Spherical law of cosines

Percentage Accurate: 73.3% → 94.1%

Time: 34.3s

Alternatives: 21

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.3% 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, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.9%

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

3. Simplified 74.9%

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

   1. cos-diff 93.9%

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

   2. +-commutative 93.9%

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

5. Applied egg-rr 93.9%

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

6. Final simplification 93.9%

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

Alternative 2: 94.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.9%

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

   1. cos-diff 93.9%

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

   2. +-commutative 93.9%

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

3. Applied egg-rr 93.9%

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

4. Final simplification 93.9%

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

Alternative 3: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -4.5 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t_0\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 3.9 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))) (t_1 (* (sin phi1) (sin phi2))))
   (if (<= phi2 -4.5e-7)
     (* R (acos (fma (sin phi1) (sin phi2) (* (cos phi1) (* (cos phi2) t_0)))))
     (if (<= phi2 3.9e-5)
       (*
        R
        (acos
         (+
          t_1
          (*
           (cos phi1)
           (+
            (* (sin lambda1) (sin lambda2))
            (* (cos lambda1) (cos lambda2)))))))
       (* R (acos (+ t_1 (* (cos phi2) (* (cos phi1) t_0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi2 <= -4.5e-7) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * (cos(phi2) * t_0))));
	} else if (phi2 <= 3.9e-5) {
		tmp = R * acos((t_1 + (cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2))))));
	} else {
		tmp = R * acos((t_1 + (cos(phi2) * (cos(phi1) * t_0))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi2 <= -4.5e-7)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * Float64(cos(phi2) * t_0)))));
	elseif (phi2 <= 3.9e-5)
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2)))))));
	else
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(phi2) * Float64(cos(phi1) * t_0)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -4.5e-7], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 3.9e-5], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -4.4999999999999998e-7

   1. Initial program 81.8%

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

   3. Simplified 81.9%

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

   if -4.4999999999999998e-7 < phi2 < 3.8999999999999999e-5

   1. Initial program 68.9%

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

      1. cos-diff 88.7%

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

      2. +-commutative 88.7%

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

   3. Applied egg-rr 88.7%

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

   4. Taylor expanded in phi2 around 0 88.6%

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

   if 3.8999999999999999e-5 < phi2

   1. Initial program 80.6%

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

      1. log1p-expm1-u 80.6%

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

      2. log1p-udef 80.5%

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

      3. *-commutative 80.5%

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

   3. Applied egg-rr 80.5%

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

      1. log1p-def 80.6%

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

      2. log1p-expm1-u 80.6%

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

      3. associate-*r* 80.7%

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

   5. Applied egg-rr 80.7%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -4.5 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 3.9 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \end{array} \]

Alternative 4: 83.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -1.12 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_0\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\left(\sqrt[3]{t_0}\right)}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))) (t_1 (* (sin phi1) (sin phi2))))
   (if (<= phi1 -1.12e-7)
     (* R (acos (+ t_1 (* (cos phi2) (* (cos phi1) t_0)))))
     (if (<= phi1 1.65e-6)
       (*
        R
        (acos
         (+
          (*
           (cos phi2)
           (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos lambda2))))
          (* phi1 (sin phi2)))))
       (*
        R
        (acos (+ t_1 (* (* (cos phi1) (cos phi2)) (pow (cbrt t_0) 3.0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi1 <= -1.12e-7) {
		tmp = R * acos((t_1 + (cos(phi2) * (cos(phi1) * t_0))));
	} else if (phi1 <= 1.65e-6) {
		tmp = R * acos(((cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))) + (phi1 * sin(phi2))));
	} else {
		tmp = R * acos((t_1 + ((cos(phi1) * cos(phi2)) * pow(cbrt(t_0), 3.0))));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double t_1 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (phi1 <= -1.12e-7) {
		tmp = R * Math.acos((t_1 + (Math.cos(phi2) * (Math.cos(phi1) * t_0))));
	} else if (phi1 <= 1.65e-6) {
		tmp = R * Math.acos(((Math.cos(phi2) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(lambda2)))) + (phi1 * Math.sin(phi2))));
	} else {
		tmp = R * Math.acos((t_1 + ((Math.cos(phi1) * Math.cos(phi2)) * Math.pow(Math.cbrt(t_0), 3.0))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi1 <= -1.12e-7)
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(phi2) * Float64(cos(phi1) * t_0)))));
	elseif (phi1 <= 1.65e-6)
		tmp = Float64(R * acos(Float64(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2)))) + Float64(phi1 * sin(phi2)))));
	else
		tmp = Float64(R * acos(Float64(t_1 + Float64(Float64(cos(phi1) * cos(phi2)) * (cbrt(t_0) ^ 3.0)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.12e-7], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 1.65e-6], N[(R * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$1 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.12e-7

   1. Initial program 83.5%

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

      1. log1p-expm1-u 83.5%

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

      2. log1p-udef 83.4%

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

      3. *-commutative 83.4%

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

   3. Applied egg-rr 83.4%

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

      1. log1p-def 83.5%

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

      2. log1p-expm1-u 83.5%

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

      3. associate-*r* 83.5%

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

   5. Applied egg-rr 83.5%

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

   if -1.12e-7 < phi1 < 1.65000000000000008e-6

   1. Initial program 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in phi1 around 0 66.0%

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

      1. cos-diff 88.0%

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

      2. +-commutative 88.0%

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

   6. Applied egg-rr 88.0%

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

   if 1.65000000000000008e-6 < phi1

   1. Initial program 81.5%

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

      1. add-cube-cbrt 81.4%

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

      2. pow3 81.4%

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

   3. Applied egg-rr 81.4%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.12 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\left(\sqrt[3]{\cos \left(\lambda_1 - \lambda_2\right)}\right)}^{3}\right)\\ \end{array} \]

Alternative 5: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -1.65 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t_0\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 3.3 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))) (t_1 (* (sin phi1) (sin phi2))))
   (if (<= phi1 -1.65e-6)
     (* R (acos (+ t_1 (* (cos phi2) (* (cos phi1) t_0)))))
     (if (<= phi1 3.3e-8)
       (*
        R
        (acos
         (+
          (*
           (cos phi2)
           (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda1) (cos lambda2))))
          (* phi1 (sin phi2)))))
       (* R (acos (+ t_1 (* (* (cos phi1) (cos phi2)) t_0))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi1 <= -1.65e-6) {
		tmp = R * acos((t_1 + (cos(phi2) * (cos(phi1) * t_0))));
	} else if (phi1 <= 3.3e-8) {
		tmp = R * acos(((cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))) + (phi1 * sin(phi2))));
	} else {
		tmp = R * acos((t_1 + ((cos(phi1) * 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((lambda1 - lambda2))
    t_1 = sin(phi1) * sin(phi2)
    if (phi1 <= (-1.65d-6)) then
        tmp = r * acos((t_1 + (cos(phi2) * (cos(phi1) * t_0))))
    else if (phi1 <= 3.3d-8) then
        tmp = r * acos(((cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))) + (phi1 * sin(phi2))))
    else
        tmp = r * acos((t_1 + ((cos(phi1) * 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((lambda1 - lambda2));
	double t_1 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (phi1 <= -1.65e-6) {
		tmp = R * Math.acos((t_1 + (Math.cos(phi2) * (Math.cos(phi1) * t_0))));
	} else if (phi1 <= 3.3e-8) {
		tmp = R * Math.acos(((Math.cos(phi2) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda1) * Math.cos(lambda2)))) + (phi1 * Math.sin(phi2))));
	} else {
		tmp = R * Math.acos((t_1 + ((Math.cos(phi1) * Math.cos(phi2)) * t_0)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	t_1 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if phi1 <= -1.65e-6:
		tmp = R * math.acos((t_1 + (math.cos(phi2) * (math.cos(phi1) * t_0))))
	elif phi1 <= 3.3e-8:
		tmp = R * math.acos(((math.cos(phi2) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda1) * math.cos(lambda2)))) + (phi1 * math.sin(phi2))))
	else:
		tmp = R * math.acos((t_1 + ((math.cos(phi1) * math.cos(phi2)) * t_0)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi1 <= -1.65e-6)
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(phi2) * Float64(cos(phi1) * t_0)))));
	elseif (phi1 <= 3.3e-8)
		tmp = Float64(R * acos(Float64(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda1) * cos(lambda2)))) + Float64(phi1 * sin(phi2)))));
	else
		tmp = Float64(R * acos(Float64(t_1 + Float64(Float64(cos(phi1) * cos(phi2)) * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	t_1 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if (phi1 <= -1.65e-6)
		tmp = R * acos((t_1 + (cos(phi2) * (cos(phi1) * t_0))));
	elseif (phi1 <= 3.3e-8)
		tmp = R * acos(((cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda1) * cos(lambda2)))) + (phi1 * sin(phi2))));
	else
		tmp = R * acos((t_1 + ((cos(phi1) * cos(phi2)) * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.65e-6], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 3.3e-8], N[(R * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$1 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.65000000000000008e-6

   1. Initial program 83.5%

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

      1. log1p-expm1-u 83.5%

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

      2. log1p-udef 83.4%

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

      3. *-commutative 83.4%

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

   3. Applied egg-rr 83.4%

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

      1. log1p-def 83.5%

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

      2. log1p-expm1-u 83.5%

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

      3. associate-*r* 83.5%

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

   5. Applied egg-rr 83.5%

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

   if -1.65000000000000008e-6 < phi1 < 3.29999999999999977e-8

   1. Initial program 66.0%

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

   3. Simplified 66.0%

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

   4. Taylor expanded in phi1 around 0 66.0%

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

      1. cos-diff 88.0%

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

      2. +-commutative 88.0%

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

   6. Applied egg-rr 88.0%

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

   if 3.29999999999999977e-8 < phi1

   1. Initial program 81.5%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.65 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 3.3 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]

Alternative 6: 57.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -4.5 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= lambda1 -4.5e-5)
     (* R (acos (+ t_0 (* (cos phi2) (cos (- lambda2 lambda1))))))
     (* R (acos (+ t_0 (* (cos phi2) (* (cos phi1) (cos lambda2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (lambda1 <= -4.5e-5) {
		tmp = R * acos((t_0 + (cos(phi2) * cos((lambda2 - lambda1)))));
	} else {
		tmp = R * acos((t_0 + (cos(phi2) * (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 = sin(phi1) * sin(phi2)
    if (lambda1 <= (-4.5d-5)) then
        tmp = r * acos((t_0 + (cos(phi2) * cos((lambda2 - lambda1)))))
    else
        tmp = r * acos((t_0 + (cos(phi2) * (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 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (lambda1 <= -4.5e-5) {
		tmp = R * Math.acos((t_0 + (Math.cos(phi2) * Math.cos((lambda2 - lambda1)))));
	} else {
		tmp = R * Math.acos((t_0 + (Math.cos(phi2) * (Math.cos(phi1) * Math.cos(lambda2)))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if lambda1 <= -4.5e-5:
		tmp = R * math.acos((t_0 + (math.cos(phi2) * math.cos((lambda2 - lambda1)))))
	else:
		tmp = R * math.acos((t_0 + (math.cos(phi2) * (math.cos(phi1) * math.cos(lambda2)))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda1 <= -4.5e-5)
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi2) * cos(Float64(lambda2 - lambda1))))));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi2) * Float64(cos(phi1) * cos(lambda2))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if (lambda1 <= -4.5e-5)
		tmp = R * acos((t_0 + (cos(phi2) * cos((lambda2 - lambda1)))));
	else
		tmp = R * acos((t_0 + (cos(phi2) * (cos(phi1) * cos(lambda2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -4.5e-5], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -4.50000000000000028e-5

   1. Initial program 55.8%

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

   2. Taylor expanded in phi1 around 0 32.6%

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

      1. sub-neg 32.6%

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

      2. neg-mul-1 32.6%

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

      3. neg-mul-1 32.6%

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

      4. remove-double-neg 32.6%

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

      5. mul-1-neg 32.6%

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

      6. distribute-neg-in 32.6%

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

      7. +-commutative 32.6%

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

      8. cos-neg 32.6%

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

      9. mul-1-neg 32.6%

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

      10. unsub-neg 32.6%

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

   4. Simplified 32.6%

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

   if -4.50000000000000028e-5 < lambda1

   1. Initial program 80.9%

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

      1. cos-diff 92.2%

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

      2. +-commutative 92.2%

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

   3. Applied egg-rr 92.2%

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

   4. Taylor expanded in lambda1 around 0 64.9%

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

      1. cos-neg 64.9%

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

      2. associate-*r* 64.9%

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

      3. *-commutative 64.9%

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

      4. *-commutative 64.9%

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

      5. cos-neg 64.9%

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

   6. Simplified 64.9%

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

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4.5 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]

Alternative 7: 62.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -3.4 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= lambda1 -3.4e-8)
     (* R (acos (+ t_0 (* (cos phi2) (* (cos phi1) (cos lambda1))))))
     (* R (acos (+ t_0 (* (cos phi2) (* (cos phi1) (cos lambda2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (lambda1 <= -3.4e-8) {
		tmp = R * acos((t_0 + (cos(phi2) * (cos(phi1) * cos(lambda1)))));
	} else {
		tmp = R * acos((t_0 + (cos(phi2) * (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 = sin(phi1) * sin(phi2)
    if (lambda1 <= (-3.4d-8)) then
        tmp = r * acos((t_0 + (cos(phi2) * (cos(phi1) * cos(lambda1)))))
    else
        tmp = r * acos((t_0 + (cos(phi2) * (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 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (lambda1 <= -3.4e-8) {
		tmp = R * Math.acos((t_0 + (Math.cos(phi2) * (Math.cos(phi1) * Math.cos(lambda1)))));
	} else {
		tmp = R * Math.acos((t_0 + (Math.cos(phi2) * (Math.cos(phi1) * Math.cos(lambda2)))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if lambda1 <= -3.4e-8:
		tmp = R * math.acos((t_0 + (math.cos(phi2) * (math.cos(phi1) * math.cos(lambda1)))))
	else:
		tmp = R * math.acos((t_0 + (math.cos(phi2) * (math.cos(phi1) * math.cos(lambda2)))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda1 <= -3.4e-8)
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi2) * Float64(cos(phi1) * cos(lambda1))))));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi2) * Float64(cos(phi1) * cos(lambda2))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if (lambda1 <= -3.4e-8)
		tmp = R * acos((t_0 + (cos(phi2) * (cos(phi1) * cos(lambda1)))));
	else
		tmp = R * acos((t_0 + (cos(phi2) * (cos(phi1) * cos(lambda2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -3.4e-8], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -3.4e-8

   1. Initial program 56.5%

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

      1. add-sqr-sqrt 20.0%

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

      2. pow2 20.0%

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

      3. *-commutative 20.0%

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

   3. Applied egg-rr 20.0%

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

   4. Taylor expanded in lambda2 around 0 55.6%

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

      1. associate-*r* 55.6%

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

      2. *-commutative 55.6%

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

   6. Simplified 55.6%

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

   if -3.4e-8 < lambda1

   1. Initial program 80.8%

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

      1. cos-diff 92.2%

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

      2. +-commutative 92.2%

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

   3. Applied egg-rr 92.1%

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

   4. Taylor expanded in lambda1 around 0 64.8%

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

      1. cos-neg 64.8%

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

      2. associate-*r* 64.8%

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

      3. *-commutative 64.8%

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

      4. *-commutative 64.8%

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

      5. cos-neg 64.8%

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

   6. Simplified 64.8%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3.4 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]

Alternative 8: 73.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (acos
   (+
    (* (sin phi1) (sin phi2))
    (* (* (cos phi1) (cos phi2)) (cos (- lambda1 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(Float64(cos(phi1) * 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[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.9%

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

2. Final simplification 74.9%

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

Alternative 9: 73.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi2) * (cos(phi1) * 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[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 74.9%

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

   1. log1p-expm1-u 74.8%

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

   2. log1p-udef 74.8%

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

   3. *-commutative 74.8%

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

3. Applied egg-rr 74.8%

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

   1. log1p-def 74.8%

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

   2. log1p-expm1-u 74.9%

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

   3. associate-*r* 74.9%

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

5. Applied egg-rr 74.9%

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

6. Final simplification 74.9%

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

Alternative 10: 53.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -0.0017:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{elif}\;\phi_1 \leq 0.0032:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0 + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= phi1 -0.0017)
     (* R (acos (+ (* (sin phi1) phi2) (* (cos phi1) t_0))))
     (if (<= phi1 0.0032)
       (* R (acos (+ (* (cos phi2) t_0) (* phi1 (sin phi2)))))
       (* R (acos (+ (* (sin phi1) (sin phi2)) (* (cos phi1) (cos phi2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -0.0017) {
		tmp = R * acos(((sin(phi1) * phi2) + (cos(phi1) * t_0)));
	} else if (phi1 <= 0.0032) {
		tmp = R * acos(((cos(phi2) * t_0) + (phi1 * sin(phi2))));
	} else {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos(phi2))));
	}
	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((lambda1 - lambda2))
    if (phi1 <= (-0.0017d0)) then
        tmp = r * acos(((sin(phi1) * phi2) + (cos(phi1) * t_0)))
    else if (phi1 <= 0.0032d0) then
        tmp = r * acos(((cos(phi2) * t_0) + (phi1 * sin(phi2))))
    else
        tmp = r * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos(phi2))))
    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((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -0.0017) {
		tmp = R * Math.acos(((Math.sin(phi1) * phi2) + (Math.cos(phi1) * t_0)));
	} else if (phi1 <= 0.0032) {
		tmp = R * Math.acos(((Math.cos(phi2) * t_0) + (phi1 * Math.sin(phi2))));
	} else {
		tmp = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + (Math.cos(phi1) * Math.cos(phi2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	tmp = 0
	if phi1 <= -0.0017:
		tmp = R * math.acos(((math.sin(phi1) * phi2) + (math.cos(phi1) * t_0)))
	elif phi1 <= 0.0032:
		tmp = R * math.acos(((math.cos(phi2) * t_0) + (phi1 * math.sin(phi2))))
	else:
		tmp = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + (math.cos(phi1) * math.cos(phi2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -0.0017)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * phi2) + Float64(cos(phi1) * t_0))));
	elseif (phi1 <= 0.0032)
		tmp = Float64(R * acos(Float64(Float64(cos(phi2) * t_0) + Float64(phi1 * sin(phi2)))));
	else
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * cos(phi2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	tmp = 0.0;
	if (phi1 <= -0.0017)
		tmp = R * acos(((sin(phi1) * phi2) + (cos(phi1) * t_0)));
	elseif (phi1 <= 0.0032)
		tmp = R * acos(((cos(phi2) * t_0) + (phi1 * sin(phi2))));
	else
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos(phi2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -0.0017], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.0032], N[(R * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $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] + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.00169999999999999991

   1. Initial program 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in phi2 around 0 41.6%

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

   if -0.00169999999999999991 < phi1 < 0.00320000000000000015

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

   4. Taylor expanded in phi1 around 0 65.5%

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

   if 0.00320000000000000015 < phi1

   1. Initial program 82.2%

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

   3. Simplified 82.2%

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

   4. Taylor expanded in lambda2 around 0 66.9%

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

   5. Taylor expanded in lambda1 around 0 41.8%

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

4. Final simplification 52.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0017:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.0032:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \]

Alternative 11: 55.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -8.2 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.011:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= phi1 -8.2e-5)
     (* R (acos (+ t_0 (* (cos phi1) (cos (- lambda2 lambda1))))))
     (if (<= phi1 0.011)
       (*
        R
        (acos
         (+ (* (cos phi2) (cos (- lambda1 lambda2))) (* phi1 (sin phi2)))))
       (* R (acos (+ t_0 (* (cos phi1) (cos phi2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi1 <= -8.2e-5) {
		tmp = R * acos((t_0 + (cos(phi1) * cos((lambda2 - lambda1)))));
	} else if (phi1 <= 0.011) {
		tmp = R * acos(((cos(phi2) * cos((lambda1 - lambda2))) + (phi1 * sin(phi2))));
	} else {
		tmp = R * acos((t_0 + (cos(phi1) * cos(phi2))));
	}
	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 = sin(phi1) * sin(phi2)
    if (phi1 <= (-8.2d-5)) then
        tmp = r * acos((t_0 + (cos(phi1) * cos((lambda2 - lambda1)))))
    else if (phi1 <= 0.011d0) then
        tmp = r * acos(((cos(phi2) * cos((lambda1 - lambda2))) + (phi1 * sin(phi2))))
    else
        tmp = r * acos((t_0 + (cos(phi1) * cos(phi2))))
    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.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (phi1 <= -8.2e-5) {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * Math.cos((lambda2 - lambda1)))));
	} else if (phi1 <= 0.011) {
		tmp = R * Math.acos(((Math.cos(phi2) * Math.cos((lambda1 - lambda2))) + (phi1 * Math.sin(phi2))));
	} else {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * Math.cos(phi2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if phi1 <= -8.2e-5:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * math.cos((lambda2 - lambda1)))))
	elif phi1 <= 0.011:
		tmp = R * math.acos(((math.cos(phi2) * math.cos((lambda1 - lambda2))) + (phi1 * math.sin(phi2))))
	else:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * math.cos(phi2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi1 <= -8.2e-5)
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * cos(Float64(lambda2 - lambda1))))));
	elseif (phi1 <= 0.011)
		tmp = Float64(R * acos(Float64(Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))) + Float64(phi1 * sin(phi2)))));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * cos(phi2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if (phi1 <= -8.2e-5)
		tmp = R * acos((t_0 + (cos(phi1) * cos((lambda2 - lambda1)))));
	elseif (phi1 <= 0.011)
		tmp = R * acos(((cos(phi2) * cos((lambda1 - lambda2))) + (phi1 * sin(phi2))));
	else
		tmp = R * acos((t_0 + (cos(phi1) * cos(phi2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -8.2e-5], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.011], N[(R * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -8.20000000000000009e-5

   1. Initial program 83.5%

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

   2. Taylor expanded in phi2 around 0 51.5%

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

      1. sub-neg 51.5%

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

      2. remove-double-neg 51.5%

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

      3. mul-1-neg 51.5%

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

      4. distribute-neg-in 51.5%

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

      5. +-commutative 51.5%

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

      6. cos-neg 51.5%

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

      7. mul-1-neg 51.5%

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

      8. unsub-neg 51.5%

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

   4. Simplified 51.5%

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

   if -8.20000000000000009e-5 < phi1 < 0.010999999999999999

   1. Initial program 65.9%

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

   3. Simplified 65.9%

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

   4. Taylor expanded in phi1 around 0 65.5%

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

   if 0.010999999999999999 < phi1

   1. Initial program 82.2%

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

   3. Simplified 82.2%

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

   4. Taylor expanded in lambda2 around 0 66.9%

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

   5. Taylor expanded in lambda1 around 0 41.8%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.2 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.011:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \]

Alternative 12: 49.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -7 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))) (t_1 (* (sin phi1) (sin phi2))))
   (if (<= phi1 -7e-5)
     (* 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 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi1 <= -7e-5) {
		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 = sin(phi1) * sin(phi2)
    if (phi1 <= (-7d-5)) 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 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (phi1 <= -7e-5) {
		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 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if phi1 <= -7e-5:
		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(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi1 <= -7e-5)
		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 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if (phi1 <= -7e-5)
		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[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -7e-5], 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 phi1 < -6.9999999999999994e-5

   1. Initial program 83.5%

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

   2. Taylor expanded in phi2 around 0 51.5%

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

      1. sub-neg 51.5%

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

      2. remove-double-neg 51.5%

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

      3. mul-1-neg 51.5%

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

      4. distribute-neg-in 51.5%

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

      5. +-commutative 51.5%

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

      6. cos-neg 51.5%

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

      7. mul-1-neg 51.5%

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

      8. unsub-neg 51.5%

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

   4. Simplified 51.5%

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

   if -6.9999999999999994e-5 < phi1

   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 48.1%

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

      1. sub-neg 48.1%

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

      2. neg-mul-1 48.1%

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

      3. neg-mul-1 48.1%

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

      4. remove-double-neg 48.1%

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

      5. mul-1-neg 48.1%

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

      6. distribute-neg-in 48.1%

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

      7. +-commutative 48.1%

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

      8. cos-neg 48.1%

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

      9. mul-1-neg 48.1%

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

      10. unsub-neg 48.1%

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

   4. Simplified 48.1%

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

4. Final simplification 49.0%

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

Alternative 13: 32.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -0.029:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\phi_1 \leq -4.8 \cdot 10^{-148}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_2 \cdot \cos \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* phi1 (sin phi2))))
   (if (<= phi1 -0.029)
     (* R (acos (* (cos phi1) (cos lambda1))))
     (if (<= phi1 -4.8e-148)
       (* R (acos (+ (cos (- lambda1 lambda2)) t_0)))
       (* R (acos (+ t_0 (* (cos phi2) (cos lambda1)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = phi1 * sin(phi2);
	double tmp;
	if (phi1 <= -0.029) {
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	} else if (phi1 <= -4.8e-148) {
		tmp = R * acos((cos((lambda1 - lambda2)) + t_0));
	} else {
		tmp = R * acos((t_0 + (cos(phi2) * cos(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) :: t_0
    real(8) :: tmp
    t_0 = phi1 * sin(phi2)
    if (phi1 <= (-0.029d0)) then
        tmp = r * acos((cos(phi1) * cos(lambda1)))
    else if (phi1 <= (-4.8d-148)) then
        tmp = r * acos((cos((lambda1 - lambda2)) + t_0))
    else
        tmp = r * acos((t_0 + (cos(phi2) * cos(lambda1))))
    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 (phi1 <= -0.029) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else if (phi1 <= -4.8e-148) {
		tmp = R * Math.acos((Math.cos((lambda1 - lambda2)) + t_0));
	} else {
		tmp = R * Math.acos((t_0 + (Math.cos(phi2) * Math.cos(lambda1))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = phi1 * math.sin(phi2)
	tmp = 0
	if phi1 <= -0.029:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda1)))
	elif phi1 <= -4.8e-148:
		tmp = R * math.acos((math.cos((lambda1 - lambda2)) + t_0))
	else:
		tmp = R * math.acos((t_0 + (math.cos(phi2) * math.cos(lambda1))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(phi1 * sin(phi2))
	tmp = 0.0
	if (phi1 <= -0.029)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda1))));
	elseif (phi1 <= -4.8e-148)
		tmp = Float64(R * acos(Float64(cos(Float64(lambda1 - lambda2)) + t_0)));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi2) * cos(lambda1)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = phi1 * sin(phi2);
	tmp = 0.0;
	if (phi1 <= -0.029)
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	elseif (phi1 <= -4.8e-148)
		tmp = R * acos((cos((lambda1 - lambda2)) + t_0));
	else
		tmp = R * acos((t_0 + (cos(phi2) * cos(lambda1))));
	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[phi1, -0.029], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, -4.8e-148], N[(R * N[ArcCos[N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.0290000000000000015

   1. Initial program 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in lambda2 around 0 59.4%

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

   5. Taylor expanded in phi2 around 0 40.2%

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

      1. *-commutative 40.2%

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

   7. Simplified 40.2%

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

   if -0.0290000000000000015 < phi1 < -4.8000000000000002e-148

   1. Initial program 65.6%

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

   3. Simplified 65.6%

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

   4. Taylor expanded in phi1 around 0 63.2%

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

   5. Taylor expanded in phi2 around 0 39.2%

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

   if -4.8000000000000002e-148 < phi1

   1. Initial program 73.0%

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

   3. Simplified 73.0%

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

   4. Taylor expanded in phi1 around 0 39.7%

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

   5. Taylor expanded in lambda2 around 0 26.9%

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

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.029:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\phi_1 \leq -4.8 \cdot 10^{-148}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \lambda_1\right)\\ \end{array} \]

Alternative 14: 44.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.061:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -0.061)
   (* R (acos (* (cos phi1) (cos lambda1))))
   (*
    R
    (acos (+ (* (cos phi2) (cos (- lambda1 lambda2))) (* phi1 (sin phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.061) {
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	} else {
		tmp = R * acos(((cos(phi2) * cos((lambda1 - lambda2))) + (phi1 * sin(phi2))));
	}
	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 <= (-0.061d0)) then
        tmp = r * acos((cos(phi1) * cos(lambda1)))
    else
        tmp = r * acos(((cos(phi2) * cos((lambda1 - lambda2))) + (phi1 * sin(phi2))))
    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 <= -0.061) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R * Math.acos(((Math.cos(phi2) * Math.cos((lambda1 - lambda2))) + (phi1 * Math.sin(phi2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -0.061:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R * math.acos(((math.cos(phi2) * math.cos((lambda1 - lambda2))) + (phi1 * math.sin(phi2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -0.061)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda1))));
	else
		tmp = Float64(R * acos(Float64(Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))) + Float64(phi1 * sin(phi2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -0.061)
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	else
		tmp = R * acos(((cos(phi2) * cos((lambda1 - lambda2))) + (phi1 * sin(phi2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -0.061], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.060999999999999999

   1. Initial program 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in lambda2 around 0 59.4%

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

   5. Taylor expanded in phi2 around 0 40.2%

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

      1. *-commutative 40.2%

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

   7. Simplified 40.2%

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

   if -0.060999999999999999 < phi1

   1. Initial program 71.9%

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

   3. Simplified 71.9%

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

   4. Taylor expanded in phi1 around 0 43.2%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.061:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]

Alternative 15: 44.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -0.00115:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0 + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= phi1 -0.00115)
     (* R (acos (+ (* (sin phi1) phi2) (* (cos phi1) t_0))))
     (* R (acos (+ (* (cos phi2) t_0) (* phi1 (sin phi2))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -0.00115) {
		tmp = R * acos(((sin(phi1) * phi2) + (cos(phi1) * t_0)));
	} else {
		tmp = R * acos(((cos(phi2) * t_0) + (phi1 * sin(phi2))));
	}
	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((lambda1 - lambda2))
    if (phi1 <= (-0.00115d0)) then
        tmp = r * acos(((sin(phi1) * phi2) + (cos(phi1) * t_0)))
    else
        tmp = r * acos(((cos(phi2) * t_0) + (phi1 * sin(phi2))))
    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((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -0.00115) {
		tmp = R * Math.acos(((Math.sin(phi1) * phi2) + (Math.cos(phi1) * t_0)));
	} else {
		tmp = R * Math.acos(((Math.cos(phi2) * t_0) + (phi1 * Math.sin(phi2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	tmp = 0
	if phi1 <= -0.00115:
		tmp = R * math.acos(((math.sin(phi1) * phi2) + (math.cos(phi1) * t_0)))
	else:
		tmp = R * math.acos(((math.cos(phi2) * t_0) + (phi1 * math.sin(phi2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -0.00115)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * phi2) + Float64(cos(phi1) * t_0))));
	else
		tmp = Float64(R * acos(Float64(Float64(cos(phi2) * t_0) + Float64(phi1 * sin(phi2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	tmp = 0.0;
	if (phi1 <= -0.00115)
		tmp = R * acos(((sin(phi1) * phi2) + (cos(phi1) * t_0)));
	else
		tmp = R * acos(((cos(phi2) * t_0) + (phi1 * sin(phi2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -0.00115], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.00115

   1. Initial program 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in phi2 around 0 41.6%

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

   if -0.00115 < phi1

   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%

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

   4. Taylor expanded in phi1 around 0 43.2%

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

4. Final simplification 42.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.00115:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]

Alternative 16: 33.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.029:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -0.029)
   (* R (acos (* (cos phi1) (cos lambda1))))
   (* R (acos (+ (* phi1 (sin phi2)) (* (cos phi2) (cos lambda2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.029) {
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	} else {
		tmp = R * acos(((phi1 * sin(phi2)) + (cos(phi2) * 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) :: tmp
    if (phi1 <= (-0.029d0)) then
        tmp = r * acos((cos(phi1) * cos(lambda1)))
    else
        tmp = r * acos(((phi1 * sin(phi2)) + (cos(phi2) * cos(lambda2))))
    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 <= -0.029) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R * Math.acos(((phi1 * Math.sin(phi2)) + (Math.cos(phi2) * Math.cos(lambda2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -0.029:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R * math.acos(((phi1 * math.sin(phi2)) + (math.cos(phi2) * math.cos(lambda2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -0.029)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda1))));
	else
		tmp = Float64(R * acos(Float64(Float64(phi1 * sin(phi2)) + Float64(cos(phi2) * cos(lambda2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -0.029)
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	else
		tmp = R * acos(((phi1 * sin(phi2)) + (cos(phi2) * cos(lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -0.029], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.0290000000000000015

   1. Initial program 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in lambda2 around 0 59.4%

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

   5. Taylor expanded in phi2 around 0 40.2%

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

      1. *-commutative 40.2%

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

   7. Simplified 40.2%

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

   if -0.0290000000000000015 < phi1

   1. Initial program 71.9%

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

   3. Simplified 71.9%

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

   4. Taylor expanded in phi1 around 0 43.2%

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

   5. Taylor expanded in lambda1 around 0 30.5%

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

      1. cos-neg 30.5%

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

   7. Simplified 30.5%

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

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.029:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]

Alternative 17: 29.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.029:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -0.029)
   (* R (acos (* (cos phi1) (cos lambda1))))
   (* R (acos (+ (cos (- lambda1 lambda2)) (* phi1 (sin phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.029) {
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	} else {
		tmp = R * acos((cos((lambda1 - lambda2)) + (phi1 * sin(phi2))));
	}
	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 <= (-0.029d0)) then
        tmp = r * acos((cos(phi1) * cos(lambda1)))
    else
        tmp = r * acos((cos((lambda1 - lambda2)) + (phi1 * sin(phi2))))
    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 <= -0.029) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R * Math.acos((Math.cos((lambda1 - lambda2)) + (phi1 * Math.sin(phi2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -0.029:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R * math.acos((math.cos((lambda1 - lambda2)) + (phi1 * math.sin(phi2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -0.029)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda1))));
	else
		tmp = Float64(R * acos(Float64(cos(Float64(lambda1 - lambda2)) + Float64(phi1 * sin(phi2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -0.029)
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	else
		tmp = R * acos((cos((lambda1 - lambda2)) + (phi1 * sin(phi2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -0.029], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] + N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.0290000000000000015

   1. Initial program 83.3%

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

   3. Simplified 83.3%

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

   4. Taylor expanded in lambda2 around 0 59.4%

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

   5. Taylor expanded in phi2 around 0 40.2%

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

      1. *-commutative 40.2%

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

   7. Simplified 40.2%

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

   if -0.0290000000000000015 < phi1

   1. Initial program 71.9%

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

   3. Simplified 71.9%

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

   4. Taylor expanded in phi1 around 0 43.2%

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

   5. Taylor expanded in phi2 around 0 24.8%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.029:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]

Alternative 18: 31.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.9%

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

3. Simplified 74.9%

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

4. Taylor expanded in lambda2 around 0 54.1%

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

5. Taylor expanded in phi2 around 0 31.9%

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

   1. *-commutative 31.9%

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

7. Simplified 31.9%

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

8. Final simplification 31.9%

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

Alternative 19: 22.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.5 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -8.5e-7) (* R (acos (cos phi1))) (* R (acos (cos lambda1)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -8.5e-7) {
		tmp = R * acos(cos(phi1));
	} else {
		tmp = R * acos(cos(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 <= (-8.5d-7)) then
        tmp = r * acos(cos(phi1))
    else
        tmp = r * acos(cos(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 <= -8.5e-7) {
		tmp = R * Math.acos(Math.cos(phi1));
	} else {
		tmp = R * Math.acos(Math.cos(lambda1));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -8.5e-7:
		tmp = R * math.acos(math.cos(phi1))
	else:
		tmp = R * math.acos(math.cos(lambda1))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -8.5e-7)
		tmp = Float64(R * acos(cos(phi1)));
	else
		tmp = Float64(R * acos(cos(lambda1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -8.5e-7)
		tmp = R * acos(cos(phi1));
	else
		tmp = R * acos(cos(lambda1));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -8.5e-7], N[(R * N[ArcCos[N[Cos[phi1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -8.50000000000000014e-7

   1. Initial program 83.5%

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

   3. Simplified 83.5%

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

   4. Taylor expanded in lambda2 around 0 60.0%

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

   5. Taylor expanded in phi2 around 0 41.1%

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

      1. *-commutative 41.1%

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

   7. Simplified 41.1%

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

   8. Taylor expanded in lambda1 around 0 34.1%

      \[\leadsto \cos^{-1} \color{blue}{\cos \phi_1} \cdot R \]

   if -8.50000000000000014e-7 < phi1

   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%

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

   4. Taylor expanded in lambda2 around 0 52.0%

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

   5. Taylor expanded in phi2 around 0 28.6%

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

      1. *-commutative 28.6%

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

   7. Simplified 28.6%

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

   8. Taylor expanded in phi1 around 0 18.2%

      \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.5 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \end{array} \]

Alternative 20: 17.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ R \cdot \cos^{-1} \cos \lambda_1 \end{array} \]

FPCore

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

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * acos(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(cos(lambda1))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.9%

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

3. Simplified 74.9%

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

4. Taylor expanded in lambda2 around 0 54.1%

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

5. Taylor expanded in phi2 around 0 31.9%

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

   1. *-commutative 31.9%

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

7. Simplified 31.9%

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

8. Taylor expanded in phi1 around 0 16.3%

   \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]

9. Final simplification 16.3%

   \[\leadsto R \cdot \cos^{-1} \cos \lambda_1 \]

Alternative 21: 5.3% accurate, 204.3× speedup

Math

\[\begin{array}{l} \\ \lambda_1 \cdot R \end{array} \]

FPCore

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

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 * 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 = lambda1 * r
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return lambda1 * R;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return lambda1 * R

Julia

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

MATLAB

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(lambda1 * R), $MachinePrecision]

Derivation

1. Initial program 74.9%

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

3. Simplified 74.9%

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

4. Taylor expanded in lambda2 around 0 54.1%

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

5. Taylor expanded in phi2 around 0 31.9%

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

   1. *-commutative 31.9%

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

7. Simplified 31.9%

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

8. Taylor expanded in phi1 around 0 16.3%

   \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]

9. Taylor expanded in lambda1 around 0 6.1%

   \[\leadsto \color{blue}{\lambda_1} \cdot R \]

10. Final simplification 6.1%

    \[\leadsto \lambda_1 \cdot R \]

Reproduce

herbie shell --seed 2023314 
(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))