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

   1. distribute-rgt-in 94.3%

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

8. Applied egg-rr 94.3%

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

9. Taylor expanded in phi1 around 0 94.3%

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

    1. fma-def 94.3%

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

11. Simplified 94.3%

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

12. Final simplification 94.3%

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

Alternative 2: 94.4% accurate, 0.5× speedup

Math

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

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
   (fma
    (sin phi1)
    (sin phi2)
    (*
     (cos phi1)
     (+
      (* (cos phi2) (* (sin lambda1) (sin lambda2)))
      (* (cos phi2) (* (cos lambda1) (cos 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(fma(sin(phi1), sin(phi2), (cos(phi1) * ((cos(phi2) * (sin(lambda1) * sin(lambda2))) + (cos(phi2) * (cos(lambda1) * cos(lambda2)))))));
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * Float64(Float64(cos(phi2) * Float64(sin(lambda1) * sin(lambda2))) + Float64(cos(phi2) * Float64(cos(lambda1) * cos(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[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. cos-diff 94.3%

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

   2. +-commutative 94.3%

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

6. Applied egg-rr 94.3%

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

   1. distribute-rgt-in 94.3%

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

8. Applied egg-rr 94.3%

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

9. Final simplification 94.3%

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

Alternative 3: 94.4% accurate, 0.6× speedup

Math

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

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
   (fma
    (sin phi1)
    (sin phi2)
    (*
     (cos phi1)
     (*
      (cos phi2)
      (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos 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(fma(sin(phi1), sin(phi2), (cos(phi1) * (cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))))));
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * Float64(cos(phi2) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(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[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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. Step-by-step derivation

   1. cos-diff 94.3%

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

   2. +-commutative 94.3%

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

6. Applied egg-rr 94.3%

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

7. Final simplification 94.3%

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

Alternative 4: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -2.15 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right), t_0\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.35 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), t_0\right)\right)\\ \end{array} \end{array} \]

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 -2.15e-5)
     (*
      R
      (-
       (* PI 0.5)
       (asin (fma (cos phi1) (* (cos phi2) (cos (- lambda2 lambda1))) t_0))))
     (if (<= phi2 1.35e-7)
       (*
        R
        (acos
         (fma
          (sin phi1)
          (sin phi2)
          (*
           (cos phi1)
           (+
            (* (sin lambda1) (sin lambda2))
            (* (cos lambda1) (cos lambda2)))))))
       (*
        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 <= -2.15e-5) {
		tmp = R * ((((double) M_PI) * 0.5) - asin(fma(cos(phi1), (cos(phi2) * cos((lambda2 - lambda1))), t_0)));
	} else if (phi2 <= 1.35e-7) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))))));
	} 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 <= -2.15e-5)
		tmp = Float64(R * Float64(Float64(pi * 0.5) - asin(fma(cos(phi1), Float64(cos(phi2) * cos(Float64(lambda2 - lambda1))), t_0))));
	elseif (phi2 <= 1.35e-7)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2)))))));
	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, -2.15e-5], N[(R * N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.35e-7], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $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 < -2.1500000000000001e-5

   1. Initial program 73.2%

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

      1. acos-asin 73.2%

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

      2. sub-neg 73.2%

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

      3. div-inv 73.2%

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

      4. metadata-eval 73.2%

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

      5. +-commutative 73.2%

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

      6. *-commutative 73.2%

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

      7. fma-def 73.2%

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

   4. Applied egg-rr 73.2%

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

      1. sub-neg 73.2%

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

      2. fma-udef 73.2%

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

      3. *-commutative 73.2%

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

      4. associate-*r* 73.2%

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

      5. fma-def 73.2%

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

      6. sub-neg 73.2%

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

      7. neg-mul-1 73.2%

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

   6. Simplified 73.2%

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

   if -2.1500000000000001e-5 < phi2 < 1.35000000000000004e-7

   1. Initial program 70.3%

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

   3. Simplified 70.3%

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

      1. cos-diff 89.1%

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

      2. +-commutative 89.1%

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

   6. Applied egg-rr 89.1%

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

   7. Taylor expanded in phi2 around 0 89.1%

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

   if 1.35000000000000004e-7 < phi2

   1. Initial program 78.1%

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

      1. +-commutative 78.1%

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

      2. associate-*l* 78.1%

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

      3. fma-def 78.2%

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

   3. Simplified 78.2%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.15 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.35 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.4% accurate, 0.7× speedup

Math

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

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
   (+
    (* (sin phi1) (sin phi2))
    (*
     (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos lambda2)))
     (* (cos phi1) (cos phi2)))))))

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(((sin(phi1) * sin(phi2)) + (((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))) * (cos(phi1) * cos(phi2)))));
}

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(((sin(phi1) * sin(phi2)) + (((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))) * (cos(phi1) * cos(phi2)))))
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.sin(phi1) * Math.sin(phi2)) + (((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(lambda2))) * (Math.cos(phi1) * Math.cos(phi2)))));
}

Python

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

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(sin(phi1) * sin(phi2)) + Float64(Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2))) * Float64(cos(phi1) * cos(phi2))))))
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(((sin(phi1) * sin(phi2)) + (((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))) * (cos(phi1) * cos(phi2)))));
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[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.1%

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

   1. cos-diff 94.3%

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

   2. +-commutative 94.3%

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

4. Applied egg-rr 94.3%

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

5. Final simplification 94.3%

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

Alternative 6: 70.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 2.25 \cdot 10^{+38}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

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 (<= lambda1 2.25e+38)
   (*
    R
    (acos
     (fma
      (cos phi1)
      (* (cos phi2) (cos (- lambda1 lambda2)))
      (* (sin phi1) (sin phi2)))))
   (*
    R
    (acos
     (+
      (* phi1 (sin phi2))
      (*
       (cos phi1)
       (fma (sin lambda1) (sin lambda2) (* (cos lambda1) (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 (lambda1 <= 2.25e+38) {
		tmp = R * acos(fma(cos(phi1), (cos(phi2) * cos((lambda1 - lambda2))), (sin(phi1) * sin(phi2))));
	} else {
		tmp = R * acos(((phi1 * sin(phi2)) + (cos(phi1) * fma(sin(lambda1), sin(lambda2), (cos(lambda1) * 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 (lambda1 <= 2.25e+38)
		tmp = Float64(R * acos(fma(cos(phi1), Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))), Float64(sin(phi1) * sin(phi2)))));
	else
		tmp = Float64(R * acos(Float64(Float64(phi1 * sin(phi2)) + Float64(cos(phi1) * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda1) * cos(lambda2)))))));
	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[lambda1, 2.25e+38], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < 2.2499999999999999e38

   1. Initial program 76.9%

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

      1. +-commutative 76.9%

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

      2. associate-*l* 76.9%

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

      3. fma-def 77.0%

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

   3. Simplified 77.0%

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

   if 2.2499999999999999e38 < lambda1

   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. Add Preprocessing

   3. Taylor expanded in phi1 around 0 34.3%

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

   4. Taylor expanded in phi2 around 0 28.0%

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

      1. sub-neg 28.0%

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

      2. remove-double-neg 28.0%

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

      3. mul-1-neg 28.0%

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

      4. distribute-neg-in 28.0%

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

      5. +-commutative 28.0%

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

      6. cos-neg 28.0%

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

      7. mul-1-neg 28.0%

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

      8. sub-neg 28.0%

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

   6. Simplified 28.0%

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

      1. cos-diff 34.6%

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

      2. *-commutative 34.6%

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

      3. *-commutative 34.6%

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

   8. Applied egg-rr 37.2%

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

      1. +-commutative 34.6%

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

      2. fma-udef 34.7%

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

   10. Simplified 37.2%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 2.25 \cdot 10^{+38}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 1.8 \cdot 10^{+38}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \end{array} \]

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 (<= lambda1 1.8e+38)
   (*
    R
    (acos
     (fma
      (cos phi1)
      (* (cos phi2) (cos (- lambda1 lambda2)))
      (* (sin phi1) (sin phi2)))))
   (*
    R
    (acos
     (+
      (*
       (cos phi1)
       (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos lambda2))))
      (* 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 tmp;
	if (lambda1 <= 1.8e+38) {
		tmp = R * acos(fma(cos(phi1), (cos(phi2) * cos((lambda1 - lambda2))), (sin(phi1) * sin(phi2))));
	} else {
		tmp = R * acos(((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))) + (phi1 * sin(phi2))));
	}
	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 <= 1.8e+38)
		tmp = Float64(R * acos(fma(cos(phi1), Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))), Float64(sin(phi1) * sin(phi2)))));
	else
		tmp = Float64(R * acos(Float64(Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2)))) + Float64(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_] := If[LessEqual[lambda1, 1.8e+38], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < 1.79999999999999985e38

   1. Initial program 76.9%

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

      1. +-commutative 76.9%

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

      2. associate-*l* 76.9%

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

      3. fma-def 77.0%

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

   3. Simplified 77.0%

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

   if 1.79999999999999985e38 < lambda1

   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. Add Preprocessing

   3. Taylor expanded in phi1 around 0 34.3%

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

   4. Taylor expanded in phi2 around 0 28.0%

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

      1. sub-neg 28.0%

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

      2. remove-double-neg 28.0%

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

      3. mul-1-neg 28.0%

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

      4. distribute-neg-in 28.0%

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

      5. +-commutative 28.0%

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

      6. cos-neg 28.0%

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

      7. mul-1-neg 28.0%

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

      8. sub-neg 28.0%

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

   6. Simplified 28.0%

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

      1. cos-diff 34.6%

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

      2. *-commutative 34.6%

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

      3. *-commutative 34.6%

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

   8. Applied egg-rr 37.2%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 1.8 \cdot 10^{+38}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 1.9 \cdot 10^{+38}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \end{array} \]

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 (<= lambda1 1.9e+38)
   (*
    R
    (acos
     (fma
      (cos phi1)
      (* (cos phi2) (cos (- lambda1 lambda2)))
      (* (sin phi1) (sin phi2)))))
   (*
    R
    (acos
     (+
      (*
       (cos phi1)
       (fma (sin lambda1) (sin lambda2) (* (cos lambda1) (cos lambda2))))
      (* phi1 phi2))))))

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 <= 1.9e+38) {
		tmp = R * acos(fma(cos(phi1), (cos(phi2) * cos((lambda1 - lambda2))), (sin(phi1) * sin(phi2))));
	} else {
		tmp = R * acos(((cos(phi1) * fma(sin(lambda1), sin(lambda2), (cos(lambda1) * cos(lambda2)))) + (phi1 * phi2)));
	}
	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 <= 1.9e+38)
		tmp = Float64(R * acos(fma(cos(phi1), Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))), Float64(sin(phi1) * sin(phi2)))));
	else
		tmp = Float64(R * acos(Float64(Float64(cos(phi1) * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda1) * cos(lambda2)))) + Float64(phi1 * 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_] := If[LessEqual[lambda1, 1.9e+38], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < 1.8999999999999999e38

   1. Initial program 76.9%

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

      1. +-commutative 76.9%

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

      2. associate-*l* 76.9%

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

      3. fma-def 77.0%

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

   3. Simplified 77.0%

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

   if 1.8999999999999999e38 < lambda1

   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. Add Preprocessing

   3. Taylor expanded in phi1 around 0 34.3%

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

   4. Taylor expanded in phi2 around 0 28.0%

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

      1. sub-neg 28.0%

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

      2. remove-double-neg 28.0%

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

      3. mul-1-neg 28.0%

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

      4. distribute-neg-in 28.0%

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

      5. +-commutative 28.0%

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

      6. cos-neg 28.0%

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

      7. mul-1-neg 28.0%

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

      8. sub-neg 28.0%

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

   6. Simplified 28.0%

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

   7. Taylor expanded in phi2 around 0 25.5%

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

      1. cos-diff 34.6%

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

      2. *-commutative 34.6%

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

      3. *-commutative 34.6%

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

   9. Applied egg-rr 34.6%

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

       1. +-commutative 34.6%

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

       2. fma-udef 34.7%

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

   11. Simplified 34.7%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 1.9 \cdot 10^{+38}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 2.25 \cdot 10^{+38}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \end{array} \]

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 (<= lambda1 2.25e+38)
   (*
    R
    (acos
     (fma
      (cos phi1)
      (* (cos phi2) (cos (- lambda1 lambda2)))
      (* (sin phi1) (sin phi2)))))
   (*
    R
    (acos
     (+
      (*
       (cos phi1)
       (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos lambda2))))
      (* phi1 phi2))))))

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+38) {
		tmp = R * acos(fma(cos(phi1), (cos(phi2) * cos((lambda1 - lambda2))), (sin(phi1) * sin(phi2))));
	} else {
		tmp = R * acos(((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))) + (phi1 * phi2)));
	}
	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+38)
		tmp = Float64(R * acos(fma(cos(phi1), Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))), Float64(sin(phi1) * sin(phi2)))));
	else
		tmp = Float64(R * acos(Float64(Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2)))) + Float64(phi1 * 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_] := If[LessEqual[lambda1, 2.25e+38], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < 2.2499999999999999e38

   1. Initial program 76.9%

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

      1. +-commutative 76.9%

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

      2. associate-*l* 76.9%

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

      3. fma-def 77.0%

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

   3. Simplified 77.0%

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

   if 2.2499999999999999e38 < lambda1

   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. Add Preprocessing

   3. Taylor expanded in phi1 around 0 34.3%

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

   4. Taylor expanded in phi2 around 0 28.0%

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

      1. sub-neg 28.0%

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

      2. remove-double-neg 28.0%

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

      3. mul-1-neg 28.0%

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

      4. distribute-neg-in 28.0%

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

      5. +-commutative 28.0%

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

      6. cos-neg 28.0%

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

      7. mul-1-neg 28.0%

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

      8. sub-neg 28.0%

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

   6. Simplified 28.0%

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

   7. Taylor expanded in phi2 around 0 25.5%

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

      1. cos-diff 34.6%

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

      2. *-commutative 34.6%

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

      3. *-commutative 34.6%

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

   9. Applied egg-rr 34.6%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 2.25 \cdot 10^{+38}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 2.25 \cdot 10^{+38}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \end{array} \]

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 (<= lambda1 2.25e+38)
   (*
    R
    (acos
     (+
      (* (sin phi1) (sin phi2))
      (* (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))
   (*
    R
    (acos
     (+
      (*
       (cos phi1)
       (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos lambda2))))
      (* phi1 phi2))))))

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+38) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * (cos(phi2) * cos((lambda1 - lambda2))))));
	} else {
		tmp = R * acos(((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))) + (phi1 * phi2)));
	}
	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+38) then
        tmp = r * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * (cos(phi2) * cos((lambda1 - lambda2))))))
    else
        tmp = r * acos(((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))) + (phi1 * phi2)))
    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+38) {
		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(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(lambda2)))) + (phi1 * phi2)));
	}
	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+38:
		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(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * math.cos(lambda2)))) + (phi1 * phi2)))
	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+38)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))))));
	else
		tmp = Float64(R * acos(Float64(Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2)))) + Float64(phi1 * phi2))));
	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+38)
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * (cos(phi2) * cos((lambda1 - lambda2))))));
	else
		tmp = R * acos(((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))) + (phi1 * phi2)));
	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+38], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < 2.2499999999999999e38

   1. Initial program 76.9%

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

   3. Taylor expanded in phi1 around 0 76.9%

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

   if 2.2499999999999999e38 < lambda1

   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. Add Preprocessing

   3. Taylor expanded in phi1 around 0 34.3%

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

   4. Taylor expanded in phi2 around 0 28.0%

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

      1. sub-neg 28.0%

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

      2. remove-double-neg 28.0%

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

      3. mul-1-neg 28.0%

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

      4. distribute-neg-in 28.0%

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

      5. +-commutative 28.0%

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

      6. cos-neg 28.0%

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

      7. mul-1-neg 28.0%

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

      8. sub-neg 28.0%

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

   6. Simplified 28.0%

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

   7. Taylor expanded in phi2 around 0 25.5%

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

      1. cos-diff 34.6%

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

      2. *-commutative 34.6%

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

      3. *-commutative 34.6%

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

   9. Applied egg-rr 34.6%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 2.25 \cdot 10^{+38}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.4% accurate, 1.0× speedup

Math

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

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 phi1) (cos phi2))) (t_1 (cos (- lambda1 lambda2))))
   (if (<= t_1 1.0)
     (* R (acos (* t_0 t_1)))
     (* R (acos (+ (* (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 t_1 = cos((lambda1 - lambda2));
	double tmp;
	if (t_1 <= 1.0) {
		tmp = R * acos((t_0 * t_1));
	} else {
		tmp = R * acos(((sin(phi1) * sin(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) :: t_1
    real(8) :: tmp
    t_0 = cos(phi1) * cos(phi2)
    t_1 = cos((lambda1 - lambda2))
    if (t_1 <= 1.0d0) then
        tmp = r * acos((t_0 * t_1))
    else
        tmp = r * acos(((sin(phi1) * sin(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(phi1) * Math.cos(phi2);
	double t_1 = Math.cos((lambda1 - lambda2));
	double tmp;
	if (t_1 <= 1.0) {
		tmp = R * Math.acos((t_0 * t_1));
	} else {
		tmp = R * Math.acos(((Math.sin(phi1) * Math.sin(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(phi1) * math.cos(phi2)
	t_1 = math.cos((lambda1 - lambda2))
	tmp = 0
	if t_1 <= 1.0:
		tmp = R * math.acos((t_0 * t_1))
	else:
		tmp = R * math.acos(((math.sin(phi1) * math.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))
	t_1 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (t_1 <= 1.0)
		tmp = Float64(R * acos(Float64(t_0 * t_1)));
	else
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(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(phi1) * cos(phi2);
	t_1 = cos((lambda1 - lambda2));
	tmp = 0.0;
	if (t_1 <= 1.0)
		tmp = R * acos((t_0 * t_1));
	else
		tmp = R * acos(((sin(phi1) * sin(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[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 1.0], N[(R * N[ArcCos[N[(t$95$0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 lambda1 lambda2)) < 1

   1. Initial program 73.1%

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

      1. add-sqr-sqrt 40.6%

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

      2. pow2 40.6%

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

   4. Applied egg-rr 40.6%

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

      1. unpow2 40.6%

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

      2. add-sqr-sqrt 73.1%

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

      3. *-commutative 73.1%

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

      4. sin-mult 58.3%

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

   6. Applied egg-rr 58.3%

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

   7. Taylor expanded in phi2 around 0 58.3%

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

      1. cos-neg 58.3%

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

      2. +-inverses 58.3%

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

   9. Simplified 58.3%

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

   if 1 < (cos.f64 (-.f64 lambda1 lambda2))

   1. Initial program 73.1%

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

   3. Taylor expanded in lambda2 around 0 49.3%

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

   4. Taylor expanded in lambda1 around 0 29.9%

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

4. Final simplification 58.3%

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

Alternative 12: 61.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.5 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \end{array} \]

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 (<= lambda2 2.5e-17)
   (*
    R
    (acos
     (+
      (* (sin phi1) (sin phi2))
      (* (cos phi1) (* (cos lambda1) (cos phi2))))))
   (* 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 (lambda2 <= 2.5e-17) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * (cos(lambda1) * cos(phi2)))));
	} 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 (lambda2 <= 2.5d-17) then
        tmp = r * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * (cos(lambda1) * cos(phi2)))))
    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 (lambda2 <= 2.5e-17) {
		tmp = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + (Math.cos(phi1) * (Math.cos(lambda1) * Math.cos(phi2)))));
	} 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 lambda2 <= 2.5e-17:
		tmp = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + (math.cos(phi1) * (math.cos(lambda1) * math.cos(phi2)))))
	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 (lambda2 <= 2.5e-17)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * Float64(cos(lambda1) * cos(phi2))))));
	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 (lambda2 <= 2.5e-17)
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * (cos(lambda1) * cos(phi2)))));
	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[lambda2, 2.5e-17], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi2], $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 lambda2 < 2.4999999999999999e-17

   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. Add Preprocessing

   3. Taylor expanded in phi1 around 0 77.3%

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

   4. Taylor expanded in lambda2 around 0 65.8%

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

   if 2.4999999999999999e-17 < lambda2

   1. Initial program 57.6%

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

      1. add-sqr-sqrt 33.5%

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

      2. pow2 33.5%

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

   4. Applied egg-rr 33.5%

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

      1. unpow2 33.5%

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

      2. add-sqr-sqrt 57.6%

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

      3. *-commutative 57.6%

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

      4. sin-mult 47.9%

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

   6. Applied egg-rr 47.9%

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

   7. Taylor expanded in phi2 around 0 48.0%

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

      1. cos-neg 48.0%

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

      2. +-inverses 48.0%

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

   9. Simplified 48.0%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.5 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 63.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq 2.5 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_2\right)\right)\\ \end{array} \end{array} \]

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 (* (sin phi1) (sin phi2))))
   (if (<= lambda2 2.5e-17)
     (* R (acos (+ t_0 (* (cos phi1) (* (cos lambda1) (cos phi2))))))
     (* R (acos (+ t_0 (* (cos phi1) (* (cos lambda2) (cos 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 = sin(phi1) * sin(phi2);
	double tmp;
	if (lambda2 <= 2.5e-17) {
		tmp = R * acos((t_0 + (cos(phi1) * (cos(lambda1) * cos(phi2)))));
	} else {
		tmp = R * acos((t_0 + (cos(phi1) * (cos(lambda2) * cos(phi2)))));
	}
	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 (lambda2 <= 2.5d-17) then
        tmp = r * acos((t_0 + (cos(phi1) * (cos(lambda1) * cos(phi2)))))
    else
        tmp = r * acos((t_0 + (cos(phi1) * (cos(lambda2) * cos(phi2)))))
    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 (lambda2 <= 2.5e-17) {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * (Math.cos(lambda1) * Math.cos(phi2)))));
	} else {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * (Math.cos(lambda2) * Math.cos(phi2)))));
	}
	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 lambda2 <= 2.5e-17:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * (math.cos(lambda1) * math.cos(phi2)))))
	else:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * (math.cos(lambda2) * math.cos(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(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda2 <= 2.5e-17)
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * Float64(cos(lambda1) * cos(phi2))))));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * Float64(cos(lambda2) * cos(phi2))))));
	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 (lambda2 <= 2.5e-17)
		tmp = R * acos((t_0 + (cos(phi1) * (cos(lambda1) * cos(phi2)))));
	else
		tmp = R * acos((t_0 + (cos(phi1) * (cos(lambda2) * cos(phi2)))));
	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[lambda2, 2.5e-17], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 2.4999999999999999e-17

   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. Add Preprocessing

   3. Taylor expanded in phi1 around 0 77.3%

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

   4. Taylor expanded in lambda2 around 0 65.8%

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

   if 2.4999999999999999e-17 < lambda2

   1. Initial program 57.6%

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

   3. Taylor expanded in phi1 around 0 57.7%

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

   4. Taylor expanded in lambda1 around 0 57.3%

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

      1. cos-neg 57.3%

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

   6. Simplified 57.3%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.5 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.0% accurate, 1.0× speedup

Math

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

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
   (+
    (* (sin phi1) (sin phi2))
    (* (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(((sin(phi1) * sin(phi2)) + (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(((sin(phi1) * sin(phi2)) + (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.sin(phi1) * Math.sin(phi2)) + (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.sin(phi1) * math.sin(phi2)) + (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(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * Float64(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(((sin(phi1) * sin(phi2)) + (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[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.1%

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

3. Taylor expanded in phi1 around 0 73.1%

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

4. Final simplification 73.1%

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

Alternative 15: 44.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -8.2 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0 + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi1 -8.2e-7)
     (* R (acos (+ (* phi1 phi2) (* (cos phi1) t_0))))
     (* R (acos (+ (* (cos phi2) t_0) (* 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((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -8.2e-7) {
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * t_0)));
	} else {
		tmp = R * acos(((cos(phi2) * t_0) + (phi1 * sin(phi2))));
	}
	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 (phi1 <= (-8.2d-7)) then
        tmp = r * acos(((phi1 * phi2) + (cos(phi1) * t_0)))
    else
        tmp = r * acos(((cos(phi2) * t_0) + (phi1 * sin(phi2))))
    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 (phi1 <= -8.2e-7) {
		tmp = R * Math.acos(((phi1 * phi2) + (Math.cos(phi1) * t_0)));
	} else {
		tmp = R * Math.acos(((Math.cos(phi2) * t_0) + (phi1 * Math.sin(phi2))));
	}
	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 phi1 <= -8.2e-7:
		tmp = R * math.acos(((phi1 * phi2) + (math.cos(phi1) * t_0)))
	else:
		tmp = R * math.acos(((math.cos(phi2) * t_0) + (phi1 * math.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 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi1 <= -8.2e-7)
		tmp = Float64(R * acos(Float64(Float64(phi1 * phi2) + Float64(cos(phi1) * t_0))));
	else
		tmp = Float64(R * acos(Float64(Float64(cos(phi2) * t_0) + Float64(phi1 * sin(phi2)))));
	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 (phi1 <= -8.2e-7)
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * t_0)));
	else
		tmp = R * acos(((cos(phi2) * t_0) + (phi1 * sin(phi2))));
	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[phi1, -8.2e-7], N[(R * N[ArcCos[N[(N[(phi1 * phi2), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -8.1999999999999998e-7

   1. Initial program 83.9%

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

   3. Taylor expanded in phi1 around 0 19.4%

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

   4. Taylor expanded in phi2 around 0 19.4%

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

      1. sub-neg 19.4%

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

      2. remove-double-neg 19.4%

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

      3. mul-1-neg 19.4%

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

      4. distribute-neg-in 19.4%

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

      5. +-commutative 19.4%

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

      6. cos-neg 19.4%

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

      7. mul-1-neg 19.4%

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

      8. sub-neg 19.4%

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

   6. Simplified 19.4%

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

   7. Taylor expanded in phi2 around 0 19.4%

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

   if -8.1999999999999998e-7 < phi1

   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. Taylor expanded in phi1 around 0 48.7%

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

   4. Taylor expanded in phi1 around 0 44.8%

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

      1. sub-neg 44.8%

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

      2. neg-mul-1 44.8%

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

      3. neg-mul-1 44.8%

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

      4. remove-double-neg 44.8%

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

      5. mul-1-neg 44.8%

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

      6. distribute-neg-in 44.8%

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

      7. +-commutative 44.8%

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

      8. cos-neg 44.8%

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

      9. mul-1-neg 44.8%

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

      10. sub-neg 44.8%

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

   6. Simplified 44.8%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.2 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 23.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq 2.5 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* phi1 (sin phi2))))
   (if (<= lambda2 2.5e-17)
     (* R (acos (+ t_0 (* (cos phi1) (cos lambda1)))))
     (* R (acos (+ t_0 (* (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 t_0 = phi1 * sin(phi2);
	double tmp;
	if (lambda2 <= 2.5e-17) {
		tmp = R * acos((t_0 + (cos(phi1) * cos(lambda1))));
	} else {
		tmp = R * acos((t_0 + (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) :: t_0
    real(8) :: tmp
    t_0 = phi1 * sin(phi2)
    if (lambda2 <= 2.5d-17) then
        tmp = r * acos((t_0 + (cos(phi1) * cos(lambda1))))
    else
        tmp = r * acos((t_0 + (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 t_0 = phi1 * Math.sin(phi2);
	double tmp;
	if (lambda2 <= 2.5e-17) {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * Math.cos(lambda1))));
	} else {
		tmp = R * Math.acos((t_0 + (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):
	t_0 = phi1 * math.sin(phi2)
	tmp = 0
	if lambda2 <= 2.5e-17:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * math.cos(lambda1))))
	else:
		tmp = R * math.acos((t_0 + (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)
	t_0 = Float64(phi1 * sin(phi2))
	tmp = 0.0
	if (lambda2 <= 2.5e-17)
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * cos(lambda1)))));
	else
		tmp = Float64(R * acos(Float64(t_0 + 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)
	t_0 = phi1 * sin(phi2);
	tmp = 0.0;
	if (lambda2 <= 2.5e-17)
		tmp = R * acos((t_0 + (cos(phi1) * cos(lambda1))));
	else
		tmp = R * acos((t_0 + (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_] := Block[{t$95$0 = N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, 2.5e-17], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 2.4999999999999999e-17

   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. Add Preprocessing

   3. Taylor expanded in phi1 around 0 42.2%

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

   4. Taylor expanded in phi2 around 0 26.4%

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

      1. sub-neg 26.4%

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

      2. remove-double-neg 26.4%

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

      3. mul-1-neg 26.4%

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

      4. distribute-neg-in 26.4%

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

      5. +-commutative 26.4%

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

      6. cos-neg 26.4%

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

      7. mul-1-neg 26.4%

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

      8. sub-neg 26.4%

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

   6. Simplified 26.4%

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

   7. Taylor expanded in lambda2 around 0 21.7%

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

      1. cos-neg 21.7%

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

   9. Simplified 21.7%

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

   if 2.4999999999999999e-17 < lambda2

   1. Initial program 57.6%

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

   3. Taylor expanded in phi1 around 0 34.2%

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

   4. Taylor expanded in phi2 around 0 23.8%

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

      1. sub-neg 23.8%

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

      2. remove-double-neg 23.8%

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

      3. mul-1-neg 23.8%

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

      4. distribute-neg-in 23.8%

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

      5. +-commutative 23.8%

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

      6. cos-neg 23.8%

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

      7. mul-1-neg 23.8%

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

      8. sub-neg 23.8%

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

   6. Simplified 23.8%

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

   7. Taylor expanded in lambda1 around 0 23.5%

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

4. Final simplification 22.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.5 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 28.2% accurate, 1.5× speedup

Math

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

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 (+ (* phi1 (sin phi2)) (* (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(((phi1 * sin(phi2)) + (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(((phi1 * sin(phi2)) + (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(((phi1 * Math.sin(phi2)) + (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(((phi1 * math.sin(phi2)) + (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(Float64(phi1 * sin(phi2)) + 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(((phi1 * sin(phi2)) + (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[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.1%

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

3. Taylor expanded in phi1 around 0 40.5%

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

4. Taylor expanded in phi2 around 0 25.8%

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

   1. sub-neg 25.8%

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

   2. remove-double-neg 25.8%

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

   3. mul-1-neg 25.8%

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

   4. distribute-neg-in 25.8%

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

   5. +-commutative 25.8%

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

   6. cos-neg 25.8%

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

   7. mul-1-neg 25.8%

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

   8. sub-neg 25.8%

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

6. Simplified 25.8%

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

7. Final simplification 25.8%

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

Alternative 18: 59.4% 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 73.1%

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

   1. add-sqr-sqrt 40.6%

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

   2. pow2 40.6%

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

4. Applied egg-rr 40.6%

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

   1. unpow2 40.6%

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

   2. add-sqr-sqrt 73.1%

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

   3. *-commutative 73.1%

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

   4. sin-mult 58.3%

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

6. Applied egg-rr 58.3%

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

7. Taylor expanded in phi2 around 0 58.3%

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

   1. cos-neg 58.3%

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

   2. +-inverses 58.3%

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

9. Simplified 58.3%

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

10. Final simplification 58.3%

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

Alternative 19: 23.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.45 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \end{array} \]

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 (<= phi1 -1.45e-6)
   (* R (acos (+ (* phi1 phi2) (* (cos phi1) (cos lambda1)))))
   (* R (acos (+ (cos (- lambda2 lambda1)) (* phi1 phi2))))))

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 (phi1 <= -1.45e-6) {
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * cos(lambda1))));
	} else {
		tmp = R * acos((cos((lambda2 - lambda1)) + (phi1 * phi2)));
	}
	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 (phi1 <= (-1.45d-6)) then
        tmp = r * acos(((phi1 * phi2) + (cos(phi1) * cos(lambda1))))
    else
        tmp = r * acos((cos((lambda2 - lambda1)) + (phi1 * phi2)))
    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 (phi1 <= -1.45e-6) {
		tmp = R * Math.acos(((phi1 * phi2) + (Math.cos(phi1) * Math.cos(lambda1))));
	} else {
		tmp = R * Math.acos((Math.cos((lambda2 - lambda1)) + (phi1 * phi2)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1.45e-6:
		tmp = R * math.acos(((phi1 * phi2) + (math.cos(phi1) * math.cos(lambda1))))
	else:
		tmp = R * math.acos((math.cos((lambda2 - lambda1)) + (phi1 * phi2)))
	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 (phi1 <= -1.45e-6)
		tmp = Float64(R * acos(Float64(Float64(phi1 * phi2) + Float64(cos(phi1) * cos(lambda1)))));
	else
		tmp = Float64(R * acos(Float64(cos(Float64(lambda2 - lambda1)) + Float64(phi1 * phi2))));
	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 (phi1 <= -1.45e-6)
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * cos(lambda1))));
	else
		tmp = R * acos((cos((lambda2 - lambda1)) + (phi1 * phi2)));
	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[phi1, -1.45e-6], N[(R * N[ArcCos[N[(N[(phi1 * phi2), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.4500000000000001e-6

   1. Initial program 83.6%

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

   3. Taylor expanded in phi1 around 0 18.3%

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

   4. Taylor expanded in phi2 around 0 18.3%

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

      1. sub-neg 18.3%

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

      2. remove-double-neg 18.3%

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

      3. mul-1-neg 18.3%

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

      4. distribute-neg-in 18.3%

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

      5. +-commutative 18.3%

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

      6. cos-neg 18.3%

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

      7. mul-1-neg 18.3%

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

      8. sub-neg 18.3%

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

   6. Simplified 18.3%

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

   7. Taylor expanded in phi2 around 0 18.3%

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

   8. Taylor expanded in lambda2 around 0 14.0%

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

      1. cos-neg 14.0%

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

   10. Simplified 14.0%

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

   if -1.4500000000000001e-6 < phi1

   1. Initial program 69.0%

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

   3. Taylor expanded in phi1 around 0 49.0%

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

   4. Taylor expanded in phi2 around 0 28.7%

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

      1. sub-neg 28.7%

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

      2. remove-double-neg 28.7%

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

      3. mul-1-neg 28.7%

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

      4. distribute-neg-in 28.7%

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

      5. +-commutative 28.7%

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

      6. cos-neg 28.7%

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

      7. mul-1-neg 28.7%

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

      8. sub-neg 28.7%

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

   6. Simplified 28.7%

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

   7. Taylor expanded in phi2 around 0 26.4%

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

   8. Taylor expanded in phi1 around 0 22.5%

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

4. Final simplification 20.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.45 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 21.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 5.5 \cdot 10^{-90}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

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 5.5e-90)
   (* R (acos (+ (* phi1 phi2) (* (cos phi1) (cos lambda1)))))
   (* R (acos (+ (* phi1 phi2) (* (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 <= 5.5e-90) {
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * cos(lambda1))));
	} else {
		tmp = R * acos(((phi1 * phi2) + (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 <= 5.5d-90) then
        tmp = r * acos(((phi1 * phi2) + (cos(phi1) * cos(lambda1))))
    else
        tmp = r * acos(((phi1 * phi2) + (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 <= 5.5e-90) {
		tmp = R * Math.acos(((phi1 * phi2) + (Math.cos(phi1) * Math.cos(lambda1))));
	} else {
		tmp = R * Math.acos(((phi1 * phi2) + (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 <= 5.5e-90:
		tmp = R * math.acos(((phi1 * phi2) + (math.cos(phi1) * math.cos(lambda1))))
	else:
		tmp = R * math.acos(((phi1 * phi2) + (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 <= 5.5e-90)
		tmp = Float64(R * acos(Float64(Float64(phi1 * phi2) + Float64(cos(phi1) * cos(lambda1)))));
	else
		tmp = Float64(R * acos(Float64(Float64(phi1 * phi2) + 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 <= 5.5e-90)
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * cos(lambda1))));
	else
		tmp = R * acos(((phi1 * phi2) + (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, 5.5e-90], N[(R * N[ArcCos[N[(N[(phi1 * phi2), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(phi1 * phi2), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 5.5000000000000003e-90

   1. Initial program 75.7%

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

   3. Taylor expanded in phi1 around 0 43.1%

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

   4. Taylor expanded in phi2 around 0 26.6%

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

      1. sub-neg 26.6%

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

      2. remove-double-neg 26.6%

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

      3. mul-1-neg 26.6%

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

      4. distribute-neg-in 26.6%

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

      5. +-commutative 26.6%

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

      6. cos-neg 26.6%

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

      7. mul-1-neg 26.6%

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

      8. sub-neg 26.6%

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

   6. Simplified 26.6%

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

   7. Taylor expanded in phi2 around 0 24.8%

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

   8. Taylor expanded in lambda2 around 0 19.9%

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

      1. cos-neg 19.9%

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

   10. Simplified 19.9%

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

   if 5.5000000000000003e-90 < lambda2

   1. Initial program 66.6%

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

   3. Taylor expanded in phi1 around 0 34.0%

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

   4. Taylor expanded in phi2 around 0 23.8%

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

      1. sub-neg 23.8%

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

      2. remove-double-neg 23.8%

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

      3. mul-1-neg 23.8%

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

      4. distribute-neg-in 23.8%

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

      5. +-commutative 23.8%

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

      6. cos-neg 23.8%

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

      7. mul-1-neg 23.8%

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

      8. sub-neg 23.8%

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

   6. Simplified 23.8%

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

   7. Taylor expanded in phi2 around 0 22.4%

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

   8. Taylor expanded in lambda1 around 0 17.0%

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

      1. *-commutative 17.0%

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

   10. Simplified 17.0%

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

4. Final simplification 19.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 5.5 \cdot 10^{-90}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 26.2% accurate, 2.0× speedup

Math

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

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 (+ (* phi1 phi2) (* (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(((phi1 * phi2) + (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(((phi1 * phi2) + (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(((phi1 * phi2) + (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(((phi1 * phi2) + (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(Float64(phi1 * phi2) + 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(((phi1 * phi2) + (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[(phi1 * phi2), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.1%

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

3. Taylor expanded in phi1 around 0 40.5%

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

4. Taylor expanded in phi2 around 0 25.8%

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

   1. sub-neg 25.8%

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

   2. remove-double-neg 25.8%

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

   3. mul-1-neg 25.8%

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

   4. distribute-neg-in 25.8%

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

   5. +-commutative 25.8%

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

   6. cos-neg 25.8%

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

   7. mul-1-neg 25.8%

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

   8. sub-neg 25.8%

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

6. Simplified 25.8%

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

7. Taylor expanded in phi2 around 0 24.1%

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

8. Final simplification 24.1%

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

Alternative 22: 18.5% accurate, 2.9× speedup

Math

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

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)) (* phi1 phi2)))))

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)) + (phi1 * phi2)));
}

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)) + (phi1 * phi2)))
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)) + (phi1 * phi2)));
}

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)) + (phi1 * phi2)))

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(Float64(lambda2 - lambda1)) + Float64(phi1 * phi2))))
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)) + (phi1 * phi2)));
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[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.1%

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

3. Taylor expanded in phi1 around 0 40.5%

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

4. Taylor expanded in phi2 around 0 25.8%

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

   1. sub-neg 25.8%

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

   2. remove-double-neg 25.8%

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

   3. mul-1-neg 25.8%

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

   4. distribute-neg-in 25.8%

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

   5. +-commutative 25.8%

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

   6. cos-neg 25.8%

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

   7. mul-1-neg 25.8%

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

   8. sub-neg 25.8%

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

6. Simplified 25.8%

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

7. Taylor expanded in phi2 around 0 24.1%

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

8. Taylor expanded in phi1 around 0 17.4%

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

9. Final simplification 17.4%

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

Reproduce

herbie shell --seed 2024012 
(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))