Spherical law of cosines

Percentage Accurate: 73.9% → 94.2%

Time: 1.8min

Alternatives: 26

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.9% 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: 94.2% accurate, 0.6× speedup

Math

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

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)
     (*
      (cos phi2)
      (+
       (+ (+ 1.0 (* (sin lambda1) (sin lambda2))) -1.0)
       (* (cos lambda1) (cos 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) * (cos(phi2) * (((1.0 + (sin(lambda1) * sin(lambda2))) + -1.0) + (cos(lambda1) * cos(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) * Float64(cos(phi2) * Float64(Float64(Float64(1.0 + Float64(sin(lambda1) * sin(lambda2))) + -1.0) + Float64(cos(lambda1) * cos(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[phi2], $MachinePrecision] * N[(N[(N[(1.0 + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 74.0%

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

3. Simplified 74.0%

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

   1. cos-diff 95.3%

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

   2. +-commutative 95.3%

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

6. Applied egg-rr 95.3%

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

   1. expm1-log1p-u 95.3%

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

   2. expm1-undefine 95.3%

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

8. Applied egg-rr 95.3%

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

   1. expm1-define 95.3%

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

10. Simplified 95.3%

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

    1. expm1-undefine 95.3%

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

    2. log1p-expm1-u 95.3%

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

    3. log1p-undefine 95.3%

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

    4. rem-exp-log 95.3%

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

    5. expm1-log1p-u 95.3%

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

12. Applied egg-rr 95.3%

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

13. Final simplification 95.3%

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

Alternative 2: 94.2% 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 74.0%

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

3. Simplified 74.0%

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

   1. cos-diff 95.3%

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

   2. +-commutative 95.3%

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

6. Applied egg-rr 95.3%

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

7. Final simplification 95.3%

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

Alternative 3: 94.2% accurate, 0.6× speedup

Math

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

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

Derivation

1. Initial program 74.0%

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

   1. cos-diff 95.3%

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

4. Applied egg-rr 95.3%

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

   1. cos-neg 95.3%

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

   2. *-commutative 95.3%

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

   3. fma-define 95.3%

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

   4. cos-neg 95.3%

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

6. Simplified 95.3%

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

7. Final simplification 95.3%

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

Alternative 4: 82.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \log \left(e^{\frac{\cos \phi_1}{\frac{1}{\cos \phi_2 \cdot t\_0}}}\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 4.5 \cdot 10^{-86}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt[3]{{\cos^{-1} \left(\mathsf{fma}\left(t\_0, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)}^{3}}\\ \end{array} \end{array} \]

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 (- lambda1 lambda2))))
   (if (<= phi1 -1.35e-6)
     (*
      R
      (acos
       (fma
        (sin phi1)
        (sin phi2)
        (log (exp (/ (cos phi1) (/ 1.0 (* (cos phi2) t_0))))))))
     (if (<= phi1 4.5e-86)
       (*
        R
        (acos
         (+
          (* phi1 (sin phi2))
          (*
           (cos phi2)
           (fma
            (sin lambda1)
            (sin lambda2)
            (* (cos lambda1) (cos lambda2)))))))
       (*
        R
        (cbrt
         (pow
          (acos (fma t_0 (* (cos phi1) (cos phi2)) (* (sin phi1) (sin phi2))))
          3.0)))))))

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((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -1.35e-6) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), log(exp((cos(phi1) / (1.0 / (cos(phi2) * t_0)))))));
	} else if (phi1 <= 4.5e-86) {
		tmp = R * acos(((phi1 * sin(phi2)) + (cos(phi2) * fma(sin(lambda1), sin(lambda2), (cos(lambda1) * cos(lambda2))))));
	} else {
		tmp = R * cbrt(pow(acos(fma(t_0, (cos(phi1) * cos(phi2)), (sin(phi1) * sin(phi2)))), 3.0));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -1.35e-6)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), log(exp(Float64(cos(phi1) / Float64(1.0 / Float64(cos(phi2) * t_0))))))));
	elseif (phi1 <= 4.5e-86)
		tmp = Float64(R * acos(Float64(Float64(phi1 * sin(phi2)) + Float64(cos(phi2) * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda1) * cos(lambda2)))))));
	else
		tmp = Float64(R * cbrt((acos(fma(t_0, Float64(cos(phi1) * cos(phi2)), Float64(sin(phi1) * sin(phi2)))) ^ 3.0)));
	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[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -1.35e-6], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[Log[N[Exp[N[(N[Cos[phi1], $MachinePrecision] / N[(1.0 / N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 4.5e-86], N[(R * N[ArcCos[N[(N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi2], $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], N[(R * N[Power[N[Power[N[ArcCos[N[(t$95$0 * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.34999999999999999e-6

   1. Initial program 74.8%

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

   3. Simplified 74.7%

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

      1. cos-mult 57.4%

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

      2. clear-num 57.4%

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

      3. associate-+r- 57.4%

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

      4. associate--r- 57.4%

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

   6. Applied egg-rr 57.4%

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

      1. add-log-exp 57.3%

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

      2. un-div-inv 57.3%

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

      3. clear-num 57.3%

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

      4. associate--l+ 57.3%

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

      5. associate-+l- 57.3%

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

      6. cos-mult 74.7%

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

   8. Applied egg-rr 74.7%

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

   if -1.34999999999999999e-6 < phi1 < 4.4999999999999998e-86

   1. Initial program 69.1%

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

   3. Simplified 69.1%

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

   5. Taylor expanded in phi1 around 0 69.1%

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

      1. cos-diff 92.0%

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

   7. Applied egg-rr 92.0%

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

      1. +-commutative 92.0%

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

      2. fma-define 92.0%

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

      3. cos-neg 92.0%

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

      4. *-commutative 92.0%

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

      5. cos-neg 92.0%

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

   9. Simplified 92.0%

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

   if 4.4999999999999998e-86 < phi1

   1. Initial program 81.8%

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

      1. add-cbrt-cube 81.6%

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

      2. pow3 81.6%

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

      3. +-commutative 81.6%

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

      4. *-commutative 81.6%

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

      5. fma-define 81.7%

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

   4. Applied egg-rr 81.7%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.35 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \log \left(e^{\frac{\cos \phi_1}{\frac{1}{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}}\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 4.5 \cdot 10^{-86}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt[3]{{\cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -6 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t\_0\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 4.5 \cdot 10^{-86}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt[3]{{\cos^{-1} \left(\mathsf{fma}\left(t\_0, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)}^{3}}\\ \end{array} \end{array} \]

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 (- lambda1 lambda2))))
   (if (<= phi1 -6e-7)
     (* R (acos (fma (sin phi1) (sin phi2) (* (cos phi1) (* (cos phi2) t_0)))))
     (if (<= phi1 4.5e-86)
       (*
        R
        (acos
         (+
          (* phi1 (sin phi2))
          (*
           (cos phi2)
           (fma
            (sin lambda1)
            (sin lambda2)
            (* (cos lambda1) (cos lambda2)))))))
       (*
        R
        (cbrt
         (pow
          (acos (fma t_0 (* (cos phi1) (cos phi2)) (* (sin phi1) (sin phi2))))
          3.0)))))))

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((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -6e-7) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * (cos(phi2) * t_0))));
	} else if (phi1 <= 4.5e-86) {
		tmp = R * acos(((phi1 * sin(phi2)) + (cos(phi2) * fma(sin(lambda1), sin(lambda2), (cos(lambda1) * cos(lambda2))))));
	} else {
		tmp = R * cbrt(pow(acos(fma(t_0, (cos(phi1) * cos(phi2)), (sin(phi1) * sin(phi2)))), 3.0));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -6e-7)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * Float64(cos(phi2) * t_0)))));
	elseif (phi1 <= 4.5e-86)
		tmp = Float64(R * acos(Float64(Float64(phi1 * sin(phi2)) + Float64(cos(phi2) * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda1) * cos(lambda2)))))));
	else
		tmp = Float64(R * cbrt((acos(fma(t_0, Float64(cos(phi1) * cos(phi2)), Float64(sin(phi1) * sin(phi2)))) ^ 3.0)));
	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[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -6e-7], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 4.5e-86], N[(R * N[ArcCos[N[(N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi2], $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], N[(R * N[Power[N[Power[N[ArcCos[N[(t$95$0 * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -5.9999999999999997e-7

   1. Initial program 74.8%

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

   3. Simplified 74.7%

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

   if -5.9999999999999997e-7 < phi1 < 4.4999999999999998e-86

   1. Initial program 69.1%

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

   3. Simplified 69.1%

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

   5. Taylor expanded in phi1 around 0 69.1%

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

      1. cos-diff 92.0%

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

   7. Applied egg-rr 92.0%

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

      1. +-commutative 92.0%

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

      2. fma-define 92.0%

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

      3. cos-neg 92.0%

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

      4. *-commutative 92.0%

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

      5. cos-neg 92.0%

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

   9. Simplified 92.0%

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

   if 4.4999999999999998e-86 < phi1

   1. Initial program 81.8%

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

      1. add-cbrt-cube 81.6%

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

      2. pow3 81.6%

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

      3. +-commutative 81.6%

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

      4. *-commutative 81.6%

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

      5. fma-define 81.7%

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

   4. Applied egg-rr 81.7%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 4.5 \cdot 10^{-86}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \sqrt[3]{{\cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.2% accurate, 0.7× speedup

Math

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

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)
      (+ (* (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(((sin(phi1) * sin(phi2)) + (cos(phi1) * (cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(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) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(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.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(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.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * math.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(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * Float64(cos(phi2) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(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) * ((sin(lambda1) * sin(lambda2)) + (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[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $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 74.0%

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

3. Simplified 74.0%

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

   1. cos-diff 95.3%

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

   2. +-commutative 95.3%

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

6. Applied egg-rr 95.3%

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

7. Taylor expanded in phi1 around 0 95.3%

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

8. Final simplification 95.3%

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

Alternative 7: 83.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.7 \cdot 10^{-6} \lor \neg \left(\phi_1 \leq 4.5 \cdot 10^{-86}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

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 (or (<= phi1 -2.7e-6) (not (<= phi1 4.5e-86)))
   (*
    R
    (acos
     (fma
      (sin phi1)
      (sin phi2)
      (* (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))
   (*
    R
    (acos
     (+
      (* phi1 (sin phi2))
      (*
       (cos phi2)
       (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 ((phi1 <= -2.7e-6) || !(phi1 <= 4.5e-86)) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * (cos(phi2) * cos((lambda1 - lambda2))))));
	} else {
		tmp = R * acos(((phi1 * sin(phi2)) + (cos(phi2) * 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 ((phi1 <= -2.7e-6) || !(phi1 <= 4.5e-86))
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))))));
	else
		tmp = Float64(R * acos(Float64(Float64(phi1 * sin(phi2)) + Float64(cos(phi2) * 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[Or[LessEqual[phi1, -2.7e-6], N[Not[LessEqual[phi1, 4.5e-86]], $MachinePrecision]], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi2], $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 phi1 < -2.69999999999999998e-6 or 4.4999999999999998e-86 < phi1

   1. Initial program 78.2%

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

   3. Simplified 78.3%

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

   if -2.69999999999999998e-6 < phi1 < 4.4999999999999998e-86

   1. Initial program 69.1%

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

   3. Simplified 69.1%

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

   5. Taylor expanded in phi1 around 0 69.1%

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

      1. cos-diff 92.0%

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

   7. Applied egg-rr 92.0%

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

      1. +-commutative 92.0%

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

      2. fma-define 92.0%

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

      3. cos-neg 92.0%

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

      4. *-commutative 92.0%

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

      5. cos-neg 92.0%

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

   9. Simplified 92.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.7 \cdot 10^{-6} \lor \neg \left(\phi_1 \leq 4.5 \cdot 10^{-86}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.6 \cdot 10^{-7} \lor \neg \left(\phi_1 \leq 4.5 \cdot 10^{-86}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \end{array} \]

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 (or (<= phi1 -3.6e-7) (not (<= phi1 4.5e-86)))
   (*
    R
    (acos
     (fma
      (sin phi1)
      (sin phi2)
      (* (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))
   (*
    R
    (acos
     (+
      (*
       (cos phi2)
       (+ (* (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 ((phi1 <= -3.6e-7) || !(phi1 <= 4.5e-86)) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * (cos(phi2) * cos((lambda1 - lambda2))))));
	} else {
		tmp = R * acos(((cos(phi2) * ((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 ((phi1 <= -3.6e-7) || !(phi1 <= 4.5e-86))
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))))));
	else
		tmp = Float64(R * acos(Float64(Float64(cos(phi2) * 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[Or[LessEqual[phi1, -3.6e-7], N[Not[LessEqual[phi1, 4.5e-86]], $MachinePrecision]], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(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] + N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.59999999999999994e-7 or 4.4999999999999998e-86 < phi1

   1. Initial program 78.2%

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

   3. Simplified 78.3%

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

   if -3.59999999999999994e-7 < phi1 < 4.4999999999999998e-86

   1. Initial program 69.1%

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

   3. Simplified 69.1%

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

   5. Taylor expanded in phi1 around 0 69.1%

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

      1. cos-diff 92.0%

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

      2. +-commutative 92.0%

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

   7. Applied egg-rr 92.0%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.6 \cdot 10^{-7} \lor \neg \left(\phi_1 \leq 4.5 \cdot 10^{-86}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -1.8 \cdot 10^{+18}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \frac{\cos \phi_1}{\frac{1}{t\_0}}\right)\\ \mathbf{elif}\;\phi_2 \leq 5.9 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos (- lambda1 lambda2)))))
   (if (<= phi2 -1.8e+18)
     (* R (acos (+ (* (sin phi1) (sin phi2)) (/ (cos phi1) (/ 1.0 t_0)))))
     (if (<= phi2 5.9e-5)
       (*
        R
        (acos
         (*
          (cos phi1)
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))))
       (* R (acos (fma (sin phi1) (sin phi2) (* (cos phi1) 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(phi2) * cos((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -1.8e+18) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) / (1.0 / t_0))));
	} else if (phi2 <= 5.9e-5) {
		tmp = R * acos((cos(phi1) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)))));
	} else {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * 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(phi2) * cos(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi2 <= -1.8e+18)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) / Float64(1.0 / t_0)))));
	elseif (phi2 <= 5.9e-5)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * t_0))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.8e+18], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 5.9e-5], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.8e18

   1. Initial program 82.9%

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

   3. Simplified 82.9%

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

      1. cos-mult 41.8%

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

      2. clear-num 41.7%

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

      3. associate-+r- 41.7%

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

      4. associate--r- 41.7%

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

   6. Applied egg-rr 41.7%

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

      1. fma-undefine 41.7%

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

      2. un-div-inv 41.7%

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

      3. clear-num 41.7%

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

      4. associate--l+ 41.7%

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

      5. associate-+l- 41.7%

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

      6. cos-mult 82.7%

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

   8. Applied egg-rr 82.7%

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

   if -1.8e18 < phi2 < 5.8999999999999998e-5

   1. Initial program 67.4%

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

   3. Simplified 67.4%

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

      1. cos-diff 92.1%

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

      2. +-commutative 92.1%

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

   6. Applied egg-rr 92.1%

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

      1. expm1-log1p-u 92.1%

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

      2. expm1-undefine 92.1%

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

   8. Applied egg-rr 92.1%

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

      1. expm1-define 92.1%

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

   10. Simplified 92.1%

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

   11. Taylor expanded in phi2 around 0 88.1%

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

       1. fma-define 88.1%

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

   13. Simplified 88.1%

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

   if 5.8999999999999998e-5 < phi2

   1. Initial program 80.9%

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

   3. Simplified 81.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.8 \cdot 10^{+18}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \frac{\cos \phi_1}{\frac{1}{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}\right)\\ \mathbf{elif}\;\phi_2 \leq 5.9 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -1.8 \cdot 10^{+18}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_1 + \frac{\cos \phi_1}{\frac{1}{t\_0}}\right)\\ \mathbf{elif}\;\phi_2 \leq 5.9 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, t\_0, t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos (- lambda1 lambda2))))
        (t_1 (* (sin phi1) (sin phi2))))
   (if (<= phi2 -1.8e+18)
     (* R (acos (+ t_1 (/ (cos phi1) (/ 1.0 t_0)))))
     (if (<= phi2 5.9e-5)
       (*
        R
        (acos
         (*
          (cos phi1)
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))))
       (* R (acos (fma (cos phi1) t_0 t_1)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos((lambda1 - lambda2));
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi2 <= -1.8e+18) {
		tmp = R * acos((t_1 + (cos(phi1) / (1.0 / t_0))));
	} else if (phi2 <= 5.9e-5) {
		tmp = R * acos((cos(phi1) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)))));
	} else {
		tmp = R * acos(fma(cos(phi1), t_0, t_1));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi2 <= -1.8e+18)
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(phi1) / Float64(1.0 / t_0)))));
	elseif (phi2 <= 5.9e-5)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = Float64(R * acos(fma(cos(phi1), t_0, t_1)));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.8e+18], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[phi1], $MachinePrecision] / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 5.9e-5], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.8e18

   1. Initial program 82.9%

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

   3. Simplified 82.9%

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

      1. cos-mult 41.8%

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

      2. clear-num 41.7%

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

      3. associate-+r- 41.7%

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

      4. associate--r- 41.7%

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

   6. Applied egg-rr 41.7%

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

      1. fma-undefine 41.7%

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

      2. un-div-inv 41.7%

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

      3. clear-num 41.7%

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

      4. associate--l+ 41.7%

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

      5. associate-+l- 41.7%

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

      6. cos-mult 82.7%

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

   8. Applied egg-rr 82.7%

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

   if -1.8e18 < phi2 < 5.8999999999999998e-5

   1. Initial program 67.4%

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

   3. Simplified 67.4%

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

      1. cos-diff 92.1%

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

      2. +-commutative 92.1%

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

   6. Applied egg-rr 92.1%

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

      1. expm1-log1p-u 92.1%

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

      2. expm1-undefine 92.1%

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

   8. Applied egg-rr 92.1%

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

      1. expm1-define 92.1%

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

   10. Simplified 92.1%

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

   11. Taylor expanded in phi2 around 0 88.1%

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

       1. fma-define 88.1%

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

   13. Simplified 88.1%

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

   if 5.8999999999999998e-5 < phi2

   1. Initial program 80.9%

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

   3. Simplified 81.0%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.8 \cdot 10^{+18}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \frac{\cos \phi_1}{\frac{1}{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}\right)\\ \mathbf{elif}\;\phi_2 \leq 5.9 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -1.8 \cdot 10^{+18}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_1 + \frac{\cos \phi_1}{\frac{1}{\cos \phi_2 \cdot t\_0}}\right)\\ \mathbf{elif}\;\phi_2 \leq 5.9 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_1 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))) (t_1 (* (sin phi1) (sin phi2))))
   (if (<= phi2 -1.8e+18)
     (* R (acos (+ t_1 (/ (cos phi1) (/ 1.0 (* (cos phi2) t_0))))))
     (if (<= phi2 5.9e-5)
       (*
        R
        (acos
         (*
          (cos phi1)
          (fma (cos lambda1) (cos lambda2) (* (sin lambda1) (sin lambda2))))))
       (* R (acos (+ t_1 (* (* (cos phi1) (cos phi2)) t_0))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi2 <= -1.8e+18) {
		tmp = R * acos((t_1 + (cos(phi1) / (1.0 / (cos(phi2) * t_0)))));
	} else if (phi2 <= 5.9e-5) {
		tmp = R * acos((cos(phi1) * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)))));
	} else {
		tmp = R * acos((t_1 + ((cos(phi1) * cos(phi2)) * t_0)));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi2 <= -1.8e+18)
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(phi1) / Float64(1.0 / Float64(cos(phi2) * t_0))))));
	elseif (phi2 <= 5.9e-5)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = Float64(R * acos(Float64(t_1 + Float64(Float64(cos(phi1) * cos(phi2)) * t_0))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.8e+18], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[phi1], $MachinePrecision] / N[(1.0 / N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 5.9e-5], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$1 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.8e18

   1. Initial program 82.9%

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

   3. Simplified 82.9%

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

      1. cos-mult 41.8%

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

      2. clear-num 41.7%

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

      3. associate-+r- 41.7%

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

      4. associate--r- 41.7%

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

   6. Applied egg-rr 41.7%

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

      1. fma-undefine 41.7%

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

      2. un-div-inv 41.7%

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

      3. clear-num 41.7%

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

      4. associate--l+ 41.7%

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

      5. associate-+l- 41.7%

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

      6. cos-mult 82.7%

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

   8. Applied egg-rr 82.7%

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

   if -1.8e18 < phi2 < 5.8999999999999998e-5

   1. Initial program 67.4%

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

   3. Simplified 67.4%

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

      1. cos-diff 92.1%

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

      2. +-commutative 92.1%

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

   6. Applied egg-rr 92.1%

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

      1. expm1-log1p-u 92.1%

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

      2. expm1-undefine 92.1%

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

   8. Applied egg-rr 92.1%

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

      1. expm1-define 92.1%

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

   10. Simplified 92.1%

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

   11. Taylor expanded in phi2 around 0 88.1%

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

       1. fma-define 88.1%

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

   13. Simplified 88.1%

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

   if 5.8999999999999998e-5 < phi2

   1. Initial program 80.9%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.8 \cdot 10^{+18}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \frac{\cos \phi_1}{\frac{1}{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}\right)\\ \mathbf{elif}\;\phi_2 \leq 5.9 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.8 \cdot 10^{+18} \lor \neg \left(\phi_2 \leq 5.9 \cdot 10^{-5}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

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 (or (<= phi2 -1.8e+18) (not (<= phi2 5.9e-5)))
   (*
    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))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -1.8e+18) || !(phi2 <= 5.9e-5)) {
		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)))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if ((phi2 <= (-1.8d+18)) .or. (.not. (phi2 <= 5.9d-5))) 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)))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -1.8e+18) || !(phi2 <= 5.9e-5)) {
		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)))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (phi2 <= -1.8e+18) or not (phi2 <= 5.9e-5):
		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)))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi2 <= -1.8e+18) || !(phi2 <= 5.9e-5))
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))));
	else
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * 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 ((phi2 <= -1.8e+18) || ~((phi2 <= 5.9e-5)))
		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)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi2, -1.8e+18], N[Not[LessEqual[phi2, 5.9e-5]], $MachinePrecision]], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.8e18 or 5.8999999999999998e-5 < phi2

   1. Initial program 81.9%

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

   if -1.8e18 < phi2 < 5.8999999999999998e-5

   1. Initial program 67.4%

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

   3. Simplified 67.4%

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

      1. cos-diff 92.1%

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

      2. +-commutative 92.1%

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

   6. Applied egg-rr 92.1%

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

      1. expm1-log1p-u 92.1%

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

      2. expm1-undefine 92.1%

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

   8. Applied egg-rr 92.1%

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

      1. expm1-define 92.1%

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

   10. Simplified 92.1%

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

   11. Taylor expanded in phi2 around 0 88.1%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.8 \cdot 10^{+18} \lor \neg \left(\phi_2 \leq 5.9 \cdot 10^{-5}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -1.8 \cdot 10^{+18}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_1 + \frac{\cos \phi_1}{\frac{1}{\cos \phi_2 \cdot t\_0}}\right)\\ \mathbf{elif}\;\phi_2 \leq 5.9 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_1 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))) (t_1 (* (sin phi1) (sin phi2))))
   (if (<= phi2 -1.8e+18)
     (* R (acos (+ t_1 (/ (cos phi1) (/ 1.0 (* (cos phi2) t_0))))))
     (if (<= phi2 5.9e-5)
       (*
        R
        (acos
         (*
          (cos phi1)
          (+
           (* (sin lambda1) (sin lambda2))
           (* (cos lambda1) (cos lambda2))))))
       (* R (acos (+ t_1 (* (* (cos phi1) (cos phi2)) t_0))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi2 <= -1.8e+18) {
		tmp = R * acos((t_1 + (cos(phi1) / (1.0 / (cos(phi2) * t_0)))));
	} else if (phi2 <= 5.9e-5) {
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	} else {
		tmp = R * acos((t_1 + ((cos(phi1) * cos(phi2)) * t_0)));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((lambda1 - lambda2))
    t_1 = sin(phi1) * sin(phi2)
    if (phi2 <= (-1.8d+18)) then
        tmp = r * acos((t_1 + (cos(phi1) / (1.0d0 / (cos(phi2) * t_0)))))
    else if (phi2 <= 5.9d-5) then
        tmp = r * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))))
    else
        tmp = r * acos((t_1 + ((cos(phi1) * cos(phi2)) * t_0)))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double t_1 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (phi2 <= -1.8e+18) {
		tmp = R * Math.acos((t_1 + (Math.cos(phi1) / (1.0 / (Math.cos(phi2) * t_0)))));
	} else if (phi2 <= 5.9e-5) {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(lambda2)))));
	} else {
		tmp = R * Math.acos((t_1 + ((Math.cos(phi1) * Math.cos(phi2)) * t_0)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	t_1 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if phi2 <= -1.8e+18:
		tmp = R * math.acos((t_1 + (math.cos(phi1) / (1.0 / (math.cos(phi2) * t_0)))))
	elif phi2 <= 5.9e-5:
		tmp = R * math.acos((math.cos(phi1) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * math.cos(lambda2)))))
	else:
		tmp = R * math.acos((t_1 + ((math.cos(phi1) * math.cos(phi2)) * t_0)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi2 <= -1.8e+18)
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(phi1) / Float64(1.0 / Float64(cos(phi2) * t_0))))));
	elseif (phi2 <= 5.9e-5)
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2))))));
	else
		tmp = Float64(R * acos(Float64(t_1 + Float64(Float64(cos(phi1) * cos(phi2)) * t_0))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	t_1 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if (phi2 <= -1.8e+18)
		tmp = R * acos((t_1 + (cos(phi1) / (1.0 / (cos(phi2) * t_0)))));
	elseif (phi2 <= 5.9e-5)
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	else
		tmp = R * acos((t_1 + ((cos(phi1) * cos(phi2)) * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.8e+18], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[phi1], $MachinePrecision] / N[(1.0 / N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 5.9e-5], N[(R * N[ArcCos[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], N[(R * N[ArcCos[N[(t$95$1 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.8e18

   1. Initial program 82.9%

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

   3. Simplified 82.9%

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

      1. cos-mult 41.8%

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

      2. clear-num 41.7%

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

      3. associate-+r- 41.7%

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

      4. associate--r- 41.7%

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

   6. Applied egg-rr 41.7%

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

      1. fma-undefine 41.7%

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

      2. un-div-inv 41.7%

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

      3. clear-num 41.7%

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

      4. associate--l+ 41.7%

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

      5. associate-+l- 41.7%

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

      6. cos-mult 82.7%

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

   8. Applied egg-rr 82.7%

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

   if -1.8e18 < phi2 < 5.8999999999999998e-5

   1. Initial program 67.4%

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

   3. Simplified 67.4%

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

      1. cos-diff 92.1%

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

      2. +-commutative 92.1%

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

   6. Applied egg-rr 92.1%

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

      1. expm1-log1p-u 92.1%

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

      2. expm1-undefine 92.1%

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

   8. Applied egg-rr 92.1%

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

      1. expm1-define 92.1%

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

   10. Simplified 92.1%

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

   11. Taylor expanded in phi2 around 0 88.1%

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

   if 5.8999999999999998e-5 < phi2

   1. Initial program 80.9%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.8 \cdot 10^{+18}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \frac{\cos \phi_1}{\frac{1}{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}\right)\\ \mathbf{elif}\;\phi_2 \leq 5.9 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.8 \cdot 10^{+18} \lor \neg \left(\phi_2 \leq 0.000118\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi2 -1.8e+18) (not (<= phi2 0.000118)))
   (*
    R
    (acos
     (+
      (* (sin phi1) (sin phi2))
      (* (cos phi1) (* (cos phi2) (cos lambda2))))))
   (*
    R
    (acos
     (*
      (cos phi1)
      (+ (* (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 ((phi2 <= -1.8e+18) || !(phi2 <= 0.000118)) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * (cos(phi2) * cos(lambda2)))));
	} else {
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * 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 ((phi2 <= (-1.8d+18)) .or. (.not. (phi2 <= 0.000118d0))) then
        tmp = r * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * (cos(phi2) * cos(lambda2)))))
    else
        tmp = r * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * 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 ((phi2 <= -1.8e+18) || !(phi2 <= 0.000118)) {
		tmp = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + (Math.cos(phi1) * (Math.cos(phi2) * Math.cos(lambda2)))));
	} else {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * 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 (phi2 <= -1.8e+18) or not (phi2 <= 0.000118):
		tmp = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + (math.cos(phi1) * (math.cos(phi2) * math.cos(lambda2)))))
	else:
		tmp = R * math.acos((math.cos(phi1) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * 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 ((phi2 <= -1.8e+18) || !(phi2 <= 0.000118))
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * Float64(cos(phi2) * cos(lambda2))))));
	else
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * 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 ((phi2 <= -1.8e+18) || ~((phi2 <= 0.000118)))
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * (cos(phi2) * cos(lambda2)))));
	else
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * 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[Or[LessEqual[phi2, -1.8e+18], N[Not[LessEqual[phi2, 0.000118]], $MachinePrecision]], 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[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[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]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.8e18 or 1.18e-4 < phi2

   1. Initial program 81.9%

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

   3. Taylor expanded in lambda1 around 0 59.3%

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

      1. *-commutative 59.3%

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

      2. cos-neg 59.3%

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

   5. Simplified 59.3%

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

   if -1.8e18 < phi2 < 1.18e-4

   1. Initial program 67.4%

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

   3. Simplified 67.4%

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

      1. cos-diff 92.1%

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

      2. +-commutative 92.1%

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

   6. Applied egg-rr 92.1%

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

      1. expm1-log1p-u 92.1%

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

      2. expm1-undefine 92.1%

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

   8. Applied egg-rr 92.1%

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

      1. expm1-define 92.1%

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

   10. Simplified 92.1%

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

   11. Taylor expanded in phi2 around 0 88.1%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.8 \cdot 10^{+18} \lor \neg \left(\phi_2 \leq 0.000118\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -0.0017:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_1 + \cos \lambda_1 \cdot t\_0\right)\\ \mathbf{elif}\;\lambda_1 \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_1 + \cos \lambda_2 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

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 (* (sin phi1) (sin phi2))))
   (if (<= lambda1 -0.0017)
     (* R (acos (+ t_1 (* (cos lambda1) t_0))))
     (if (<= lambda1 1.8e-6)
       (* R (acos (+ t_1 (* (cos lambda2) t_0))))
       (*
        R
        (acos
         (*
          (cos phi1)
          (+
           (* (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 t_0 = cos(phi1) * cos(phi2);
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if (lambda1 <= -0.0017) {
		tmp = R * acos((t_1 + (cos(lambda1) * t_0)));
	} else if (lambda1 <= 1.8e-6) {
		tmp = R * acos((t_1 + (cos(lambda2) * t_0)));
	} else {
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * 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) :: t_1
    real(8) :: tmp
    t_0 = cos(phi1) * cos(phi2)
    t_1 = sin(phi1) * sin(phi2)
    if (lambda1 <= (-0.0017d0)) then
        tmp = r * acos((t_1 + (cos(lambda1) * t_0)))
    else if (lambda1 <= 1.8d-6) then
        tmp = r * acos((t_1 + (cos(lambda2) * t_0)))
    else
        tmp = r * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * 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 = Math.cos(phi1) * Math.cos(phi2);
	double t_1 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (lambda1 <= -0.0017) {
		tmp = R * Math.acos((t_1 + (Math.cos(lambda1) * t_0)));
	} else if (lambda1 <= 1.8e-6) {
		tmp = R * Math.acos((t_1 + (Math.cos(lambda2) * t_0)));
	} else {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * 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 = math.cos(phi1) * math.cos(phi2)
	t_1 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if lambda1 <= -0.0017:
		tmp = R * math.acos((t_1 + (math.cos(lambda1) * t_0)))
	elif lambda1 <= 1.8e-6:
		tmp = R * math.acos((t_1 + (math.cos(lambda2) * t_0)))
	else:
		tmp = R * math.acos((math.cos(phi1) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * 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(cos(phi1) * cos(phi2))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda1 <= -0.0017)
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(lambda1) * t_0))));
	elseif (lambda1 <= 1.8e-6)
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(lambda2) * t_0))));
	else
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * 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 = cos(phi1) * cos(phi2);
	t_1 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if (lambda1 <= -0.0017)
		tmp = R * acos((t_1 + (cos(lambda1) * t_0)));
	elseif (lambda1 <= 1.8e-6)
		tmp = R * acos((t_1 + (cos(lambda2) * t_0)));
	else
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * 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[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -0.0017], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[lambda1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, 1.8e-6], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[lambda2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[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]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -0.00169999999999999991

   1. Initial program 58.3%

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

   3. Taylor expanded in lambda2 around 0 58.0%

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

   if -0.00169999999999999991 < lambda1 < 1.79999999999999992e-6

   1. Initial program 90.1%

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

   3. Taylor expanded in lambda1 around 0 89.5%

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

      1. cos-neg 89.5%

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

   5. Simplified 89.5%

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

   if 1.79999999999999992e-6 < lambda1

   1. Initial program 58.4%

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

   3. Simplified 58.5%

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

      1. cos-diff 99.1%

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

      2. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. expm1-log1p-u 99.1%

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

      2. expm1-undefine 99.1%

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

   8. Applied egg-rr 99.1%

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

      1. expm1-define 99.1%

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

   10. Simplified 99.1%

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

   11. Taylor expanded in phi2 around 0 61.4%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.0017:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 1.8 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \lambda_2 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 74.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -0.0017:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 1.12 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

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 (<= lambda1 -0.0017)
     (* R (acos (+ t_0 (* (cos lambda1) (* (cos phi1) (cos phi2))))))
     (if (<= lambda1 1.12e-6)
       (* R (acos (+ t_0 (* (cos phi1) (* (cos phi2) (cos lambda2))))))
       (*
        R
        (acos
         (*
          (cos phi1)
          (+
           (* (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 t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (lambda1 <= -0.0017) {
		tmp = R * acos((t_0 + (cos(lambda1) * (cos(phi1) * cos(phi2)))));
	} else if (lambda1 <= 1.12e-6) {
		tmp = R * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda2)))));
	} else {
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * 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 = sin(phi1) * sin(phi2)
    if (lambda1 <= (-0.0017d0)) then
        tmp = r * acos((t_0 + (cos(lambda1) * (cos(phi1) * cos(phi2)))))
    else if (lambda1 <= 1.12d-6) then
        tmp = r * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda2)))))
    else
        tmp = r * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * 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 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (lambda1 <= -0.0017) {
		tmp = R * Math.acos((t_0 + (Math.cos(lambda1) * (Math.cos(phi1) * Math.cos(phi2)))));
	} else if (lambda1 <= 1.12e-6) {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * (Math.cos(phi2) * Math.cos(lambda2)))));
	} else {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * 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 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if lambda1 <= -0.0017:
		tmp = R * math.acos((t_0 + (math.cos(lambda1) * (math.cos(phi1) * math.cos(phi2)))))
	elif lambda1 <= 1.12e-6:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * (math.cos(phi2) * math.cos(lambda2)))))
	else:
		tmp = R * math.acos((math.cos(phi1) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * 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(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda1 <= -0.0017)
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(lambda1) * Float64(cos(phi1) * cos(phi2))))));
	elseif (lambda1 <= 1.12e-6)
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * Float64(cos(phi2) * cos(lambda2))))));
	else
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * 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 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if (lambda1 <= -0.0017)
		tmp = R * acos((t_0 + (cos(lambda1) * (cos(phi1) * cos(phi2)))));
	elseif (lambda1 <= 1.12e-6)
		tmp = R * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda2)))));
	else
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * 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[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -0.0017], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, 1.12e-6], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[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]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -0.00169999999999999991

   1. Initial program 58.3%

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

   3. Taylor expanded in lambda2 around 0 58.0%

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

   if -0.00169999999999999991 < lambda1 < 1.12000000000000008e-6

   1. Initial program 90.1%

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

   3. Taylor expanded in lambda1 around 0 89.5%

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

      1. *-commutative 89.5%

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

      2. cos-neg 89.5%

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

   5. Simplified 89.5%

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

   if 1.12000000000000008e-6 < lambda1

   1. Initial program 58.4%

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

   3. Simplified 58.5%

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

      1. cos-diff 99.1%

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

      2. +-commutative 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. expm1-log1p-u 99.1%

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

      2. expm1-undefine 99.1%

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

   8. Applied egg-rr 99.1%

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

      1. expm1-define 99.1%

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

   10. Simplified 99.1%

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

   11. Taylor expanded in phi2 around 0 61.4%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.0017:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 1.12 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 68.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.0165:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 0.0165)
   (*
    R
    (acos
     (*
      (cos phi1)
      (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos lambda2))))))
   (* R (expm1 (log1p (acos (* (cos phi2) (cos (- lambda2 lambda1)))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.0165) {
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))));
	} else {
		tmp = R * expm1(log1p(acos((cos(phi2) * cos((lambda2 - lambda1))))));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.0165) {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(lambda2)))));
	} else {
		tmp = R * Math.expm1(Math.log1p(Math.acos((Math.cos(phi2) * Math.cos((lambda2 - lambda1))))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 0.0165:
		tmp = R * math.acos((math.cos(phi1) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * math.cos(lambda2)))))
	else:
		tmp = R * math.expm1(math.log1p(math.acos((math.cos(phi2) * math.cos((lambda2 - lambda1))))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 0.0165)
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2))))));
	else
		tmp = Float64(R * expm1(log1p(acos(Float64(cos(phi2) * cos(Float64(lambda2 - lambda1)))))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 0.0165], N[(R * N[ArcCos[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], N[(R * N[(Exp[N[Log[1 + N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.016500000000000001

   1. Initial program 71.8%

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

   3. Simplified 71.8%

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

      1. cos-diff 94.1%

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

      2. +-commutative 94.1%

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

   6. Applied egg-rr 94.1%

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

      1. expm1-log1p-u 94.0%

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

      2. expm1-undefine 94.0%

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

   8. Applied egg-rr 94.0%

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

      1. expm1-define 94.0%

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

   10. Simplified 94.0%

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

   11. Taylor expanded in phi2 around 0 67.9%

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

   if 0.016500000000000001 < phi2

   1. Initial program 80.9%

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

   3. Simplified 81.0%

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

   5. Taylor expanded in phi1 around 0 40.1%

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

      1. expm1-log1p-u 40.0%

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

      2. +-commutative 40.0%

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

      3. fma-define 40.0%

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

   7. Applied egg-rr 40.0%

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

   8. Taylor expanded in phi1 around 0 49.0%

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

      1. sub-neg 49.0%

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

      2. neg-mul-1 49.0%

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

      3. neg-mul-1 49.0%

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

      4. remove-double-neg 49.0%

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

      5. mul-1-neg 49.0%

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

      6. distribute-neg-in 49.0%

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

      7. +-commutative 49.0%

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

      8. cos-neg 49.0%

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

      9. mul-1-neg 49.0%

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

      10. sub-neg 49.0%

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

   10. Simplified 49.0%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}\right)\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.0165:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 58.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi2 2e-5)
     (* R (acos (* (cos phi1) t_0)))
     (* R (expm1 (log1p (acos (* (cos phi2) t_0))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 2e-5) {
		tmp = R * acos((cos(phi1) * t_0));
	} else {
		tmp = R * expm1(log1p(acos((cos(phi2) * t_0))));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 2e-5) {
		tmp = R * Math.acos((Math.cos(phi1) * t_0));
	} else {
		tmp = R * Math.expm1(Math.log1p(Math.acos((Math.cos(phi2) * t_0))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	tmp = 0
	if phi2 <= 2e-5:
		tmp = R * math.acos((math.cos(phi1) * t_0))
	else:
		tmp = R * math.expm1(math.log1p(math.acos((math.cos(phi2) * t_0))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi2 <= 2e-5)
		tmp = Float64(R * acos(Float64(cos(phi1) * t_0)));
	else
		tmp = Float64(R * expm1(log1p(acos(Float64(cos(phi2) * t_0)))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 2e-5], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(Exp[N[Log[1 + N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.00000000000000016e-5

   1. Initial program 72.1%

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

   3. Simplified 72.1%

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

      1. add-sqr-sqrt 59.1%

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

      2. sqrt-unprod 62.5%

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

      3. pow2 62.5%

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

   6. Applied egg-rr 62.5%

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

   7. Taylor expanded in phi2 around 0 51.5%

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

      1. sub-neg 51.5%

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

      2. remove-double-neg 51.5%

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

      3. mul-1-neg 51.5%

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

      4. distribute-neg-in 51.5%

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

      5. +-commutative 51.5%

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

      6. cos-neg 51.5%

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

      7. mul-1-neg 51.5%

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

      8. sub-neg 51.5%

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

   9. Simplified 51.5%

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

   if 2.00000000000000016e-5 < phi2

   1. Initial program 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in phi1 around 0 39.7%

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

      1. expm1-log1p-u 39.7%

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

      2. +-commutative 39.7%

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

      3. fma-define 39.7%

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

   7. Applied egg-rr 39.7%

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

   8. Taylor expanded in phi1 around 0 48.5%

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

      1. sub-neg 48.5%

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

      2. neg-mul-1 48.5%

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

      3. neg-mul-1 48.5%

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

      4. remove-double-neg 48.5%

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

      5. mul-1-neg 48.5%

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

      6. distribute-neg-in 48.5%

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

      7. +-commutative 48.5%

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

      8. cos-neg 48.5%

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

      9. mul-1-neg 48.5%

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

      10. sub-neg 48.5%

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

   10. Simplified 48.5%

       \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)}\right)\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 57.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.00044:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \left(\sin \phi_2 + \frac{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}{\phi_1}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -0.00044)
   (* R (acos (* (cos phi1) (cos (- lambda2 lambda1)))))
   (*
    R
    (acos
     (*
      phi1
      (+ (sin phi2) (/ (* (cos phi2) (cos (- lambda1 lambda2))) phi1)))))))

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 <= -0.00044) {
		tmp = R * acos((cos(phi1) * cos((lambda2 - lambda1))));
	} else {
		tmp = R * acos((phi1 * (sin(phi2) + ((cos(phi2) * cos((lambda1 - lambda2))) / phi1))));
	}
	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 <= (-0.00044d0)) then
        tmp = r * acos((cos(phi1) * cos((lambda2 - lambda1))))
    else
        tmp = r * acos((phi1 * (sin(phi2) + ((cos(phi2) * cos((lambda1 - lambda2))) / phi1))))
    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 <= -0.00044) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos((lambda2 - lambda1))));
	} else {
		tmp = R * Math.acos((phi1 * (Math.sin(phi2) + ((Math.cos(phi2) * Math.cos((lambda1 - lambda2))) / phi1))));
	}
	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 <= -0.00044:
		tmp = R * math.acos((math.cos(phi1) * math.cos((lambda2 - lambda1))))
	else:
		tmp = R * math.acos((phi1 * (math.sin(phi2) + ((math.cos(phi2) * math.cos((lambda1 - lambda2))) / phi1))))
	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 <= -0.00044)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(Float64(lambda2 - lambda1)))));
	else
		tmp = Float64(R * acos(Float64(phi1 * Float64(sin(phi2) + Float64(Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))) / phi1)))));
	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 <= -0.00044)
		tmp = R * acos((cos(phi1) * cos((lambda2 - lambda1))));
	else
		tmp = R * acos((phi1 * (sin(phi2) + ((cos(phi2) * cos((lambda1 - lambda2))) / phi1))));
	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, -0.00044], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(phi1 * N[(N[Sin[phi2], $MachinePrecision] + N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.40000000000000016e-4

   1. Initial program 74.8%

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

   3. Simplified 74.7%

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

      1. add-sqr-sqrt 43.9%

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

      2. sqrt-unprod 51.9%

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

      3. pow2 51.9%

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

   6. Applied egg-rr 51.9%

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

   7. Taylor expanded in phi2 around 0 49.1%

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

      1. sub-neg 49.1%

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

      2. remove-double-neg 49.1%

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

      3. mul-1-neg 49.1%

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

      4. distribute-neg-in 49.1%

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

      5. +-commutative 49.1%

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

      6. cos-neg 49.1%

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

      7. mul-1-neg 49.1%

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

      8. sub-neg 49.1%

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

   9. Simplified 49.1%

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

   if -4.40000000000000016e-4 < phi1

   1. Initial program 73.7%

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

   3. Simplified 73.7%

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

   5. Taylor expanded in phi1 around 0 49.8%

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

      1. add-cube-cbrt 49.3%

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

      2. pow3 49.3%

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

      3. *-commutative 49.3%

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

   7. Applied egg-rr 49.3%

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

   8. Taylor expanded in phi1 around inf 49.9%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.00044:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \left(\sin \phi_2 + \frac{\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)}{\phi_1}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 57.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0036:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \end{array} \]

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 (<= phi1 -0.0036)
   (* R (acos (* (cos phi1) (cos (- lambda2 lambda1)))))
   (*
    R
    (acos (+ (* (cos phi2) (cos (- lambda1 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 (phi1 <= -0.0036) {
		tmp = R * acos((cos(phi1) * cos((lambda2 - lambda1))));
	} else {
		tmp = R * acos(((cos(phi2) * cos((lambda1 - lambda2))) + (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) :: tmp
    if (phi1 <= (-0.0036d0)) then
        tmp = r * acos((cos(phi1) * cos((lambda2 - lambda1))))
    else
        tmp = r * acos(((cos(phi2) * cos((lambda1 - lambda2))) + (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 tmp;
	if (phi1 <= -0.0036) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos((lambda2 - lambda1))));
	} else {
		tmp = R * Math.acos(((Math.cos(phi2) * Math.cos((lambda1 - lambda2))) + (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):
	tmp = 0
	if phi1 <= -0.0036:
		tmp = R * math.acos((math.cos(phi1) * math.cos((lambda2 - lambda1))))
	else:
		tmp = R * math.acos(((math.cos(phi2) * math.cos((lambda1 - lambda2))) + (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)
	tmp = 0.0
	if (phi1 <= -0.0036)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(Float64(lambda2 - lambda1)))));
	else
		tmp = Float64(R * acos(Float64(Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))) + 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)
	tmp = 0.0;
	if (phi1 <= -0.0036)
		tmp = R * acos((cos(phi1) * cos((lambda2 - lambda1))));
	else
		tmp = R * acos(((cos(phi2) * cos((lambda1 - lambda2))) + (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_] := If[LessEqual[phi1, -0.0036], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.0035999999999999999

   1. Initial program 74.8%

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

   3. Simplified 74.7%

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

      1. add-sqr-sqrt 43.9%

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

      2. sqrt-unprod 51.9%

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

      3. pow2 51.9%

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

   6. Applied egg-rr 51.9%

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

   7. Taylor expanded in phi2 around 0 49.1%

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

      1. sub-neg 49.1%

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

      2. remove-double-neg 49.1%

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

      3. mul-1-neg 49.1%

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

      4. distribute-neg-in 49.1%

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

      5. +-commutative 49.1%

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

      6. cos-neg 49.1%

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

      7. mul-1-neg 49.1%

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

      8. sub-neg 49.1%

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

   9. Simplified 49.1%

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

   if -0.0035999999999999999 < phi1

   1. Initial program 73.7%

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

   3. Simplified 73.7%

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

   5. Taylor expanded in phi1 around 0 49.8%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0036:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 49.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2 \cdot 10^{+26}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2e+26)
   (* R (acos (* (cos phi1) (cos (- lambda2 lambda1)))))
   (* R (acos (+ (* phi1 (sin phi2)) (* (cos phi2) (cos lambda1)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2e+26) {
		tmp = R * acos((cos(phi1) * cos((lambda2 - lambda1))));
	} else {
		tmp = R * acos(((phi1 * sin(phi2)) + (cos(phi2) * cos(lambda1))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 2d+26) then
        tmp = r * acos((cos(phi1) * cos((lambda2 - lambda1))))
    else
        tmp = r * acos(((phi1 * sin(phi2)) + (cos(phi2) * cos(lambda1))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2e+26) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos((lambda2 - lambda1))));
	} else {
		tmp = R * Math.acos(((phi1 * Math.sin(phi2)) + (Math.cos(phi2) * Math.cos(lambda1))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2e+26:
		tmp = R * math.acos((math.cos(phi1) * math.cos((lambda2 - lambda1))))
	else:
		tmp = R * math.acos(((phi1 * math.sin(phi2)) + (math.cos(phi2) * math.cos(lambda1))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2e+26)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(Float64(lambda2 - lambda1)))));
	else
		tmp = Float64(R * acos(Float64(Float64(phi1 * sin(phi2)) + Float64(cos(phi2) * cos(lambda1)))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 2e+26)
		tmp = R * acos((cos(phi1) * cos((lambda2 - lambda1))));
	else
		tmp = R * acos(((phi1 * sin(phi2)) + (cos(phi2) * cos(lambda1))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2e+26], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.0000000000000001e26

   1. Initial program 72.2%

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

   3. Simplified 72.2%

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

      1. add-sqr-sqrt 59.0%

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

      2. sqrt-unprod 62.4%

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

      3. pow2 62.4%

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

   6. Applied egg-rr 62.4%

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

   7. Taylor expanded in phi2 around 0 50.8%

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

      1. sub-neg 50.8%

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

      2. remove-double-neg 50.8%

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

      3. mul-1-neg 50.8%

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

      4. distribute-neg-in 50.8%

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

      5. +-commutative 50.8%

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

      6. cos-neg 50.8%

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

      7. mul-1-neg 50.8%

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

      8. sub-neg 50.8%

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

   9. Simplified 50.8%

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

   if 2.0000000000000001e26 < phi2

   1. Initial program 80.0%

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

   3. Simplified 80.0%

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

   5. Taylor expanded in phi1 around 0 40.5%

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

   6. Taylor expanded in lambda2 around 0 26.8%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2 \cdot 10^{+26}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 47.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.65 \cdot 10^{-128}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \end{array} \]

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 (<= phi1 -1.65e-128)
   (* R (acos (* (cos phi1) (cos (- lambda2 lambda1)))))
   (* R (acos (+ (* (cos phi2) (cos (- lambda1 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 (phi1 <= -1.65e-128) {
		tmp = R * acos((cos(phi1) * cos((lambda2 - lambda1))));
	} else {
		tmp = R * acos(((cos(phi2) * cos((lambda1 - 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 (phi1 <= (-1.65d-128)) then
        tmp = r * acos((cos(phi1) * cos((lambda2 - lambda1))))
    else
        tmp = r * acos(((cos(phi2) * cos((lambda1 - 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 (phi1 <= -1.65e-128) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos((lambda2 - lambda1))));
	} else {
		tmp = R * Math.acos(((Math.cos(phi2) * Math.cos((lambda1 - 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 phi1 <= -1.65e-128:
		tmp = R * math.acos((math.cos(phi1) * math.cos((lambda2 - lambda1))))
	else:
		tmp = R * math.acos(((math.cos(phi2) * math.cos((lambda1 - 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 (phi1 <= -1.65e-128)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(Float64(lambda2 - lambda1)))));
	else
		tmp = Float64(R * acos(Float64(Float64(cos(phi2) * cos(Float64(lambda1 - 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 (phi1 <= -1.65e-128)
		tmp = R * acos((cos(phi1) * cos((lambda2 - lambda1))));
	else
		tmp = R * acos(((cos(phi2) * cos((lambda1 - 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[phi1, -1.65e-128], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.65e-128

   1. Initial program 72.9%

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

   3. Simplified 72.8%

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

      1. add-sqr-sqrt 49.6%

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

      2. sqrt-unprod 55.7%

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

      3. pow2 55.7%

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

   6. Applied egg-rr 55.7%

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

   7. Taylor expanded in phi2 around 0 46.5%

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

      1. sub-neg 46.5%

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

      2. remove-double-neg 46.5%

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

      3. mul-1-neg 46.5%

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

      4. distribute-neg-in 46.5%

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

      5. +-commutative 46.5%

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

      6. cos-neg 46.5%

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

      7. mul-1-neg 46.5%

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

      8. sub-neg 46.5%

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

   9. Simplified 46.5%

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

   if -1.65e-128 < phi1

   1. Initial program 74.6%

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

   3. Simplified 74.7%

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

   5. Taylor expanded in phi1 around 0 47.4%

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

   6. Taylor expanded in phi2 around 0 37.3%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.65 \cdot 10^{-128}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 42.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

3. Simplified 74.0%

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

   1. add-sqr-sqrt 58.2%

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

   2. sqrt-unprod 62.2%

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

   3. pow2 62.2%

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

6. Applied egg-rr 62.2%

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

7. Taylor expanded in phi2 around 0 43.6%

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

   1. sub-neg 43.6%

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

   2. remove-double-neg 43.6%

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

   3. mul-1-neg 43.6%

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

   4. distribute-neg-in 43.6%

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

   5. +-commutative 43.6%

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

   6. cos-neg 43.6%

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

   7. mul-1-neg 43.6%

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

   8. sub-neg 43.6%

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

9. Simplified 43.6%

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

10. Final simplification 43.6%

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

Alternative 24: 20.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.4 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2\right)\\ \end{array} \end{array} \]

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 (<= phi2 4.4e-15)
   (* R (acos (+ (cos (- lambda1 lambda2)) (* phi1 phi2))))
   (* R (acos (* 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 (phi2 <= 4.4e-15) {
		tmp = R * acos((cos((lambda1 - lambda2)) + (phi1 * phi2)));
	} else {
		tmp = R * acos((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) :: tmp
    if (phi2 <= 4.4d-15) then
        tmp = r * acos((cos((lambda1 - lambda2)) + (phi1 * phi2)))
    else
        tmp = r * acos((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 tmp;
	if (phi2 <= 4.4e-15) {
		tmp = R * Math.acos((Math.cos((lambda1 - lambda2)) + (phi1 * phi2)));
	} else {
		tmp = R * Math.acos((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):
	tmp = 0
	if phi2 <= 4.4e-15:
		tmp = R * math.acos((math.cos((lambda1 - lambda2)) + (phi1 * phi2)))
	else:
		tmp = R * math.acos((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)
	tmp = 0.0
	if (phi2 <= 4.4e-15)
		tmp = Float64(R * acos(Float64(cos(Float64(lambda1 - lambda2)) + Float64(phi1 * phi2))));
	else
		tmp = Float64(R * acos(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)
	tmp = 0.0;
	if (phi2 <= 4.4e-15)
		tmp = R * acos((cos((lambda1 - lambda2)) + (phi1 * phi2)));
	else
		tmp = R * acos((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_] := If[LessEqual[phi2, 4.4e-15], N[(R * N[ArcCos[N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.39999999999999971e-15

   1. Initial program 72.0%

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

   3. Simplified 72.0%

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

   5. Taylor expanded in phi1 around 0 36.9%

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

   6. Taylor expanded in phi2 around 0 26.5%

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

   7. Taylor expanded in phi2 around 0 25.5%

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

   if 4.39999999999999971e-15 < phi2

   1. Initial program 80.2%

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

   3. Simplified 80.2%

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

   5. Taylor expanded in phi1 around 0 39.1%

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

      1. add-cube-cbrt 38.8%

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

      2. pow3 38.8%

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

      3. *-commutative 38.8%

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

   7. Applied egg-rr 38.8%

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

   8. Taylor expanded in phi1 around inf 11.5%

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

4. Final simplification 22.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.4 \cdot 10^{-15}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 12.3% accurate, 3.0× speedup

Math

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

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 (* 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) {
	return R * acos((phi1 * sin(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((phi1 * sin(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((phi1 * Math.sin(phi2)));
}

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)))

Julia

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

Derivation

1. Initial program 74.0%

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

3. Simplified 74.0%

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

5. Taylor expanded in phi1 around 0 37.5%

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

   1. add-cube-cbrt 37.1%

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

   2. pow3 37.1%

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

   3. *-commutative 37.1%

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

7. Applied egg-rr 37.1%

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

8. Taylor expanded in phi1 around inf 13.0%

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

9. Final simplification 13.0%

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

Alternative 26: 9.5% accurate, 5.8× speedup

Math

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

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))))

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));
}

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))
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));
}

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))

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * acos(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((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[(phi1 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.0%

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

3. Simplified 74.0%

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

5. Taylor expanded in phi1 around 0 37.5%

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

   1. add-cube-cbrt 37.1%

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

   2. pow3 37.1%

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

   3. *-commutative 37.1%

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

7. Applied egg-rr 37.1%

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

8. Taylor expanded in phi1 around inf 13.0%

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

9. Taylor expanded in phi2 around 0 10.7%

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

10. Final simplification 10.7%

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

Reproduce

herbie shell --seed 2024116 
(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))