Spherical law of cosines

Percentage Accurate: 73.6% → 94.1%

Time: 35.1s

Alternatives: 23

Speedup: 0.9×

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: 73.6% 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: 94.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + t_0 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right) \cdot R \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2))))
   (*
    (acos
     (fma
      (sin phi1)
      (sin phi2)
      (+
       (* t_0 (* (cos lambda1) (cos lambda2)))
       (* t_0 (* (sin lambda1) (sin lambda2))))))
    R)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * cos(phi2);
	return acos(fma(sin(phi1), sin(phi2), ((t_0 * (cos(lambda1) * cos(lambda2))) + (t_0 * (sin(lambda1) * sin(lambda2)))))) * R;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	return Float64(acos(fma(sin(phi1), sin(phi2), Float64(Float64(t_0 * Float64(cos(lambda1) * cos(lambda2))) + Float64(t_0 * Float64(sin(lambda1) * sin(lambda2)))))) * R)
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, N[(N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[(t$95$0 * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Initial program 71.8%

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

3. Simplified 71.8%

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

   1. associate-*r* 71.8%

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

   2. cos-diff 93.3%

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

   3. distribute-lft-in 93.3%

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

5. Applied egg-rr 93.3%

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

6. Final simplification 93.3%

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

Alternative 2: 94.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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. cos-diff 93.3%

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

   2. +-commutative 93.3%

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

3. Applied egg-rr 93.3%

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

   1. *-commutative 93.3%

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

   2. fma-def 93.3%

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

5. Applied egg-rr 93.3%

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

6. Final simplification 93.3%

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

Alternative 3: 94.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.8%

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

3. Simplified 71.8%

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

   1. cos-diff 93.3%

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

   2. +-commutative 93.3%

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

5. Applied egg-rr 93.3%

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

6. Final simplification 93.3%

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

Alternative 4: 94.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

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 = r * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda1) * sin(lambda2))))))
end function

Java

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

Python

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) * math.cos(lambda2)) + (math.sin(lambda1) * math.sin(lambda2))))))

Julia

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(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda1) * sin(lambda2)))))))
end

MATLAB

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

Wolfram

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[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

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. cos-diff 93.3%

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

   2. +-commutative 93.3%

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

3. Applied egg-rr 93.3%

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

4. Final simplification 93.3%

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

Alternative 5: 82.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{if}\;\phi_1 \leq -1.1 \cdot 10^{+42}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, t_0\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.93:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))
   (if (<= phi1 -1.1e+42)
     (* R (acos (fma (sin phi1) (sin phi2) t_0)))
     (if (<= phi1 0.93)
       (*
        R
        (acos
         (+
          (*
           (* (cos phi1) (cos phi2))
           (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))))
          (* phi1 (sin phi2)))))
       (* R (acos (+ (* (sin phi1) (sin phi2)) t_0)))))))

C

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 <= -1.1e+42) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), t_0));
	} else if (phi1 <= 0.93) {
		tmp = R * acos((((cos(phi1) * cos(phi2)) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda1) * sin(lambda2)))) + (phi1 * sin(phi2))));
	} else {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + t_0));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))
	tmp = 0.0
	if (phi1 <= -1.1e+42)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), t_0)));
	elseif (phi1 <= 0.93)
		tmp = Float64(R * acos(Float64(Float64(Float64(cos(phi1) * cos(phi2)) * Float64(Float64(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda1) * sin(lambda2)))) + Float64(phi1 * sin(phi2)))));
	else
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + t_0)));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.1e+42], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.93], N[(R * N[ArcCos[N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $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[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.1000000000000001e42

   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 69.2%

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

   if -1.1000000000000001e42 < phi1 < 0.930000000000000049

   1. Initial program 67.1%

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

   2. Taylor expanded in phi1 around 0 66.7%

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

      1. cos-diff 87.0%

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

      2. distribute-lft-out 87.0%

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

      3. associate-*l* 87.0%

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

   4. Applied egg-rr 87.0%

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

      1. +-commutative 87.0%

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

      2. associate-*r* 87.0%

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

      3. distribute-lft-out 87.0%

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

   6. Simplified 87.0%

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

   if 0.930000000000000049 < phi1

   1. Initial program 81.7%

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

   2. Taylor expanded in phi1 around 0 81.7%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.1 \cdot 10^{+42}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.93:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \end{array} \]

Alternative 6: 68.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -41:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 -41.0)
   (*
    R
    (acos
     (+
      (* phi1 (sin phi2))
      (*
       (cos phi1)
       (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2)))))))
   (*
    R
    (acos
     (fma
      (sin phi1)
      (sin phi2)
      (* (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -41.0) {
		tmp = R * acos(((phi1 * sin(phi2)) + (cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2))))));
	} else {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * (cos(phi2) * cos((lambda1 - lambda2))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= -41.0)
		tmp = Float64(R * acos(Float64(Float64(phi1 * sin(phi2)) + Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2)))))));
	else
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, -41.0], N[(R * N[ArcCos[N[(N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $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], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -41

   1. Initial program 59.5%

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

   2. Taylor expanded in phi1 around 0 33.7%

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

   3. Taylor expanded in phi2 around 0 26.3%

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

      1. sub-neg 26.3%

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

      2. remove-double-neg 26.3%

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

      3. mul-1-neg 26.3%

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

      4. distribute-neg-in 26.3%

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

      5. +-commutative 26.3%

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

      6. cos-neg 26.3%

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

      7. mul-1-neg 26.3%

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

      8. sub-neg 26.3%

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

   5. Simplified 26.3%

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

      1. cos-diff 42.2%

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

      2. *-commutative 42.2%

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

      3. fma-def 42.2%

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

   7. Applied egg-rr 42.2%

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

   if -41 < lambda2

   1. Initial program 76.2%

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

   3. Simplified 76.2%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -41:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \]

Alternative 7: 68.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -36000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 -36000.0)
   (*
    R
    (acos
     (+
      (* phi1 (sin phi2))
      (*
       (cos phi1)
       (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2)))))))
   (*
    R
    (acos
     (fma
      (sin phi1)
      (sin phi2)
      (* (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -36000.0) {
		tmp = R * acos(((phi1 * sin(phi2)) + (cos(phi1) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda1) * sin(lambda2))))));
	} else {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * (cos(phi2) * cos((lambda1 - lambda2))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= -36000.0)
		tmp = Float64(R * acos(Float64(Float64(phi1 * sin(phi2)) + Float64(cos(phi1) * Float64(Float64(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda1) * sin(lambda2)))))));
	else
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, -36000.0], N[(R * N[ArcCos[N[(N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -36000

   1. Initial program 58.9%

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

   2. Taylor expanded in phi1 around 0 32.7%

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

   3. Taylor expanded in phi2 around 0 25.2%

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

      1. sub-neg 25.2%

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

      2. remove-double-neg 25.2%

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

      3. mul-1-neg 25.2%

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

      4. distribute-neg-in 25.2%

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

      5. +-commutative 25.2%

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

      6. cos-neg 25.2%

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

      7. mul-1-neg 25.2%

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

      8. sub-neg 25.2%

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

   5. Simplified 25.2%

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

      1. cos-diff 41.3%

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

      2. *-commutative 41.3%

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

      3. *-commutative 41.3%

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

      4. +-commutative 41.3%

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

   7. Applied egg-rr 41.3%

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

   if -36000 < lambda2

   1. Initial program 76.3%

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

   3. Simplified 76.3%

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

4. Final simplification 67.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -36000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \]

Alternative 8: 73.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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. +-commutative 71.8%

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

   2. associate-*l* 71.8%

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

   3. fma-def 71.8%

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

3. Simplified 71.8%

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

4. Final simplification 71.8%

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

Alternative 9: 73.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 71.8%

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

3. Simplified 71.8%

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

4. Final simplification 71.8%

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

Alternative 10: 58.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 0.00185:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 0.00185)
   (*
    R
    (acos
     (+
      (* (sin phi1) (sin phi2))
      (* (* (cos phi1) (cos phi2)) (cos lambda1)))))
   (*
    R
    (acos
     (fma (sin phi1) (sin phi2) (* (cos phi2) (cos (- lambda2 lambda1))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 0.00185) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos(lambda1))));
	} else {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi2) * cos((lambda2 - lambda1)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 0.00185)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(lambda1)))));
	else
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi2) * cos(Float64(lambda2 - lambda1))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 0.00185], 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[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 0.0018500000000000001

   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. Taylor expanded in lambda2 around 0 60.9%

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

   if 0.0018500000000000001 < lambda2

   1. Initial program 61.5%

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

   3. Simplified 61.6%

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

   4. Taylor expanded in phi1 around 0 44.0%

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

      1. sub-neg 44.0%

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

      2. neg-mul-1 44.0%

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

      3. neg-mul-1 44.0%

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

      4. remove-double-neg 44.0%

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

      5. mul-1-neg 44.0%

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

      6. distribute-neg-in 44.0%

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

      7. +-commutative 44.0%

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

      8. cos-neg 44.0%

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

      9. mul-1-neg 44.0%

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

      10. unsub-neg 44.0%

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

   6. Simplified 44.0%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 0.00185:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \end{array} \]

Alternative 11: 63.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + t_0 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + t_0 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2))) (t_1 (* (sin phi1) (sin phi2))))
   (if (<= lambda1 -4.2e-6)
     (* R (acos (+ t_1 (* t_0 (cos lambda1)))))
     (* R (acos (+ t_1 (* t_0 (cos lambda2))))))))

C

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 <= -4.2e-6) {
		tmp = R * acos((t_1 + (t_0 * cos(lambda1))));
	} else {
		tmp = R * acos((t_1 + (t_0 * cos(lambda2))));
	}
	return tmp;
}

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
    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 <= (-4.2d-6)) then
        tmp = r * acos((t_1 + (t_0 * cos(lambda1))))
    else
        tmp = r * acos((t_1 + (t_0 * cos(lambda2))))
    end if
    code = tmp
end function

Java

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 <= -4.2e-6) {
		tmp = R * Math.acos((t_1 + (t_0 * Math.cos(lambda1))));
	} else {
		tmp = R * Math.acos((t_1 + (t_0 * Math.cos(lambda2))));
	}
	return tmp;
}

Python

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 <= -4.2e-6:
		tmp = R * math.acos((t_1 + (t_0 * math.cos(lambda1))))
	else:
		tmp = R * math.acos((t_1 + (t_0 * math.cos(lambda2))))
	return tmp

Julia

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 <= -4.2e-6)
		tmp = Float64(R * acos(Float64(t_1 + Float64(t_0 * cos(lambda1)))));
	else
		tmp = Float64(R * acos(Float64(t_1 + Float64(t_0 * cos(lambda2)))));
	end
	return tmp
end

MATLAB

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 <= -4.2e-6)
		tmp = R * acos((t_1 + (t_0 * cos(lambda1))));
	else
		tmp = R * acos((t_1 + (t_0 * cos(lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

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, -4.2e-6], N[(R * N[ArcCos[N[(t$95$1 + N[(t$95$0 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$1 + N[(t$95$0 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -4.1999999999999996e-6

   1. Initial program 53.4%

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

   2. Taylor expanded in lambda2 around 0 53.5%

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

   if -4.1999999999999996e-6 < lambda1

   1. Initial program 79.3%

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

   2. Taylor expanded in lambda1 around 0 66.0%

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

      1. cos-neg 66.0%

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

   4. Simplified 66.0%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 12: 73.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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

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

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. Taylor expanded in phi1 around 0 71.8%

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

3. Final simplification 71.8%

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

Alternative 13: 46.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -1.4:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi1 -1.4)
     (* R (acos (fma (sin phi1) (sin phi2) (* (cos phi1) t_0))))
     (* R (acos (+ (* phi1 (sin phi2)) (* (cos phi2) t_0)))))))

C

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

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi1 <= -1.4)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * t_0))));
	else
		tmp = Float64(R * acos(Float64(Float64(phi1 * sin(phi2)) + Float64(cos(phi2) * t_0))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -1.4], 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[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.3999999999999999

   1. Initial program 68.5%

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

   3. Simplified 68.6%

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

   4. Taylor expanded in phi2 around 0 41.4%

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

      1. sub-neg 41.4%

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

      2. remove-double-neg 41.4%

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

      3. mul-1-neg 41.4%

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

      4. distribute-neg-in 41.4%

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

      5. +-commutative 41.4%

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

      6. cos-neg 41.4%

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

      7. mul-1-neg 41.4%

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

      8. unsub-neg 41.4%

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

   6. Simplified 41.4%

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

   if -1.3999999999999999 < phi1

   1. Initial program 72.9%

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

   2. Taylor expanded in phi1 around 0 46.8%

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

   3. Taylor expanded in phi1 around 0 42.5%

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

      1. sub-neg 42.5%

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

      2. remove-double-neg 42.5%

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

      3. mul-1-neg 42.5%

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

      4. distribute-neg-in 42.5%

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

      5. +-commutative 42.5%

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

      6. cos-neg 42.5%

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

      7. mul-1-neg 42.5%

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

      8. sub-neg 42.5%

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

   5. Simplified 42.5%

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

4. Final simplification 42.2%

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

Alternative 14: 50.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -5 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot t_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi1 -5e-6)
     (* R (acos (fma (sin phi1) (sin phi2) (* (cos phi1) t_0))))
     (* R (acos (fma (sin phi1) (sin phi2) (* (cos phi2) t_0)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -5e-6) {
		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

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi1 <= -5e-6)
		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

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -5e-6], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 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 phi1 < -5.00000000000000041e-6

   1. Initial program 67.7%

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

   3. Simplified 67.8%

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

   4. Taylor expanded in phi2 around 0 41.0%

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

      1. sub-neg 41.0%

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

      2. remove-double-neg 41.0%

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

      3. mul-1-neg 41.0%

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

      4. distribute-neg-in 41.0%

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

      5. +-commutative 41.0%

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

      6. cos-neg 41.0%

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

      7. mul-1-neg 41.0%

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

      8. unsub-neg 41.0%

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

   6. Simplified 41.0%

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

   if -5.00000000000000041e-6 < phi1

   1. Initial program 73.1%

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

   3. Simplified 73.2%

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

   4. Taylor expanded in phi1 around 0 48.1%

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

      1. sub-neg 48.1%

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

      2. neg-mul-1 48.1%

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

      3. neg-mul-1 48.1%

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

      4. remove-double-neg 48.1%

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

      5. mul-1-neg 48.1%

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

      6. distribute-neg-in 48.1%

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

      7. +-commutative 48.1%

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

      8. cos-neg 48.1%

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

      9. mul-1-neg 48.1%

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

      10. unsub-neg 48.1%

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

   6. Simplified 48.1%

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

4. Final simplification 46.3%

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

Alternative 15: 46.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -5.2 \cdot 10^{+36}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 \cdot \cos \lambda_1 + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + t_0 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2))))
   (if (<= phi1 -5.2e+36)
     (* R (acos (+ (* t_0 (cos lambda1)) (* (sin phi1) phi2))))
     (* R (acos (+ (* phi1 (sin phi2)) (* t_0 (cos (- lambda1 lambda2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * cos(phi2);
	double tmp;
	if (phi1 <= -5.2e+36) {
		tmp = R * acos(((t_0 * cos(lambda1)) + (sin(phi1) * phi2)));
	} else {
		tmp = R * acos(((phi1 * sin(phi2)) + (t_0 * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(phi1) * cos(phi2)
    if (phi1 <= (-5.2d+36)) then
        tmp = r * acos(((t_0 * cos(lambda1)) + (sin(phi1) * phi2)))
    else
        tmp = r * acos(((phi1 * sin(phi2)) + (t_0 * cos((lambda1 - lambda2)))))
    end if
    code = tmp
end function

Java

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 (phi1 <= -5.2e+36) {
		tmp = R * Math.acos(((t_0 * Math.cos(lambda1)) + (Math.sin(phi1) * phi2)));
	} else {
		tmp = R * Math.acos(((phi1 * Math.sin(phi2)) + (t_0 * Math.cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.cos(phi2)
	tmp = 0
	if phi1 <= -5.2e+36:
		tmp = R * math.acos(((t_0 * math.cos(lambda1)) + (math.sin(phi1) * phi2)))
	else:
		tmp = R * math.acos(((phi1 * math.sin(phi2)) + (t_0 * math.cos((lambda1 - lambda2)))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if (phi1 <= -5.2e+36)
		tmp = Float64(R * acos(Float64(Float64(t_0 * cos(lambda1)) + Float64(sin(phi1) * phi2))));
	else
		tmp = Float64(R * acos(Float64(Float64(phi1 * sin(phi2)) + Float64(t_0 * cos(Float64(lambda1 - lambda2))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * cos(phi2);
	tmp = 0.0;
	if (phi1 <= -5.2e+36)
		tmp = R * acos(((t_0 * cos(lambda1)) + (sin(phi1) * phi2)));
	else
		tmp = R * acos(((phi1 * sin(phi2)) + (t_0 * cos((lambda1 - lambda2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -5.2e+36], N[(R * N[ArcCos[N[(N[(t$95$0 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(phi1 * N[Sin[phi2], $MachinePrecision]), $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 phi1 < -5.2000000000000003e36

   1. Initial program 68.3%

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

      1. log1p-expm1-u 68.2%

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

      2. log1p-udef 68.2%

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

   3. Applied egg-rr 68.2%

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

   4. Taylor expanded in lambda2 around 0 52.2%

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

   5. Taylor expanded in phi2 around 0 24.9%

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

      1. *-commutative 24.9%

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

   7. Simplified 24.9%

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

   if -5.2000000000000003e36 < phi1

   1. Initial program 72.9%

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

   2. Taylor expanded in phi1 around 0 47.2%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.2 \cdot 10^{+36}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1 + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]

Alternative 16: 47.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.175:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 0.175)
   (*
    R
    (acos
     (+
      (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))
      (* (sin phi1) phi2))))
   (*
    R
    (acos (+ (* phi1 (sin phi2)) (* (cos phi2) (cos (- lambda2 lambda1))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.175) {
		tmp = R * acos((((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))) + (sin(phi1) * phi2)));
	} else {
		tmp = R * acos(((phi1 * sin(phi2)) + (cos(phi2) * cos((lambda2 - lambda1)))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (phi2 <= 0.175d0) then
        tmp = r * acos((((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))) + (sin(phi1) * phi2)))
    else
        tmp = r * acos(((phi1 * sin(phi2)) + (cos(phi2) * cos((lambda2 - lambda1)))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.175) {
		tmp = R * Math.acos((((Math.cos(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2))) + (Math.sin(phi1) * phi2)));
	} else {
		tmp = R * Math.acos(((phi1 * Math.sin(phi2)) + (Math.cos(phi2) * Math.cos((lambda2 - lambda1)))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 0.175:
		tmp = R * math.acos((((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2))) + (math.sin(phi1) * phi2)))
	else:
		tmp = R * math.acos(((phi1 * math.sin(phi2)) + (math.cos(phi2) * math.cos((lambda2 - lambda1)))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 0.175)
		tmp = Float64(R * acos(Float64(Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))) + Float64(sin(phi1) * phi2))));
	else
		tmp = Float64(R * acos(Float64(Float64(phi1 * sin(phi2)) + Float64(cos(phi2) * cos(Float64(lambda2 - lambda1))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 0.175)
		tmp = R * acos((((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))) + (sin(phi1) * phi2)));
	else
		tmp = R * acos(((phi1 * sin(phi2)) + (cos(phi2) * cos((lambda2 - lambda1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 0.175], N[(R * N[ArcCos[N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.17499999999999999

   1. Initial program 69.3%

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

   2. Taylor expanded in phi2 around 0 47.9%

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

   if 0.17499999999999999 < phi2

   1. Initial program 77.9%

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

   2. Taylor expanded in phi1 around 0 33.2%

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

   3. Taylor expanded in phi1 around 0 33.2%

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

      1. sub-neg 33.2%

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

      2. remove-double-neg 33.2%

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

      3. mul-1-neg 33.2%

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

      4. distribute-neg-in 33.2%

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

      5. +-commutative 33.2%

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

      6. cos-neg 33.2%

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

      7. mul-1-neg 33.2%

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

      8. sub-neg 33.2%

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

   5. Simplified 33.2%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.175:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]

Alternative 17: 42.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.45:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1 + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.45)
   (*
    R
    (acos (+ (* (* (cos phi1) (cos phi2)) (cos lambda1)) (* (sin phi1) phi2))))
   (*
    R
    (acos (+ (* phi1 (sin phi2)) (* (cos phi2) (cos (- lambda2 lambda1))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.45) {
		tmp = R * acos((((cos(phi1) * cos(phi2)) * cos(lambda1)) + (sin(phi1) * phi2)));
	} else {
		tmp = R * acos(((phi1 * sin(phi2)) + (cos(phi2) * cos((lambda2 - lambda1)))));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (phi1 <= (-1.45d0)) then
        tmp = r * acos((((cos(phi1) * cos(phi2)) * cos(lambda1)) + (sin(phi1) * phi2)))
    else
        tmp = r * acos(((phi1 * sin(phi2)) + (cos(phi2) * cos((lambda2 - lambda1)))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.45) {
		tmp = R * Math.acos((((Math.cos(phi1) * Math.cos(phi2)) * Math.cos(lambda1)) + (Math.sin(phi1) * phi2)));
	} else {
		tmp = R * Math.acos(((phi1 * Math.sin(phi2)) + (Math.cos(phi2) * Math.cos((lambda2 - lambda1)))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1.45:
		tmp = R * math.acos((((math.cos(phi1) * math.cos(phi2)) * math.cos(lambda1)) + (math.sin(phi1) * phi2)))
	else:
		tmp = R * math.acos(((phi1 * math.sin(phi2)) + (math.cos(phi2) * math.cos((lambda2 - lambda1)))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.45)
		tmp = Float64(R * acos(Float64(Float64(Float64(cos(phi1) * cos(phi2)) * cos(lambda1)) + Float64(sin(phi1) * phi2))));
	else
		tmp = Float64(R * acos(Float64(Float64(phi1 * sin(phi2)) + Float64(cos(phi2) * cos(Float64(lambda2 - lambda1))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1.45)
		tmp = R * acos((((cos(phi1) * cos(phi2)) * cos(lambda1)) + (sin(phi1) * phi2)));
	else
		tmp = R * acos(((phi1 * sin(phi2)) + (cos(phi2) * cos((lambda2 - lambda1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.45], N[(R * N[ArcCos[N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.44999999999999996

   1. Initial program 68.5%

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

      1. log1p-expm1-u 68.4%

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

      2. log1p-udef 68.4%

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

   3. Applied egg-rr 68.4%

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

   4. Taylor expanded in lambda2 around 0 51.8%

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

   5. Taylor expanded in phi2 around 0 25.9%

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

      1. *-commutative 25.9%

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

   7. Simplified 25.9%

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

   if -1.44999999999999996 < phi1

   1. Initial program 72.9%

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

   2. Taylor expanded in phi1 around 0 46.8%

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

   3. Taylor expanded in phi1 around 0 42.5%

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

      1. sub-neg 42.5%

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

      2. remove-double-neg 42.5%

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

      3. mul-1-neg 42.5%

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

      4. distribute-neg-in 42.5%

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

      5. +-commutative 42.5%

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

      6. cos-neg 42.5%

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

      7. mul-1-neg 42.5%

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

      8. sub-neg 42.5%

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

   5. Simplified 42.5%

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

4. Final simplification 38.4%

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

Alternative 18: 34.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq 5 \cdot 10^{-132}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))) (t_1 (* phi1 (sin phi2))))
   (if (<= phi2 5e-132)
     (* R (acos (+ t_1 (* (cos phi1) t_0))))
     (* R (acos (+ t_1 (* (cos phi2) t_0)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = phi1 * sin(phi2);
	double tmp;
	if (phi2 <= 5e-132) {
		tmp = R * acos((t_1 + (cos(phi1) * t_0)));
	} else {
		tmp = R * acos((t_1 + (cos(phi2) * t_0)));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((lambda2 - lambda1))
    t_1 = phi1 * sin(phi2)
    if (phi2 <= 5d-132) then
        tmp = r * acos((t_1 + (cos(phi1) * t_0)))
    else
        tmp = r * acos((t_1 + (cos(phi2) * t_0)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda2 - lambda1));
	double t_1 = phi1 * Math.sin(phi2);
	double tmp;
	if (phi2 <= 5e-132) {
		tmp = R * Math.acos((t_1 + (Math.cos(phi1) * t_0)));
	} else {
		tmp = R * Math.acos((t_1 + (Math.cos(phi2) * t_0)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	t_1 = phi1 * math.sin(phi2)
	tmp = 0
	if phi2 <= 5e-132:
		tmp = R * math.acos((t_1 + (math.cos(phi1) * t_0)))
	else:
		tmp = R * math.acos((t_1 + (math.cos(phi2) * t_0)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = Float64(phi1 * sin(phi2))
	tmp = 0.0
	if (phi2 <= 5e-132)
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(phi1) * t_0))));
	else
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(phi2) * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	t_1 = phi1 * sin(phi2);
	tmp = 0.0;
	if (phi2 <= 5e-132)
		tmp = R * acos((t_1 + (cos(phi1) * t_0)));
	else
		tmp = R * acos((t_1 + (cos(phi2) * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, 5e-132], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.9999999999999999e-132

   1. Initial program 70.5%

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

   2. Taylor expanded in phi1 around 0 42.1%

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

   3. Taylor expanded in phi2 around 0 33.6%

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

      1. sub-neg 33.6%

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

      2. remove-double-neg 33.6%

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

      3. mul-1-neg 33.6%

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

      4. distribute-neg-in 33.6%

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

      5. +-commutative 33.6%

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

      6. cos-neg 33.6%

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

      7. mul-1-neg 33.6%

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

      8. sub-neg 33.6%

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

   5. Simplified 33.6%

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

   if 4.9999999999999999e-132 < phi2

   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 34.0%

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

   3. Taylor expanded in phi1 around 0 33.8%

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

      1. sub-neg 33.8%

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

      2. remove-double-neg 33.8%

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

      3. mul-1-neg 33.8%

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

      4. distribute-neg-in 33.8%

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

      5. +-commutative 33.8%

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

      6. cos-neg 33.8%

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

      7. mul-1-neg 33.8%

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

      8. sub-neg 33.8%

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

   5. Simplified 33.8%

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

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5 \cdot 10^{-132}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]

Alternative 19: 22.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -5.5 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* phi1 (sin phi2))))
   (if (<= lambda1 -5.5e-6)
     (* R (acos (+ t_0 (* (cos phi1) (cos lambda1)))))
     (* R (acos (+ t_0 (* (cos phi1) (cos lambda2))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = phi1 * sin(phi2);
	double tmp;
	if (lambda1 <= -5.5e-6) {
		tmp = R * acos((t_0 + (cos(phi1) * cos(lambda1))));
	} else {
		tmp = R * acos((t_0 + (cos(phi1) * cos(lambda2))));
	}
	return tmp;
}

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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = phi1 * sin(phi2)
    if (lambda1 <= (-5.5d-6)) then
        tmp = r * acos((t_0 + (cos(phi1) * cos(lambda1))))
    else
        tmp = r * acos((t_0 + (cos(phi1) * cos(lambda2))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = phi1 * Math.sin(phi2);
	double tmp;
	if (lambda1 <= -5.5e-6) {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * Math.cos(lambda1))));
	} else {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * Math.cos(lambda2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = phi1 * math.sin(phi2)
	tmp = 0
	if lambda1 <= -5.5e-6:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * math.cos(lambda1))))
	else:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * math.cos(lambda2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(phi1 * sin(phi2))
	tmp = 0.0
	if (lambda1 <= -5.5e-6)
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * cos(lambda1)))));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * cos(lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = phi1 * sin(phi2);
	tmp = 0.0;
	if (lambda1 <= -5.5e-6)
		tmp = R * acos((t_0 + (cos(phi1) * cos(lambda1))));
	else
		tmp = R * acos((t_0 + (cos(phi1) * cos(lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -5.5e-6], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -5.4999999999999999e-6

   1. Initial program 53.4%

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

   2. Taylor expanded in phi1 around 0 30.5%

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

   3. Taylor expanded in phi2 around 0 25.3%

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

      1. sub-neg 25.3%

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

      2. remove-double-neg 25.3%

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

      3. mul-1-neg 25.3%

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

      4. distribute-neg-in 25.3%

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

      5. +-commutative 25.3%

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

      6. cos-neg 25.3%

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

      7. mul-1-neg 25.3%

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

      8. sub-neg 25.3%

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

   5. Simplified 25.3%

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

   6. Taylor expanded in lambda2 around 0 25.4%

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

      1. cos-neg 25.4%

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

   8. Simplified 25.4%

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

   if -5.4999999999999999e-6 < lambda1

   1. Initial program 79.3%

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

   2. Taylor expanded in phi1 around 0 42.1%

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

   3. Taylor expanded in phi2 around 0 26.5%

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

      1. sub-neg 26.5%

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

      2. remove-double-neg 26.5%

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

      3. mul-1-neg 26.5%

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

      4. distribute-neg-in 26.5%

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

      5. +-commutative 26.5%

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

      6. cos-neg 26.5%

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

      7. mul-1-neg 26.5%

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

      8. sub-neg 26.5%

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

   5. Simplified 26.5%

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

   6. Taylor expanded in lambda1 around 0 21.3%

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

4. Final simplification 22.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -5.5 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 20: 27.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

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 = r * acos(((phi1 * sin(phi2)) + (cos(phi1) * cos((lambda2 - lambda1)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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. Taylor expanded in phi1 around 0 38.7%

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

3. Taylor expanded in phi2 around 0 26.1%

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

   1. sub-neg 26.1%

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

   2. remove-double-neg 26.1%

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

   3. mul-1-neg 26.1%

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

   4. distribute-neg-in 26.1%

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

   5. +-commutative 26.1%

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

   6. cos-neg 26.1%

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

   7. mul-1-neg 26.1%

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

   8. sub-neg 26.1%

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

5. Simplified 26.1%

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

6. Final simplification 26.1%

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

Alternative 21: 18.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

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 = r * acos(((phi1 * sin(phi2)) + (cos(phi1) * cos(lambda1))))
end function

Java

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(lambda1))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Taylor expanded in phi1 around 0 38.7%

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

3. Taylor expanded in phi2 around 0 26.1%

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

   1. sub-neg 26.1%

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

   2. remove-double-neg 26.1%

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

   3. mul-1-neg 26.1%

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

   4. distribute-neg-in 26.1%

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

   5. +-commutative 26.1%

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

   6. cos-neg 26.1%

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

   7. mul-1-neg 26.1%

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

   8. sub-neg 26.1%

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

5. Simplified 26.1%

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

6. Taylor expanded in lambda2 around 0 16.0%

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

   1. cos-neg 16.0%

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

8. Simplified 16.0%

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

9. Final simplification 16.0%

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

Alternative 22: 16.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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 = r * acos(((cos(phi1) * cos(lambda1)) + (phi1 * phi2)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Taylor expanded in phi1 around 0 38.7%

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

3. Taylor expanded in phi2 around 0 26.1%

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

   1. sub-neg 26.1%

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

   2. remove-double-neg 26.1%

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

   3. mul-1-neg 26.1%

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

   4. distribute-neg-in 26.1%

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

   5. +-commutative 26.1%

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

   6. cos-neg 26.1%

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

   7. mul-1-neg 26.1%

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

   8. sub-neg 26.1%

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

5. Simplified 26.1%

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

6. Taylor expanded in lambda2 around 0 16.0%

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

   1. cos-neg 16.0%

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

8. Simplified 16.0%

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

9. Taylor expanded in phi2 around 0 15.1%

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

10. Final simplification 15.1%

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

Alternative 23: 7.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

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 = r * acos((cos(phi1) + (phi1 * sin(phi2))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Taylor expanded in phi1 around 0 38.7%

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

3. Taylor expanded in phi2 around 0 26.1%

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

   1. sub-neg 26.1%

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

   2. remove-double-neg 26.1%

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

   3. mul-1-neg 26.1%

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

   4. distribute-neg-in 26.1%

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

   5. +-commutative 26.1%

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

   6. cos-neg 26.1%

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

   7. mul-1-neg 26.1%

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

   8. sub-neg 26.1%

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

5. Simplified 26.1%

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

6. Taylor expanded in lambda2 around 0 16.0%

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

   1. cos-neg 16.0%

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

8. Simplified 16.0%

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

9. Taylor expanded in lambda1 around 0 6.0%

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

10. Final simplification 6.0%

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

Reproduce

herbie shell --seed 2023319 
(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))