Spherical law of cosines

Percentage Accurate: 73.9% → 94.3%

Time: 37.0s

Alternatives: 28

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: 94.3% accurate, 0.5× speedup

Math

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

FPCore

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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)
    (fma
     (cos lambda1)
     (* (* (cos phi1) (cos phi2)) (cos lambda2))
     (* (cos phi1) (* (* (cos phi2) (sin lambda1)) (sin lambda2))))))
  R))

C

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

Julia

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

Wolfram

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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[lambda1], $MachinePrecision] * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]

Derivation

1. Initial program 70.7%

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

3. Simplified 70.7%

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

   1. associate-*r* 70.7%

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

   2. cos-diff 91.9%

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

   3. distribute-lft-in 91.9%

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

5. Applied egg-rr 91.9%

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

6. Taylor expanded in phi1 around 0 91.9%

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

   1. fma-def 91.9%

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

   2. associate-*r* 91.9%

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

   3. *-commutative 91.9%

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

   4. associate-*l* 91.9%

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

   5. *-commutative 91.9%

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

   6. associate-*r* 91.9%

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

8. Simplified 91.9%

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

9. Taylor expanded in phi1 around 0 91.9%

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

    1. cos-neg 91.9%

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

    2. *-commutative 91.9%

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

    3. cos-neg 91.9%

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

    4. associate-*r* 91.9%

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

11. Simplified 91.9%

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

12. Final simplification 91.9%

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

Alternative 2: 94.3% accurate, 0.5× speedup

Math

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

FPCore

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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)
    (fma
     (cos lambda1)
     (* (* (cos phi1) (cos phi2)) (cos lambda2))
     (* (cos phi2) (* (sin lambda1) (* (cos phi1) (sin lambda2)))))))))

C

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

Julia

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

Wolfram

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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[lambda1], $MachinePrecision] * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.7%

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

3. Simplified 70.7%

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

   1. associate-*r* 70.7%

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

   2. cos-diff 91.9%

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

   3. distribute-lft-in 91.9%

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

5. Applied egg-rr 91.9%

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

6. Taylor expanded in phi1 around 0 91.9%

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

   1. fma-def 91.9%

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

   2. associate-*r* 91.9%

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

   3. *-commutative 91.9%

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

   4. associate-*l* 91.9%

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

   5. *-commutative 91.9%

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

   6. associate-*r* 91.9%

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

8. Simplified 91.9%

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

9. Final simplification 91.9%

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

Alternative 3: 94.3% accurate, 0.5× speedup

Math

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

FPCore

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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 phi1) (cos phi2)) (cos lambda2))))))))

C

assert(lambda1 < lambda2);
assert(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(phi1) * cos(phi2)) * cos(lambda2))))));
}

Julia

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

Wolfram

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[phi2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.7%

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

3. Simplified 70.7%

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

   1. associate-*r* 70.7%

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

   2. cos-diff 91.9%

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

   3. distribute-lft-in 91.9%

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

5. Applied egg-rr 91.9%

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

6. Taylor expanded in phi1 around 0 91.9%

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

   1. fma-def 91.9%

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

   2. associate-*r* 91.9%

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

   3. *-commutative 91.9%

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

   4. associate-*l* 91.9%

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

   5. *-commutative 91.9%

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

   6. associate-*r* 91.9%

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

8. Simplified 91.9%

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

9. Taylor expanded in phi1 around 0 91.9%

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

    1. cos-neg 91.9%

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

    2. *-commutative 91.9%

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

    3. cos-neg 91.9%

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

    4. associate-*r* 91.9%

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

11. Simplified 91.9%

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

    1. fma-udef 91.9%

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

13. Applied egg-rr 91.9%

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

14. Final simplification 91.9%

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

Alternative 4: 94.3% accurate, 0.6× speedup

Math

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

FPCore

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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(lambda1 < lambda2);
assert(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

lambda1, lambda2 = sort([lambda1, lambda2])
phi1, phi2 = sort([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: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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 70.7%

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

   1. cos-diff 47.6%

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

3. Applied egg-rr 91.9%

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

   1. cos-neg 47.6%

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

   2. *-commutative 47.6%

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

   3. fma-def 47.6%

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

   4. cos-neg 47.6%

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

5. Simplified 91.9%

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

6. Final simplification 91.9%

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

Alternative 5: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ [phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -0.0115:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_0\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 5.9 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(t_0, t_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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))) (t_1 (* (cos phi1) (cos phi2))))
   (if (<= phi1 -0.0115)
     (* R (acos (fma (sin phi1) (sin phi2) (* (cos phi2) (* (cos phi1) t_0)))))
     (if (<= phi1 5.9e-9)
       (*
        R
        (acos
         (+
          (*
           t_1
           (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2))))
          (* phi1 (sin phi2)))))
       (* R (- (* PI 0.5) (asin (fma t_0 t_1 (* (sin phi1) (sin phi2))))))))))

C

assert(lambda1 < lambda2);
assert(phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = cos(phi1) * cos(phi2);
	double tmp;
	if (phi1 <= -0.0115) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi2) * (cos(phi1) * t_0))));
	} else if (phi1 <= 5.9e-9) {
		tmp = R * acos(((t_1 * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2)))) + (phi1 * sin(phi2))));
	} else {
		tmp = R * ((((double) M_PI) * 0.5) - asin(fma(t_0, t_1, (sin(phi1) * sin(phi2)))));
	}
	return tmp;
}

Julia

lambda1, lambda2 = sort([lambda1, lambda2])
phi1, phi2 = sort([phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if (phi1 <= -0.0115)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi2) * Float64(cos(phi1) * t_0)))));
	elseif (phi1 <= 5.9e-9)
		tmp = Float64(R * acos(Float64(Float64(t_1 * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2)))) + Float64(phi1 * sin(phi2)))));
	else
		tmp = Float64(R * Float64(Float64(pi * 0.5) - asin(fma(t_0, t_1, Float64(sin(phi1) * sin(phi2))))));
	end
	return tmp
end

Wolfram

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -0.0115], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 5.9e-9], N[(R * N[ArcCos[N[(N[(t$95$1 * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(t$95$0 * t$95$1 + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.0115

   1. Initial program 84.1%

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

      1. log1p-expm1-u 84.0%

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

      2. log1p-udef 84.0%

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

   3. Applied egg-rr 84.0%

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

   4. Taylor expanded in phi1 around 0 84.0%

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

   6. Simplified 84.1%

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

   if -0.0115 < phi1 < 5.8999999999999999e-9

   1. Initial program 63.9%

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

   2. Taylor expanded in phi1 around 0 63.9%

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

      1. cos-diff 67.1%

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

   4. Applied egg-rr 85.4%

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

      1. cos-neg 67.1%

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

      2. *-commutative 67.1%

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

      3. fma-def 67.1%

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

      4. cos-neg 67.1%

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

   6. Simplified 85.4%

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

   if 5.8999999999999999e-9 < phi1

   1. Initial program 74.0%

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

      1. acos-asin 74.1%

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

      2. sub-neg 74.1%

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

      3. div-inv 74.1%

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

      4. metadata-eval 74.1%

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

      5. +-commutative 74.1%

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

      6. *-commutative 74.1%

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

      7. fma-def 74.1%

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

   3. Applied egg-rr 74.1%

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

      1. sub-neg 74.1%

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

      2. sub-neg 74.1%

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

      3. remove-double-neg 74.1%

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

      4. mul-1-neg 74.1%

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

      5. distribute-neg-in 74.1%

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

      6. +-commutative 74.1%

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

      7. cos-neg 74.1%

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

      8. mul-1-neg 74.1%

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

      9. unsub-neg 74.1%

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

   5. Simplified 74.1%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0115:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 5.9 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\\ \end{array} \]

Alternative 6: 94.3% accurate, 0.7× speedup

Math

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

FPCore

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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(lambda1 < lambda2);
assert(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: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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 lambda1 < lambda2;
assert 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

[lambda1, lambda2] = sort([lambda1, lambda2])
[phi1, phi2] = sort([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

lambda1, lambda2 = sort([lambda1, lambda2])
phi1, phi2 = sort([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)) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2)))))))
end

MATLAB

lambda1, lambda2 = num2cell(sort([lambda1, lambda2])){:}
phi1, phi2 = num2cell(sort([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: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.7%

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

   1. cos-diff 35.5%

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

   2. +-commutative 35.5%

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

3. Applied egg-rr 91.9%

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

4. Final simplification 91.9%

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

Alternative 7: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} [lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ [phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -0.011:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_0\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 5.9 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(t_0, t_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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))) (t_1 (* (cos phi1) (cos phi2))))
   (if (<= phi1 -0.011)
     (* R (acos (fma (sin phi1) (sin phi2) (* (cos phi2) (* (cos phi1) t_0)))))
     (if (<= phi1 5.9e-9)
       (*
        R
        (acos
         (+
          (*
           t_1
           (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos lambda2))))
          (* phi1 (sin phi2)))))
       (* R (- (* PI 0.5) (asin (fma t_0 t_1 (* (sin phi1) (sin phi2))))))))))

C

assert(lambda1 < lambda2);
assert(phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = cos(phi1) * cos(phi2);
	double tmp;
	if (phi1 <= -0.011) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi2) * (cos(phi1) * t_0))));
	} else if (phi1 <= 5.9e-9) {
		tmp = R * acos(((t_1 * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))) + (phi1 * sin(phi2))));
	} else {
		tmp = R * ((((double) M_PI) * 0.5) - asin(fma(t_0, t_1, (sin(phi1) * sin(phi2)))));
	}
	return tmp;
}

Julia

lambda1, lambda2 = sort([lambda1, lambda2])
phi1, phi2 = sort([phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if (phi1 <= -0.011)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi2) * Float64(cos(phi1) * t_0)))));
	elseif (phi1 <= 5.9e-9)
		tmp = Float64(R * acos(Float64(Float64(t_1 * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2)))) + Float64(phi1 * sin(phi2)))));
	else
		tmp = Float64(R * Float64(Float64(pi * 0.5) - asin(fma(t_0, t_1, Float64(sin(phi1) * sin(phi2))))));
	end
	return tmp
end

Wolfram

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -0.011], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 5.9e-9], N[(R * N[ArcCos[N[(N[(t$95$1 * 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], N[(R * N[(N[(Pi * 0.5), $MachinePrecision] - N[ArcSin[N[(t$95$0 * t$95$1 + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.010999999999999999

   1. Initial program 84.1%

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

      1. log1p-expm1-u 84.0%

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

      2. log1p-udef 84.0%

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

   3. Applied egg-rr 84.0%

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

   4. Taylor expanded in phi1 around 0 84.0%

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

   6. Simplified 84.1%

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

   if -0.010999999999999999 < phi1 < 5.8999999999999999e-9

   1. Initial program 63.9%

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

   2. Taylor expanded in phi1 around 0 63.9%

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

      1. cos-diff 44.1%

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

      2. +-commutative 44.1%

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

   4. Applied egg-rr 85.4%

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

   if 5.8999999999999999e-9 < phi1

   1. Initial program 74.0%

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

      1. acos-asin 74.1%

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

      2. sub-neg 74.1%

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

      3. div-inv 74.1%

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

      4. metadata-eval 74.1%

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

      5. +-commutative 74.1%

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

      6. *-commutative 74.1%

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

      7. fma-def 74.1%

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

   3. Applied egg-rr 74.1%

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

      1. sub-neg 74.1%

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

      2. sub-neg 74.1%

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

      3. remove-double-neg 74.1%

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

      4. mul-1-neg 74.1%

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

      5. distribute-neg-in 74.1%

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

      6. +-commutative 74.1%

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

      7. cos-neg 74.1%

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

      8. mul-1-neg 74.1%

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

      9. unsub-neg 74.1%

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

   5. Simplified 74.1%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.011:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 5.9 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\pi \cdot 0.5 - \sin^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\\ \end{array} \]

Alternative 8: 74.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ [phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.1 \cdot 10^{-15} \lor \neg \left(\phi_1 \leq 1.22 \cdot 10^{-157}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -2.1e-15) (not (<= phi1 1.22e-157)))
   (*
    R
    (acos
     (fma
      (sin phi1)
      (sin phi2)
      (* (cos phi2) (* (cos phi1) (cos (- lambda2 lambda1)))))))
   (*
    R
    (acos
     (+
      (*
       (* (cos phi1) (cos phi2))
       (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2))))
      (* phi1 phi2))))))

C

assert(lambda1 < lambda2);
assert(phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -2.1e-15) || !(phi1 <= 1.22e-157)) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi2) * (cos(phi1) * cos((lambda2 - lambda1))))));
	} else {
		tmp = R * acos((((cos(phi1) * cos(phi2)) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2)))) + (phi1 * phi2)));
	}
	return tmp;
}

Julia

lambda1, lambda2 = sort([lambda1, lambda2])
phi1, phi2 = sort([phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -2.1e-15) || !(phi1 <= 1.22e-157))
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi2) * Float64(cos(phi1) * cos(Float64(lambda2 - lambda1)))))));
	else
		tmp = Float64(R * acos(Float64(Float64(Float64(cos(phi1) * cos(phi2)) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2)))) + Float64(phi1 * phi2))));
	end
	return tmp
end

Wolfram

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -2.1e-15], N[Not[LessEqual[phi1, 1.22e-157]], $MachinePrecision]], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(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] + N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.09999999999999981e-15 or 1.22e-157 < phi1

   1. Initial program 75.8%

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

      1. log1p-expm1-u 75.8%

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

      2. log1p-udef 75.8%

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

   3. Applied egg-rr 75.8%

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

   4. Taylor expanded in phi1 around 0 75.7%

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

   6. Simplified 75.8%

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

   if -2.09999999999999981e-15 < phi1 < 1.22e-157

   1. Initial program 62.6%

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

   2. Taylor expanded in phi1 around 0 62.6%

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

   3. Taylor expanded in phi2 around 0 50.5%

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

      1. cos-diff 69.3%

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

   5. Applied egg-rr 69.3%

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

      1. cos-neg 69.3%

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

      2. *-commutative 69.3%

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

      3. fma-def 69.3%

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

      4. cos-neg 69.3%

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

   7. Simplified 69.3%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.1 \cdot 10^{-15} \lor \neg \left(\phi_1 \leq 1.22 \cdot 10^{-157}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \]

Alternative 9: 74.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ [phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.5 \cdot 10^{-14} \lor \neg \left(\phi_1 \leq 9.6 \cdot 10^{-163}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -3.5e-14) (not (<= phi1 9.6e-163)))
   (*
    R
    (acos
     (fma
      (sin phi1)
      (sin phi2)
      (* (cos phi2) (* (cos phi1) (cos (- lambda2 lambda1)))))))
   (*
    R
    (acos
     (+
      (*
       (* (cos phi1) (cos phi2))
       (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos lambda2))))
      (* phi1 phi2))))))

C

assert(lambda1 < lambda2);
assert(phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -3.5e-14) || !(phi1 <= 9.6e-163)) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi2) * (cos(phi1) * cos((lambda2 - lambda1))))));
	} else {
		tmp = R * acos((((cos(phi1) * cos(phi2)) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))) + (phi1 * phi2)));
	}
	return tmp;
}

Julia

lambda1, lambda2 = sort([lambda1, lambda2])
phi1, phi2 = sort([phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -3.5e-14) || !(phi1 <= 9.6e-163))
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi2) * Float64(cos(phi1) * cos(Float64(lambda2 - lambda1)))))));
	else
		tmp = Float64(R * acos(Float64(Float64(Float64(cos(phi1) * cos(phi2)) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2)))) + Float64(phi1 * phi2))));
	end
	return tmp
end

Wolfram

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -3.5e-14], N[Not[LessEqual[phi1, 9.6e-163]], $MachinePrecision]], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.5000000000000002e-14 or 9.6000000000000003e-163 < phi1

   1. Initial program 75.0%

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

      1. log1p-expm1-u 75.0%

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

      2. log1p-udef 74.9%

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

   3. Applied egg-rr 74.9%

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

   4. Taylor expanded in phi1 around 0 74.9%

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

   6. Simplified 75.0%

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

   if -3.5000000000000002e-14 < phi1 < 9.6000000000000003e-163

   1. Initial program 63.7%

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

   2. Taylor expanded in phi1 around 0 63.7%

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

   3. Taylor expanded in phi2 around 0 51.3%

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

      1. cos-diff 42.5%

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

      2. +-commutative 42.5%

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

   5. Applied egg-rr 69.7%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.5 \cdot 10^{-14} \lor \neg \left(\phi_1 \leq 9.6 \cdot 10^{-163}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \]

Alternative 10: 75.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ [phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.05 \cdot 10^{-228} \lor \neg \left(\phi_2 \leq 6.6 \cdot 10^{-64}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi2 -1.05e-228) (not (<= phi2 6.6e-64)))
   (*
    R
    (acos
     (fma
      (sin phi1)
      (sin phi2)
      (* (cos phi2) (* (cos phi1) (cos (- lambda2 lambda1)))))))
   (*
    R
    (acos
     (+
      (* phi1 phi2)
      (*
       (cos phi1)
       (+
        (* (sin lambda1) (sin lambda2))
        (* (cos lambda1) (cos lambda2)))))))))

C

assert(lambda1 < lambda2);
assert(phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -1.05e-228) || !(phi2 <= 6.6e-64)) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi2) * (cos(phi1) * cos((lambda2 - lambda1))))));
	} else {
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))))));
	}
	return tmp;
}

Julia

lambda1, lambda2 = sort([lambda1, lambda2])
phi1, phi2 = sort([phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi2 <= -1.05e-228) || !(phi2 <= 6.6e-64))
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi2) * Float64(cos(phi1) * cos(Float64(lambda2 - lambda1)))))));
	else
		tmp = Float64(R * acos(Float64(Float64(phi1 * phi2) + Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2)))))));
	end
	return tmp
end

Wolfram

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi2, -1.05e-228], N[Not[LessEqual[phi2, 6.6e-64]], $MachinePrecision]], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(phi1 * phi2), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.04999999999999995e-228 or 6.5999999999999999e-64 < phi2

   1. Initial program 71.8%

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

      1. log1p-expm1-u 71.8%

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

      2. log1p-udef 71.8%

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

   3. Applied egg-rr 71.8%

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

   4. Taylor expanded in phi1 around 0 71.8%

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

   6. Simplified 71.8%

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

   if -1.04999999999999995e-228 < phi2 < 6.5999999999999999e-64

   1. Initial program 67.9%

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

   2. Taylor expanded in phi1 around 0 51.0%

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

   3. Taylor expanded in phi2 around 0 51.0%

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

   4. Taylor expanded in phi2 around 0 51.0%

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

      1. cos-diff 72.1%

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

      2. +-commutative 72.1%

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

   6. Applied egg-rr 72.1%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.05 \cdot 10^{-228} \lor \neg \left(\phi_2 \leq 6.6 \cdot 10^{-64}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]

Alternative 11: 75.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ [phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -2.35 \cdot 10^{-228}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.4 \cdot 10^{-63}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right), t_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= phi2 -2.35e-228)
     (*
      R
      (acos (+ t_0 (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))))
     (if (<= phi2 1.4e-63)
       (*
        R
        (acos
         (+
          (* phi1 phi2)
          (*
           (cos phi1)
           (+
            (* (sin lambda1) (sin lambda2))
            (* (cos lambda1) (cos lambda2)))))))
       (*
        R
        (acos
         (fma (cos phi1) (* (cos phi2) (cos (- lambda2 lambda1))) t_0)))))))

C

assert(lambda1 < lambda2);
assert(phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi2 <= -2.35e-228) {
		tmp = R * acos((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	} else if (phi2 <= 1.4e-63) {
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))))));
	} else {
		tmp = R * acos(fma(cos(phi1), (cos(phi2) * cos((lambda2 - lambda1))), t_0));
	}
	return tmp;
}

Julia

lambda1, lambda2 = sort([lambda1, lambda2])
phi1, phi2 = sort([phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi2 <= -2.35e-228)
		tmp = Float64(R * acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))));
	elseif (phi2 <= 1.4e-63)
		tmp = Float64(R * acos(Float64(Float64(phi1 * phi2) + Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2)))))));
	else
		tmp = Float64(R * acos(fma(cos(phi1), Float64(cos(phi2) * cos(Float64(lambda2 - lambda1))), t_0)));
	end
	return tmp
end

Wolfram

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -2.35e-228], N[(R * N[ArcCos[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.4e-63], N[(R * N[ArcCos[N[(N[(phi1 * phi2), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -2.3500000000000001e-228

   1. Initial program 70.2%

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

   if -2.3500000000000001e-228 < phi2 < 1.4000000000000001e-63

   1. Initial program 67.9%

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

   2. Taylor expanded in phi1 around 0 51.0%

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

   3. Taylor expanded in phi2 around 0 51.0%

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

   4. Taylor expanded in phi2 around 0 51.0%

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

      1. cos-diff 72.1%

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

      2. +-commutative 72.1%

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

   6. Applied egg-rr 72.1%

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

   if 1.4000000000000001e-63 < phi2

   1. Initial program 73.6%

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

   2. Taylor expanded in phi1 around 0 73.6%

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

      1. associate-*r* 73.6%

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

      2. fma-udef 73.6%

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

      3. sub-neg 73.6%

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

      4. neg-mul-1 73.6%

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

      5. fma-udef 73.6%

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

      6. associate-*r* 73.6%

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

      7. fma-def 73.7%

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

   4. Simplified 73.7%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.35 \cdot 10^{-228}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.4 \cdot 10^{-63}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]

Alternative 12: 75.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ [phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.65 \cdot 10^{-228} \lor \neg \left(\phi_2 \leq 4.5 \cdot 10^{-63}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi2 -1.65e-228) (not (<= phi2 4.5e-63)))
   (*
    R
    (acos
     (+
      (* (sin phi1) (sin phi2))
      (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))))
   (*
    R
    (acos
     (+
      (* phi1 phi2)
      (*
       (cos phi1)
       (+
        (* (sin lambda1) (sin lambda2))
        (* (cos lambda1) (cos lambda2)))))))))

C

assert(lambda1 < lambda2);
assert(phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -1.65e-228) || !(phi2 <= 4.5e-63)) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	} else {
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))))));
	}
	return tmp;
}

Fortran

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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 <= (-1.65d-228)) .or. (.not. (phi2 <= 4.5d-63))) then
        tmp = r * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))))
    else
        tmp = r * acos(((phi1 * phi2) + (cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))))))
    end if
    code = tmp
end function

Java

assert lambda1 < lambda2;
assert phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi2 <= -1.65e-228) || !(phi2 <= 4.5e-63)) {
		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(((phi1 * phi2) + (Math.cos(phi1) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(lambda2))))));
	}
	return tmp;
}

Python

[lambda1, lambda2] = sort([lambda1, lambda2])
[phi1, phi2] = sort([phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (phi2 <= -1.65e-228) or not (phi2 <= 4.5e-63):
		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(((phi1 * phi2) + (math.cos(phi1) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * math.cos(lambda2))))))
	return tmp

Julia

lambda1, lambda2 = sort([lambda1, lambda2])
phi1, phi2 = sort([phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi2 <= -1.65e-228) || !(phi2 <= 4.5e-63))
		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(Float64(phi1 * phi2) + Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2)))))));
	end
	return tmp
end

MATLAB

lambda1, lambda2 = num2cell(sort([lambda1, lambda2])){:}
phi1, phi2 = num2cell(sort([phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((phi2 <= -1.65e-228) || ~((phi2 <= 4.5e-63)))
		tmp = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	else
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi2, -1.65e-228], N[Not[LessEqual[phi2, 4.5e-63]], $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[(phi1 * phi2), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.65000000000000003e-228 or 4.5e-63 < phi2

   1. Initial program 71.8%

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

   if -1.65000000000000003e-228 < phi2 < 4.5e-63

   1. Initial program 67.9%

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

   2. Taylor expanded in phi1 around 0 51.0%

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

   3. Taylor expanded in phi2 around 0 51.0%

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

   4. Taylor expanded in phi2 around 0 51.0%

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

      1. cos-diff 72.1%

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

      2. +-commutative 72.1%

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

   6. Applied egg-rr 72.1%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.65 \cdot 10^{-228} \lor \neg \left(\phi_2 \leq 4.5 \cdot 10^{-63}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]

Alternative 13: 63.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ [phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\lambda_2 \leq 1.4 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.06 \cdot 10^{+137}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot t_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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 (<= lambda2 1.4e-14)
     (*
      R
      (acos
       (+
        (* (sin phi1) (sin phi2))
        (* (cos lambda1) (* (cos phi1) (cos phi2))))))
     (if (<= lambda2 1.06e+137)
       (* R (acos (fma (sin phi1) (sin phi2) (* (cos phi2) t_0))))
       (* R (acos (fma (sin phi1) (sin phi2) (* (cos phi1) t_0))))))))

C

assert(lambda1 < lambda2);
assert(phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (lambda2 <= 1.4e-14) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(lambda1) * (cos(phi1) * cos(phi2)))));
	} else if (lambda2 <= 1.06e+137) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi2) * t_0)));
	} else {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * t_0)));
	}
	return tmp;
}

Julia

lambda1, lambda2 = sort([lambda1, lambda2])
phi1, phi2 = sort([phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (lambda2 <= 1.4e-14)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(lambda1) * Float64(cos(phi1) * cos(phi2))))));
	elseif (lambda2 <= 1.06e+137)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi2) * t_0))));
	else
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * t_0))));
	end
	return tmp
end

Wolfram

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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[lambda2, 1.4e-14], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 1.06e+137], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $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 lambda2 < 1.4e-14

   1. Initial program 73.5%

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

   2. Taylor expanded in lambda2 around 0 64.7%

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

   if 1.4e-14 < lambda2 < 1.06000000000000006e137

   1. Initial program 56.6%

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

   3. Simplified 56.6%

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

   4. Taylor expanded in phi1 around 0 35.2%

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

      1. sub-neg 35.2%

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

      2. neg-mul-1 35.2%

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

      3. neg-mul-1 35.2%

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

      4. remove-double-neg 35.2%

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

      5. mul-1-neg 35.2%

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

      6. distribute-neg-in 35.2%

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

      7. +-commutative 35.2%

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

      8. cos-neg 35.2%

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

      9. mul-1-neg 35.2%

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

      10. unsub-neg 35.2%

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

   6. Simplified 35.2%

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

   if 1.06000000000000006e137 < lambda2

   1. Initial program 67.2%

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

   3. Simplified 67.1%

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

   4. Taylor expanded in phi2 around 0 46.7%

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

      1. sub-neg 46.7%

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

      2. remove-double-neg 46.7%

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

      3. mul-1-neg 46.7%

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

      4. distribute-neg-in 46.7%

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

      5. +-commutative 46.7%

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

      6. cos-neg 46.7%

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

      7. mul-1-neg 46.7%

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

      8. unsub-neg 46.7%

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

   6. Simplified 46.7%

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

4. Final simplification 58.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.4 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.06 \cdot 10^{+137}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \end{array} \]

Alternative 14: 73.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ [phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -3 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \lambda_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot \cos \lambda_2 + t_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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 -3e-5)
     (* R (acos (+ t_1 (* (cos lambda1) t_0))))
     (* R (acos (+ (* t_0 (cos lambda2)) t_1))))))

C

assert(lambda1 < lambda2);
assert(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 <= -3e-5) {
		tmp = R * acos((t_1 + (cos(lambda1) * t_0)));
	} else {
		tmp = R * acos(((t_0 * cos(lambda2)) + t_1));
	}
	return tmp;
}

Fortran

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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 <= (-3d-5)) then
        tmp = r * acos((t_1 + (cos(lambda1) * t_0)))
    else
        tmp = r * acos(((t_0 * cos(lambda2)) + t_1))
    end if
    code = tmp
end function

Java

assert lambda1 < lambda2;
assert 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 <= -3e-5) {
		tmp = R * Math.acos((t_1 + (Math.cos(lambda1) * t_0)));
	} else {
		tmp = R * Math.acos(((t_0 * Math.cos(lambda2)) + t_1));
	}
	return tmp;
}

Python

[lambda1, lambda2] = sort([lambda1, lambda2])
[phi1, phi2] = sort([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 <= -3e-5:
		tmp = R * math.acos((t_1 + (math.cos(lambda1) * t_0)))
	else:
		tmp = R * math.acos(((t_0 * math.cos(lambda2)) + t_1))
	return tmp

Julia

lambda1, lambda2 = sort([lambda1, lambda2])
phi1, phi2 = sort([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 <= -3e-5)
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(lambda1) * t_0))));
	else
		tmp = Float64(R * acos(Float64(Float64(t_0 * cos(lambda2)) + t_1)));
	end
	return tmp
end

MATLAB

lambda1, lambda2 = num2cell(sort([lambda1, lambda2])){:}
phi1, phi2 = num2cell(sort([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 <= -3e-5)
		tmp = R * acos((t_1 + (cos(lambda1) * t_0)));
	else
		tmp = R * acos(((t_0 * cos(lambda2)) + t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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, -3e-5], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[lambda1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(t$95$0 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -3.00000000000000008e-5

   1. Initial program 60.1%

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

   2. Taylor expanded in lambda2 around 0 60.5%

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

   if -3.00000000000000008e-5 < lambda1

   1. Initial program 74.7%

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

   2. Taylor expanded in lambda1 around 0 59.9%

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

      1. cos-neg 20.6%

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

   4. Simplified 59.9%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2 + \sin \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]

Alternative 15: 73.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

lambda1, lambda2 = sort([lambda1, lambda2])
phi1, phi2 = sort([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)) * cos(Float64(lambda1 - lambda2))))))
end

MATLAB

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

Wolfram

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.7%

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

2. Final simplification 70.7%

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

Alternative 16: 57.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ [phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.96:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -0.96)
   (*
    R
    (acos
     (fma (sin phi1) (sin phi2) (* (cos phi1) (cos (- lambda2 lambda1))))))
   (*
    R
    (acos
     (+
      (* phi1 (sin phi2))
      (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))))))

C

assert(lambda1 < lambda2);
assert(phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.96) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * cos((lambda2 - lambda1)))));
	} else {
		tmp = R * acos(((phi1 * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Julia

lambda1, lambda2 = sort([lambda1, lambda2])
phi1, phi2 = sort([phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -0.96)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * cos(Float64(lambda2 - lambda1))))));
	else
		tmp = Float64(R * acos(Float64(Float64(phi1 * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))));
	end
	return tmp
end

Wolfram

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -0.96], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(phi1 * 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]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.95999999999999996

   1. Initial program 83.7%

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

   3. Simplified 83.8%

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

   4. Taylor expanded in phi2 around 0 48.6%

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

      1. sub-neg 48.6%

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

      2. remove-double-neg 48.6%

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

      3. mul-1-neg 48.6%

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

      4. distribute-neg-in 48.6%

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

      5. +-commutative 48.6%

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

      6. cos-neg 48.6%

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

      7. mul-1-neg 48.6%

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

      8. unsub-neg 48.6%

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

   6. Simplified 48.6%

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

   if -0.95999999999999996 < phi1

   1. Initial program 67.5%

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

   2. Taylor expanded in phi1 around 0 50.0%

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

4. Final simplification 49.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.96:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]

Alternative 17: 58.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ [phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 6.2 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot t_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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 6.2e-10)
     (* R (acos (fma (sin phi1) (sin phi2) (* (cos phi1) t_0))))
     (* R (acos (fma (sin phi1) (sin phi2) (* (cos phi2) t_0)))))))

C

assert(lambda1 < lambda2);
assert(phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 6.2e-10) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * t_0)));
	} else {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi2) * t_0)));
	}
	return tmp;
}

Julia

lambda1, lambda2 = sort([lambda1, lambda2])
phi1, phi2 = sort([phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi2 <= 6.2e-10)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * t_0))));
	else
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi2) * t_0))));
	end
	return tmp
end

Wolfram

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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, 6.2e-10], 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], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 6.2000000000000003e-10

   1. Initial program 69.2%

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

   3. Simplified 69.1%

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

   4. Taylor expanded in phi2 around 0 51.3%

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

      1. sub-neg 51.3%

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

      2. remove-double-neg 51.3%

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

      3. mul-1-neg 51.3%

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

      4. distribute-neg-in 51.3%

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

      5. +-commutative 51.3%

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

      6. cos-neg 51.3%

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

      7. mul-1-neg 51.3%

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

      8. unsub-neg 51.3%

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

   6. Simplified 51.3%

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

   if 6.2000000000000003e-10 < phi2

   1. Initial program 74.7%

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

   3. Simplified 74.8%

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

   4. Taylor expanded in phi1 around 0 47.9%

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

      1. sub-neg 47.9%

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

      2. neg-mul-1 47.9%

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

      3. neg-mul-1 47.9%

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

      4. remove-double-neg 47.9%

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

      5. mul-1-neg 47.9%

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

      6. distribute-neg-in 47.9%

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

      7. +-commutative 47.9%

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

      8. cos-neg 47.9%

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

      9. mul-1-neg 47.9%

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

      10. unsub-neg 47.9%

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

   6. Simplified 47.9%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 6.2 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \end{array} \]

Alternative 18: 54.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} [lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ [phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \\ \begin{array}{l} t_0 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -5000:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + t_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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)) (cos (- lambda1 lambda2)))))
   (if (<= phi1 -5000.0)
     (* R (acos (+ t_0 (* (sin phi1) phi2))))
     (* R (acos (+ (* phi1 (sin phi2)) t_0))))))

C

assert(lambda1 < lambda2);
assert(phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -5000.0) {
		tmp = R * acos((t_0 + (sin(phi1) * phi2)));
	} else {
		tmp = R * acos(((phi1 * sin(phi2)) + t_0));
	}
	return tmp;
}

Fortran

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))
    if (phi1 <= (-5000.0d0)) then
        tmp = r * acos((t_0 + (sin(phi1) * phi2)))
    else
        tmp = r * acos(((phi1 * sin(phi2)) + t_0))
    end if
    code = tmp
end function

Java

assert lambda1 < lambda2;
assert 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)) * Math.cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -5000.0) {
		tmp = R * Math.acos((t_0 + (Math.sin(phi1) * phi2)));
	} else {
		tmp = R * Math.acos(((phi1 * Math.sin(phi2)) + t_0));
	}
	return tmp;
}

Python

[lambda1, lambda2] = sort([lambda1, lambda2])
[phi1, phi2] = sort([phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2))
	tmp = 0
	if phi1 <= -5000.0:
		tmp = R * math.acos((t_0 + (math.sin(phi1) * phi2)))
	else:
		tmp = R * math.acos(((phi1 * math.sin(phi2)) + t_0))
	return tmp

Julia

lambda1, lambda2 = sort([lambda1, lambda2])
phi1, phi2 = sort([phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi1 <= -5000.0)
		tmp = Float64(R * acos(Float64(t_0 + Float64(sin(phi1) * phi2))));
	else
		tmp = Float64(R * acos(Float64(Float64(phi1 * sin(phi2)) + t_0)));
	end
	return tmp
end

MATLAB

lambda1, lambda2 = num2cell(sort([lambda1, lambda2])){:}
phi1, phi2 = num2cell(sort([phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2));
	tmp = 0.0;
	if (phi1 <= -5000.0)
		tmp = R * acos((t_0 + (sin(phi1) * phi2)));
	else
		tmp = R * acos(((phi1 * sin(phi2)) + t_0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -5000.0], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -5e3

   1. Initial program 83.4%

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

   2. Taylor expanded in phi2 around 0 41.4%

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

   if -5e3 < phi1

   1. Initial program 67.7%

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

   2. Taylor expanded in phi1 around 0 49.7%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]

Alternative 19: 40.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} [lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ [phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\lambda_2 \leq 3.25 \cdot 10^{-12}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \lambda_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2))))
   (if (<= lambda2 3.25e-12)
     (* R (acos (+ (* phi1 (sin phi2)) (* (cos lambda1) t_0))))
     (* R (acos (+ (* phi1 phi2) (* t_0 (cos (- lambda1 lambda2)))))))))

C

assert(lambda1 < lambda2);
assert(phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * cos(phi2);
	double tmp;
	if (lambda2 <= 3.25e-12) {
		tmp = R * acos(((phi1 * sin(phi2)) + (cos(lambda1) * t_0)));
	} else {
		tmp = R * acos(((phi1 * phi2) + (t_0 * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Fortran

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi1) * cos(phi2)
    if (lambda2 <= 3.25d-12) then
        tmp = r * acos(((phi1 * sin(phi2)) + (cos(lambda1) * t_0)))
    else
        tmp = r * acos(((phi1 * phi2) + (t_0 * cos((lambda1 - lambda2)))))
    end if
    code = tmp
end function

Java

assert lambda1 < lambda2;
assert 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 tmp;
	if (lambda2 <= 3.25e-12) {
		tmp = R * Math.acos(((phi1 * Math.sin(phi2)) + (Math.cos(lambda1) * t_0)));
	} else {
		tmp = R * Math.acos(((phi1 * phi2) + (t_0 * Math.cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Python

[lambda1, lambda2] = sort([lambda1, lambda2])
[phi1, phi2] = sort([phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.cos(phi2)
	tmp = 0
	if lambda2 <= 3.25e-12:
		tmp = R * math.acos(((phi1 * math.sin(phi2)) + (math.cos(lambda1) * t_0)))
	else:
		tmp = R * math.acos(((phi1 * phi2) + (t_0 * math.cos((lambda1 - lambda2)))))
	return tmp

Julia

lambda1, lambda2 = sort([lambda1, lambda2])
phi1, phi2 = sort([phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if (lambda2 <= 3.25e-12)
		tmp = Float64(R * acos(Float64(Float64(phi1 * sin(phi2)) + Float64(cos(lambda1) * t_0))));
	else
		tmp = Float64(R * acos(Float64(Float64(phi1 * phi2) + Float64(t_0 * cos(Float64(lambda1 - lambda2))))));
	end
	return tmp
end

MATLAB

lambda1, lambda2 = num2cell(sort([lambda1, lambda2])){:}
phi1, phi2 = num2cell(sort([phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * cos(phi2);
	tmp = 0.0;
	if (lambda2 <= 3.25e-12)
		tmp = R * acos(((phi1 * sin(phi2)) + (cos(lambda1) * t_0)));
	else
		tmp = R * acos(((phi1 * phi2) + (t_0 * cos((lambda1 - lambda2)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, 3.25e-12], N[(R * N[ArcCos[N[(N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(phi1 * phi2), $MachinePrecision] + N[(t$95$0 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 3.2500000000000001e-12

   1. Initial program 73.7%

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

   2. Taylor expanded in phi1 around 0 45.2%

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

   3. Taylor expanded in lambda2 around 0 38.2%

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

   if 3.2500000000000001e-12 < lambda2

   1. Initial program 61.9%

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

   2. Taylor expanded in phi1 around 0 38.5%

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

   3. Taylor expanded in phi2 around 0 36.0%

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

4. Final simplification 37.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 3.25 \cdot 10^{-12}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]

Alternative 20: 43.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ [phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -0.00031:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \lambda_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot \cos \lambda_2 + t_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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 (* phi1 (sin phi2))))
   (if (<= lambda1 -0.00031)
     (* R (acos (+ t_1 (* (cos lambda1) t_0))))
     (* R (acos (+ (* t_0 (cos lambda2)) t_1))))))

C

assert(lambda1 < lambda2);
assert(phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * cos(phi2);
	double t_1 = phi1 * sin(phi2);
	double tmp;
	if (lambda1 <= -0.00031) {
		tmp = R * acos((t_1 + (cos(lambda1) * t_0)));
	} else {
		tmp = R * acos(((t_0 * cos(lambda2)) + t_1));
	}
	return tmp;
}

Fortran

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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 = phi1 * sin(phi2)
    if (lambda1 <= (-0.00031d0)) then
        tmp = r * acos((t_1 + (cos(lambda1) * t_0)))
    else
        tmp = r * acos(((t_0 * cos(lambda2)) + t_1))
    end if
    code = tmp
end function

Java

assert lambda1 < lambda2;
assert 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 = phi1 * Math.sin(phi2);
	double tmp;
	if (lambda1 <= -0.00031) {
		tmp = R * Math.acos((t_1 + (Math.cos(lambda1) * t_0)));
	} else {
		tmp = R * Math.acos(((t_0 * Math.cos(lambda2)) + t_1));
	}
	return tmp;
}

Python

[lambda1, lambda2] = sort([lambda1, lambda2])
[phi1, phi2] = sort([phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.cos(phi2)
	t_1 = phi1 * math.sin(phi2)
	tmp = 0
	if lambda1 <= -0.00031:
		tmp = R * math.acos((t_1 + (math.cos(lambda1) * t_0)))
	else:
		tmp = R * math.acos(((t_0 * math.cos(lambda2)) + t_1))
	return tmp

Julia

lambda1, lambda2 = sort([lambda1, lambda2])
phi1, phi2 = sort([phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	t_1 = Float64(phi1 * sin(phi2))
	tmp = 0.0
	if (lambda1 <= -0.00031)
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(lambda1) * t_0))));
	else
		tmp = Float64(R * acos(Float64(Float64(t_0 * cos(lambda2)) + t_1)));
	end
	return tmp
end

MATLAB

lambda1, lambda2 = num2cell(sort([lambda1, lambda2])){:}
phi1, phi2 = num2cell(sort([phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * cos(phi2);
	t_1 = phi1 * sin(phi2);
	tmp = 0.0;
	if (lambda1 <= -0.00031)
		tmp = R * acos((t_1 + (cos(lambda1) * t_0)));
	else
		tmp = R * acos(((t_0 * cos(lambda2)) + t_1));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -0.00031], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[lambda1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(t$95$0 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -3.1e-4

   1. Initial program 60.1%

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

   2. Taylor expanded in phi1 around 0 36.2%

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

   3. Taylor expanded in lambda2 around 0 36.5%

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

   if -3.1e-4 < lambda1

   1. Initial program 74.7%

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

   2. Taylor expanded in phi1 around 0 46.2%

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

   3. Taylor expanded in lambda1 around 0 38.2%

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

      1. cos-neg 20.6%

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

   5. Simplified 38.2%

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

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.00031:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2 + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]

Alternative 21: 43.4% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

2. Taylor expanded in phi1 around 0 43.5%

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

3. Final simplification 43.5%

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

Alternative 22: 33.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} [lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ [phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\lambda_1 \leq -0.00043:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \lambda_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot \cos \lambda_2 + \phi_1 \cdot \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2))))
   (if (<= lambda1 -0.00043)
     (* R (acos (+ (* phi1 phi2) (* (cos lambda1) t_0))))
     (* R (acos (+ (* t_0 (cos lambda2)) (* phi1 phi2)))))))

C

assert(lambda1 < lambda2);
assert(phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * cos(phi2);
	double tmp;
	if (lambda1 <= -0.00043) {
		tmp = R * acos(((phi1 * phi2) + (cos(lambda1) * t_0)));
	} else {
		tmp = R * acos(((t_0 * cos(lambda2)) + (phi1 * phi2)));
	}
	return tmp;
}

Fortran

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi1) * cos(phi2)
    if (lambda1 <= (-0.00043d0)) then
        tmp = r * acos(((phi1 * phi2) + (cos(lambda1) * t_0)))
    else
        tmp = r * acos(((t_0 * cos(lambda2)) + (phi1 * phi2)))
    end if
    code = tmp
end function

Java

assert lambda1 < lambda2;
assert 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 tmp;
	if (lambda1 <= -0.00043) {
		tmp = R * Math.acos(((phi1 * phi2) + (Math.cos(lambda1) * t_0)));
	} else {
		tmp = R * Math.acos(((t_0 * Math.cos(lambda2)) + (phi1 * phi2)));
	}
	return tmp;
}

Python

[lambda1, lambda2] = sort([lambda1, lambda2])
[phi1, phi2] = sort([phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.cos(phi2)
	tmp = 0
	if lambda1 <= -0.00043:
		tmp = R * math.acos(((phi1 * phi2) + (math.cos(lambda1) * t_0)))
	else:
		tmp = R * math.acos(((t_0 * math.cos(lambda2)) + (phi1 * phi2)))
	return tmp

Julia

lambda1, lambda2 = sort([lambda1, lambda2])
phi1, phi2 = sort([phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if (lambda1 <= -0.00043)
		tmp = Float64(R * acos(Float64(Float64(phi1 * phi2) + Float64(cos(lambda1) * t_0))));
	else
		tmp = Float64(R * acos(Float64(Float64(t_0 * cos(lambda2)) + Float64(phi1 * phi2))));
	end
	return tmp
end

MATLAB

lambda1, lambda2 = num2cell(sort([lambda1, lambda2])){:}
phi1, phi2 = num2cell(sort([phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * cos(phi2);
	tmp = 0.0;
	if (lambda1 <= -0.00043)
		tmp = R * acos(((phi1 * phi2) + (cos(lambda1) * t_0)));
	else
		tmp = R * acos(((t_0 * cos(lambda2)) + (phi1 * phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -0.00043], N[(R * N[ArcCos[N[(N[(phi1 * phi2), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(t$95$0 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -4.29999999999999989e-4

   1. Initial program 60.1%

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

   2. Taylor expanded in phi1 around 0 36.2%

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

   3. Taylor expanded in phi2 around 0 29.0%

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

   4. Taylor expanded in lambda2 around 0 29.3%

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

   if -4.29999999999999989e-4 < lambda1

   1. Initial program 74.7%

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

   2. Taylor expanded in phi1 around 0 46.2%

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

   3. Taylor expanded in phi2 around 0 36.9%

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

   4. Taylor expanded in lambda1 around 0 30.2%

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

      1. cos-neg 20.6%

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

   6. Simplified 30.2%

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

4. Final simplification 29.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.00043:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2 + \phi_1 \cdot \phi_2\right)\\ \end{array} \]

Alternative 23: 33.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

2. Taylor expanded in phi1 around 0 43.5%

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

3. Taylor expanded in phi2 around 0 34.7%

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

4. Final simplification 34.7%

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

Alternative 24: 33.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ [phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -2 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_2 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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 -2e-8)
     (* R (acos (+ (* phi1 phi2) (* (cos phi1) t_0))))
     (* R (acos (+ (* phi1 phi2) (* (cos phi2) t_0)))))))

C

assert(lambda1 < lambda2);
assert(phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -2e-8) {
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * t_0)));
	} else {
		tmp = R * acos(((phi1 * phi2) + (cos(phi2) * t_0)));
	}
	return tmp;
}

Fortran

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((lambda1 - lambda2))
    if (phi1 <= (-2d-8)) then
        tmp = r * acos(((phi1 * phi2) + (cos(phi1) * t_0)))
    else
        tmp = r * acos(((phi1 * phi2) + (cos(phi2) * t_0)))
    end if
    code = tmp
end function

Java

assert lambda1 < lambda2;
assert phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -2e-8) {
		tmp = R * Math.acos(((phi1 * phi2) + (Math.cos(phi1) * t_0)));
	} else {
		tmp = R * Math.acos(((phi1 * phi2) + (Math.cos(phi2) * t_0)));
	}
	return tmp;
}

Python

[lambda1, lambda2] = sort([lambda1, lambda2])
[phi1, phi2] = sort([phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	tmp = 0
	if phi1 <= -2e-8:
		tmp = R * math.acos(((phi1 * phi2) + (math.cos(phi1) * t_0)))
	else:
		tmp = R * math.acos(((phi1 * phi2) + (math.cos(phi2) * t_0)))
	return tmp

Julia

lambda1, lambda2 = sort([lambda1, lambda2])
phi1, phi2 = sort([phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -2e-8)
		tmp = Float64(R * acos(Float64(Float64(phi1 * phi2) + Float64(cos(phi1) * t_0))));
	else
		tmp = Float64(R * acos(Float64(Float64(phi1 * phi2) + Float64(cos(phi2) * t_0))));
	end
	return tmp
end

MATLAB

lambda1, lambda2 = num2cell(sort([lambda1, lambda2])){:}
phi1, phi2 = num2cell(sort([phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	tmp = 0.0;
	if (phi1 <= -2e-8)
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * t_0)));
	else
		tmp = R * acos(((phi1 * phi2) + (cos(phi2) * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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, -2e-8], N[(R * N[ArcCos[N[(N[(phi1 * phi2), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(phi1 * phi2), $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2e-8

   1. Initial program 84.1%

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

   2. Taylor expanded in phi1 around 0 17.1%

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

   3. Taylor expanded in phi2 around 0 16.5%

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

   4. Taylor expanded in phi2 around 0 16.5%

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

   if -2e-8 < phi1

   1. Initial program 67.4%

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

   2. Taylor expanded in phi1 around 0 50.1%

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

   3. Taylor expanded in phi2 around 0 39.3%

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

   4. Taylor expanded in phi1 around 0 34.1%

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

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]

Alternative 25: 23.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ [phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -105:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \lambda_1 \cdot \cos \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -105.0)
   (* R (acos (+ (* phi1 phi2) (* (cos lambda1) (cos phi1)))))
   (* R (acos (+ (cos (- lambda2 lambda1)) (* phi1 phi2))))))

C

assert(lambda1 < lambda2);
assert(phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -105.0) {
		tmp = R * acos(((phi1 * phi2) + (cos(lambda1) * cos(phi1))));
	} else {
		tmp = R * acos((cos((lambda2 - lambda1)) + (phi1 * phi2)));
	}
	return tmp;
}

Fortran

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: 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 <= (-105.0d0)) then
        tmp = r * acos(((phi1 * phi2) + (cos(lambda1) * cos(phi1))))
    else
        tmp = r * acos((cos((lambda2 - lambda1)) + (phi1 * phi2)))
    end if
    code = tmp
end function

Java

assert lambda1 < lambda2;
assert phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -105.0) {
		tmp = R * Math.acos(((phi1 * phi2) + (Math.cos(lambda1) * Math.cos(phi1))));
	} else {
		tmp = R * Math.acos((Math.cos((lambda2 - lambda1)) + (phi1 * phi2)));
	}
	return tmp;
}

Python

[lambda1, lambda2] = sort([lambda1, lambda2])
[phi1, phi2] = sort([phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -105.0:
		tmp = R * math.acos(((phi1 * phi2) + (math.cos(lambda1) * math.cos(phi1))))
	else:
		tmp = R * math.acos((math.cos((lambda2 - lambda1)) + (phi1 * phi2)))
	return tmp

Julia

lambda1, lambda2 = sort([lambda1, lambda2])
phi1, phi2 = sort([phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -105.0)
		tmp = Float64(R * acos(Float64(Float64(phi1 * phi2) + Float64(cos(lambda1) * cos(phi1)))));
	else
		tmp = Float64(R * acos(Float64(cos(Float64(lambda2 - lambda1)) + Float64(phi1 * phi2))));
	end
	return tmp
end

MATLAB

lambda1, lambda2 = num2cell(sort([lambda1, lambda2])){:}
phi1, phi2 = num2cell(sort([phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -105.0)
		tmp = R * acos(((phi1 * phi2) + (cos(lambda1) * cos(phi1))));
	else
		tmp = R * acos((cos((lambda2 - lambda1)) + (phi1 * phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -105.0], N[(R * N[ArcCos[N[(N[(phi1 * phi2), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -105

   1. Initial program 83.4%

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

   2. Taylor expanded in phi1 around 0 17.2%

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

   3. Taylor expanded in phi2 around 0 17.2%

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

   4. Taylor expanded in phi2 around 0 17.2%

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

   5. Taylor expanded in lambda2 around 0 13.5%

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

   if -105 < phi1

   1. Initial program 67.7%

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

   2. Taylor expanded in phi1 around 0 49.7%

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

   3. Taylor expanded in phi2 around 0 38.9%

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

   4. Taylor expanded in phi2 around 0 28.6%

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

   5. Taylor expanded in phi1 around 0 23.5%

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

      1. cos-neg 23.5%

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

      2. sub-neg 23.5%

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

      3. +-commutative 23.5%

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

      4. distribute-neg-in 23.5%

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

      5. remove-double-neg 23.5%

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

      6. sub-neg 23.5%

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

   7. Simplified 23.5%

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

4. Final simplification 21.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -105:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \lambda_1 \cdot \cos \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \]

Alternative 26: 25.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} [lambda1, lambda2] = \mathsf{sort}([lambda1, lambda2])\\ [phi1, phi2] = \mathsf{sort}([phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 5.5 \cdot 10^{-36}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \lambda_1 \cdot \cos \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 5.5e-36)
   (* R (acos (+ (* phi1 phi2) (* (cos lambda1) (cos phi1)))))
   (* R (acos (+ (* phi1 phi2) (* (cos phi1) (cos lambda2)))))))

C

assert(lambda1 < lambda2);
assert(phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 5.5e-36) {
		tmp = R * acos(((phi1 * phi2) + (cos(lambda1) * cos(phi1))));
	} else {
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * cos(lambda2))));
	}
	return tmp;
}

Fortran

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 5.5d-36) then
        tmp = r * acos(((phi1 * phi2) + (cos(lambda1) * cos(phi1))))
    else
        tmp = r * acos(((phi1 * phi2) + (cos(phi1) * cos(lambda2))))
    end if
    code = tmp
end function

Java

assert lambda1 < lambda2;
assert phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 5.5e-36) {
		tmp = R * Math.acos(((phi1 * phi2) + (Math.cos(lambda1) * Math.cos(phi1))));
	} else {
		tmp = R * Math.acos(((phi1 * phi2) + (Math.cos(phi1) * Math.cos(lambda2))));
	}
	return tmp;
}

Python

[lambda1, lambda2] = sort([lambda1, lambda2])
[phi1, phi2] = sort([phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 5.5e-36:
		tmp = R * math.acos(((phi1 * phi2) + (math.cos(lambda1) * math.cos(phi1))))
	else:
		tmp = R * math.acos(((phi1 * phi2) + (math.cos(phi1) * math.cos(lambda2))))
	return tmp

Julia

lambda1, lambda2 = sort([lambda1, lambda2])
phi1, phi2 = sort([phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 5.5e-36)
		tmp = Float64(R * acos(Float64(Float64(phi1 * phi2) + Float64(cos(lambda1) * cos(phi1)))));
	else
		tmp = Float64(R * acos(Float64(Float64(phi1 * phi2) + Float64(cos(phi1) * cos(lambda2)))));
	end
	return tmp
end

MATLAB

lambda1, lambda2 = num2cell(sort([lambda1, lambda2])){:}
phi1, phi2 = num2cell(sort([phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 5.5e-36)
		tmp = R * acos(((phi1 * phi2) + (cos(lambda1) * cos(phi1))));
	else
		tmp = R * acos(((phi1 * phi2) + (cos(phi1) * cos(lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: lambda1 and lambda2 should be sorted in increasing order before calling this function.
NOTE: phi1 and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 5.5e-36], N[(R * N[ArcCos[N[(N[(phi1 * phi2), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(phi1 * phi2), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 5.49999999999999984e-36

   1. Initial program 73.7%

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

   2. Taylor expanded in phi1 around 0 45.1%

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

   3. Taylor expanded in phi2 around 0 34.5%

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

   4. Taylor expanded in phi2 around 0 26.9%

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

   5. Taylor expanded in lambda2 around 0 22.4%

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

   if 5.49999999999999984e-36 < lambda2

   1. Initial program 62.9%

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

   2. Taylor expanded in phi1 around 0 39.4%

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

   3. Taylor expanded in phi2 around 0 35.4%

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

   4. Taylor expanded in phi2 around 0 25.2%

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

   5. Taylor expanded in lambda1 around 0 25.1%

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

      1. cos-neg 25.1%

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

   7. Simplified 25.1%

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

4. Final simplification 23.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 5.5 \cdot 10^{-36}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \lambda_1 \cdot \cos \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 27: 26.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

2. Taylor expanded in phi1 around 0 43.5%

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

3. Taylor expanded in phi2 around 0 34.7%

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

4. Taylor expanded in phi2 around 0 26.4%

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

5. Final simplification 26.4%

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

Alternative 28: 18.0% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.7%

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

2. Taylor expanded in phi1 around 0 43.5%

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

3. Taylor expanded in phi2 around 0 34.7%

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

4. Taylor expanded in phi2 around 0 26.4%

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

5. Taylor expanded in phi1 around 0 19.9%

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

   1. cos-neg 19.9%

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

   2. sub-neg 19.9%

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

   3. +-commutative 19.9%

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

   4. distribute-neg-in 19.9%

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

   5. remove-double-neg 19.9%

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

   6. sub-neg 19.9%

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

7. Simplified 19.9%

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

8. Final simplification 19.9%

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

Reproduce

herbie shell --seed 2023299 
(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))