Spherical law of cosines

Percentage Accurate: 73.8% → 94.2%

Time: 33.6s

Alternatives: 22

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.2% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 74.0%

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

3. Simplified 74.0%

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

   1. cos-diff 95.7%

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

   2. +-commutative 95.7%

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

5. Applied egg-rr 95.7%

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

   1. *-commutative 95.7%

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

   2. fma-def 95.7%

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

7. Applied egg-rr 95.7%

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

8. Final simplification 95.7%

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

Alternative 2: 94.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

3. Simplified 74.0%

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

   1. cos-diff 95.7%

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

   2. +-commutative 95.7%

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

5. Applied egg-rr 95.7%

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

6. Final simplification 95.7%

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

Alternative 3: 94.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * acos(((sin(phi1) * sin(phi2)) + (((cos(lambda1) * cos(lambda2)) + (sin(lambda2) * sin(lambda1))) * (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)) + (((cos(lambda1) * cos(lambda2)) + (sin(lambda2) * sin(lambda1))) * (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.cos(lambda1) * Math.cos(lambda2)) + (Math.sin(lambda2) * Math.sin(lambda1))) * (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.cos(lambda1) * math.cos(lambda2)) + (math.sin(lambda2) * math.sin(lambda1))) * (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(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda2) * sin(lambda1))) * Float64(cos(phi1) * cos(phi2))))))
end

MATLAB

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

Derivation

1. Initial program 74.0%

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

   1. cos-diff 95.7%

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

   2. +-commutative 95.7%

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

3. Applied egg-rr 95.7%

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

4. Final simplification 95.7%

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

Alternative 4: 83.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t_0, t_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 9.5 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \left(e^{\mathsf{log1p}\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)} + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))) (t_1 (* (sin phi1) (sin phi2))))
   (if (<= phi1 -2.6e-25)
     (* R (acos (fma (cos phi1) (* (cos phi2) t_0) t_1)))
     (if (<= phi1 9.5e-9)
       (*
        R
        (acos
         (+
          (*
           (cos phi2)
           (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda2) (sin lambda1))))
          (* phi1 (sin phi2)))))
       (*
        R
        (acos
         (+ t_1 (+ (exp (log1p (* (* (cos phi1) (cos phi2)) t_0))) -1.0))))))))

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 <= -2.6e-25) {
		tmp = R * acos(fma(cos(phi1), (cos(phi2) * t_0), t_1));
	} else if (phi1 <= 9.5e-9) {
		tmp = R * acos(((cos(phi2) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda2) * sin(lambda1)))) + (phi1 * sin(phi2))));
	} else {
		tmp = R * acos((t_1 + (exp(log1p(((cos(phi1) * cos(phi2)) * t_0))) + -1.0)));
	}
	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 <= -2.6e-25)
		tmp = Float64(R * acos(fma(cos(phi1), Float64(cos(phi2) * t_0), t_1)));
	elseif (phi1 <= 9.5e-9)
		tmp = Float64(R * acos(Float64(Float64(cos(phi2) * Float64(Float64(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda2) * sin(lambda1)))) + Float64(phi1 * sin(phi2)))));
	else
		tmp = Float64(R * acos(Float64(t_1 + Float64(exp(log1p(Float64(Float64(cos(phi1) * cos(phi2)) * t_0))) + -1.0))));
	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, -2.6e-25], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 9.5e-9], N[(R * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Exp[N[Log[1 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.6e-25

   1. Initial program 77.4%

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

      1. +-commutative 77.4%

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

      2. associate-*l* 77.4%

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

      3. fma-def 77.4%

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

   3. Simplified 77.4%

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

   if -2.6e-25 < phi1 < 9.5000000000000007e-9

   1. Initial program 67.2%

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

      1. +-commutative 67.2%

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

      2. associate-*l* 67.2%

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

      3. fma-def 67.2%

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

   3. Simplified 67.2%

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

   4. Taylor expanded in phi1 around 0 67.2%

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

      1. cos-diff 91.4%

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

      2. +-commutative 91.4%

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

   6. Applied egg-rr 91.4%

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

   if 9.5000000000000007e-9 < phi1

   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. expm1-log1p-u 80.7%

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

      2. expm1-udef 80.8%

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

      3. *-commutative 80.8%

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

   3. Applied egg-rr 80.8%

      \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(e^{\mathsf{log1p}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)} - 1\right)}\right) \cdot R \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 9.5 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(e^{\mathsf{log1p}\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} + -1\right)\right)\\ \end{array} \]

Alternative 5: 83.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t_0, t_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \sqrt[3]{{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))) (t_1 (* (sin phi1) (sin phi2))))
   (if (<= phi1 -2.6e-25)
     (* R (acos (fma (cos phi1) (* (cos phi2) t_0) t_1)))
     (if (<= phi1 4e-6)
       (*
        R
        (acos
         (+
          (*
           (cos phi2)
           (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda2) (sin lambda1))))
          (* phi1 (sin phi2)))))
       (*
        R
        (acos (+ t_1 (cbrt (pow (* (* (cos phi1) (cos phi2)) 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 <= -2.6e-25) {
		tmp = R * acos(fma(cos(phi1), (cos(phi2) * t_0), t_1));
	} else if (phi1 <= 4e-6) {
		tmp = R * acos(((cos(phi2) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda2) * sin(lambda1)))) + (phi1 * sin(phi2))));
	} else {
		tmp = R * acos((t_1 + cbrt(pow(((cos(phi1) * cos(phi2)) * 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 <= -2.6e-25)
		tmp = Float64(R * acos(fma(cos(phi1), Float64(cos(phi2) * t_0), t_1)));
	elseif (phi1 <= 4e-6)
		tmp = Float64(R * acos(Float64(Float64(cos(phi2) * Float64(Float64(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda2) * sin(lambda1)))) + Float64(phi1 * sin(phi2)))));
	else
		tmp = Float64(R * acos(Float64(t_1 + cbrt((Float64(Float64(cos(phi1) * cos(phi2)) * 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, -2.6e-25], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 4e-6], N[(R * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$1 + N[Power[N[Power[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.6e-25

   1. Initial program 77.4%

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

      1. +-commutative 77.4%

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

      2. associate-*l* 77.4%

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

      3. fma-def 77.4%

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

   3. Simplified 77.4%

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

   if -2.6e-25 < phi1 < 3.99999999999999982e-6

   1. Initial program 67.2%

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

      1. +-commutative 67.2%

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

      2. associate-*l* 67.2%

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

      3. fma-def 67.2%

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

   3. Simplified 67.2%

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

   4. Taylor expanded in phi1 around 0 67.2%

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

      1. cos-diff 91.4%

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

      2. +-commutative 91.4%

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

   6. Applied egg-rr 91.4%

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

   if 3.99999999999999982e-6 < phi1

   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. add-cbrt-cube 80.6%

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

      2. pow3 80.6%

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

      3. *-commutative 80.6%

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

   3. Applied egg-rr 80.6%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 4 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \sqrt[3]{{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}^{3}}\right)\\ \end{array} \]

Alternative 6: 83.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t_0, t_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 8.5 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))) (t_1 (* (sin phi1) (sin phi2))))
   (if (<= phi1 -2.6e-25)
     (* R (acos (fma (cos phi1) (* (cos phi2) t_0) t_1)))
     (if (<= phi1 8.5e-14)
       (*
        R
        (acos
         (+
          (*
           (cos phi2)
           (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda2) (sin lambda1))))
          (* 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 <= -2.6e-25) {
		tmp = R * acos(fma(cos(phi1), (cos(phi2) * t_0), t_1));
	} else if (phi1 <= 8.5e-14) {
		tmp = R * acos(((cos(phi2) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda2) * sin(lambda1)))) + (phi1 * sin(phi2))));
	} else {
		tmp = R * acos((t_1 + ((cos(phi1) * 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 <= -2.6e-25)
		tmp = Float64(R * acos(fma(cos(phi1), Float64(cos(phi2) * t_0), t_1)));
	elseif (phi1 <= 8.5e-14)
		tmp = Float64(R * acos(Float64(Float64(cos(phi2) * Float64(Float64(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda2) * sin(lambda1)))) + Float64(phi1 * sin(phi2)))));
	else
		tmp = Float64(R * acos(Float64(t_1 + Float64(Float64(cos(phi1) * cos(phi2)) * 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[phi1, -2.6e-25], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 8.5e-14], N[(R * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $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 < -2.6e-25

   1. Initial program 77.4%

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

      1. +-commutative 77.4%

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

      2. associate-*l* 77.4%

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

      3. fma-def 77.4%

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

   3. Simplified 77.4%

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

   if -2.6e-25 < phi1 < 8.50000000000000038e-14

   1. Initial program 67.5%

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

      1. +-commutative 67.5%

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

      2. associate-*l* 67.5%

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

      3. fma-def 67.5%

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

   3. Simplified 67.5%

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

   4. Taylor expanded in phi1 around 0 67.5%

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

      1. cos-diff 92.1%

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

      2. +-commutative 92.1%

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

   6. Applied egg-rr 92.1%

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

   if 8.50000000000000038e-14 < phi1

   1. Initial program 80.1%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 8.5 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]

Alternative 7: 68.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -3.7 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 -3.7e-9)
   (*
    R
    (acos
     (+
      (*
       (cos phi2)
       (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda2) (sin lambda1))))
      (* phi1 phi2))))
   (*
    R
    (acos
     (fma
      (cos phi1)
      (* (cos phi2) (cos (- lambda1 lambda2)))
      (* (sin phi1) (sin phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -3.7e-9) {
		tmp = R * acos(((cos(phi2) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda2) * sin(lambda1)))) + (phi1 * phi2)));
	} else {
		tmp = R * acos(fma(cos(phi1), (cos(phi2) * cos((lambda1 - lambda2))), (sin(phi1) * sin(phi2))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= -3.7e-9)
		tmp = Float64(R * acos(Float64(Float64(cos(phi2) * Float64(Float64(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda2) * sin(lambda1)))) + Float64(phi1 * phi2))));
	else
		tmp = Float64(R * acos(fma(cos(phi1), Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))), Float64(sin(phi1) * sin(phi2)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, -3.7e-9], N[(R * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -3.7e-9

   1. Initial program 54.4%

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

      1. +-commutative 54.4%

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

      2. associate-*l* 54.4%

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

      3. fma-def 54.4%

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

   3. Simplified 54.4%

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

   4. Taylor expanded in phi1 around 0 30.9%

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

   5. Taylor expanded in phi2 around 0 23.5%

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

      1. cos-diff 98.4%

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

      2. +-commutative 98.4%

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

   7. Applied egg-rr 43.1%

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

   if -3.7e-9 < lambda2

   1. Initial program 82.7%

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

      1. +-commutative 82.7%

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

      2. associate-*l* 82.7%

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

      3. fma-def 82.7%

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

   3. Simplified 82.7%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -3.7 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]

Alternative 8: 68.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -3.7 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 -3.7e-9)
   (*
    R
    (acos
     (+
      (*
       (cos phi2)
       (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda2) (sin lambda1))))
      (* phi1 phi2))))
   (*
    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) {
	double tmp;
	if (lambda2 <= -3.7e-9) {
		tmp = R * acos(((cos(phi2) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda2) * sin(lambda1)))) + (phi1 * phi2)));
	} else {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= (-3.7d-9)) then
        tmp = r * acos(((cos(phi2) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda2) * sin(lambda1)))) + (phi1 * phi2)))
    else
        tmp = r * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -3.7e-9) {
		tmp = R * Math.acos(((Math.cos(phi2) * ((Math.cos(lambda1) * Math.cos(lambda2)) + (Math.sin(lambda2) * Math.sin(lambda1)))) + (phi1 * phi2)));
	} else {
		tmp = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= -3.7e-9:
		tmp = R * math.acos(((math.cos(phi2) * ((math.cos(lambda1) * math.cos(lambda2)) + (math.sin(lambda2) * math.sin(lambda1)))) + (phi1 * phi2)))
	else:
		tmp = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + ((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2)))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= -3.7e-9)
		tmp = Float64(R * acos(Float64(Float64(cos(phi2) * Float64(Float64(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda2) * sin(lambda1)))) + Float64(phi1 * phi2))));
	else
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= -3.7e-9)
		tmp = R * acos(((cos(phi2) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda2) * sin(lambda1)))) + (phi1 * phi2)));
	else
		tmp = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, -3.7e-9], N[(R * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -3.7e-9

   1. Initial program 54.4%

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

      1. +-commutative 54.4%

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

      2. associate-*l* 54.4%

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

      3. fma-def 54.4%

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

   3. Simplified 54.4%

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

   4. Taylor expanded in phi1 around 0 30.9%

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

   5. Taylor expanded in phi2 around 0 23.5%

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

      1. cos-diff 98.4%

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

      2. +-commutative 98.4%

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

   7. Applied egg-rr 43.1%

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

   if -3.7e-9 < lambda2

   1. Initial program 82.7%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -3.7 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]

Alternative 9: 61.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -3.7 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \phi_2\right)\\ \mathbf{elif}\;\lambda_2 \leq 0.059:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 -3.7e-9)
   (*
    R
    (acos
     (+
      (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda2) (sin lambda1)))
      (* phi1 phi2))))
   (if (<= lambda2 0.059)
     (*
      R
      (acos
       (+
        (* (sin phi1) (sin phi2))
        (* (cos lambda1) (* (cos phi1) (cos phi2))))))
     (* R (acos (fma (sin phi1) (sin phi2) (* (cos phi1) (cos lambda2))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -3.7e-9) {
		tmp = R * acos((((cos(lambda1) * cos(lambda2)) + (sin(lambda2) * sin(lambda1))) + (phi1 * phi2)));
	} else if (lambda2 <= 0.059) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(lambda1) * (cos(phi1) * cos(phi2)))));
	} else {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * cos(lambda2))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= -3.7e-9)
		tmp = Float64(R * acos(Float64(Float64(Float64(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda2) * sin(lambda1))) + Float64(phi1 * phi2))));
	elseif (lambda2 <= 0.059)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(lambda1) * Float64(cos(phi1) * cos(phi2))))));
	else
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * cos(lambda2)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, -3.7e-9], N[(R * N[ArcCos[N[(N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 0.059], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -3.7e-9

   1. Initial program 54.4%

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

      1. +-commutative 54.4%

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

      2. associate-*l* 54.4%

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

      3. fma-def 54.4%

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

   3. Simplified 54.4%

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

   4. Taylor expanded in phi1 around 0 30.9%

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

   5. Taylor expanded in phi2 around 0 23.5%

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

   6. Taylor expanded in phi2 around 0 17.6%

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

      1. cos-neg 17.6%

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

      2. sub-neg 17.6%

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

      3. +-commutative 17.6%

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

      4. distribute-neg-in 17.6%

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

      5. remove-double-neg 17.6%

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

      6. sub-neg 17.6%

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

   8. Simplified 17.6%

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

      1. cos-diff 32.0%

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

      2. *-commutative 32.0%

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

   10. Applied egg-rr 32.0%

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

   if -3.7e-9 < lambda2 < 0.058999999999999997

   1. Initial program 91.6%

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

   2. Taylor expanded in lambda2 around 0 91.6%

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

   if 0.058999999999999997 < lambda2

   1. Initial program 65.7%

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

   3. Simplified 65.8%

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

   4. Taylor expanded in lambda1 around 0 66.1%

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

      1. cos-neg 66.1%

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

   6. Simplified 66.1%

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

   7. Taylor expanded in phi2 around 0 43.5%

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

      1. *-commutative 43.5%

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

   9. Simplified 43.5%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -3.7 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \phi_2\right)\\ \mathbf{elif}\;\lambda_2 \leq 0.059:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]

Alternative 10: 65.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -3.7 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \phi_2\right)\\ \mathbf{elif}\;\lambda_2 \leq 0.013:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \lambda_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \lambda_2 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2))) (t_1 (* (sin phi1) (sin phi2))))
   (if (<= lambda2 -3.7e-9)
     (*
      R
      (acos
       (+
        (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda2) (sin lambda1)))
        (* phi1 phi2))))
     (if (<= lambda2 0.013)
       (* R (acos (+ t_1 (* (cos lambda1) t_0))))
       (* R (acos (+ t_1 (* (cos lambda2) t_0))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * cos(phi2);
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if (lambda2 <= -3.7e-9) {
		tmp = R * acos((((cos(lambda1) * cos(lambda2)) + (sin(lambda2) * sin(lambda1))) + (phi1 * phi2)));
	} else if (lambda2 <= 0.013) {
		tmp = R * acos((t_1 + (cos(lambda1) * t_0)));
	} else {
		tmp = R * acos((t_1 + (cos(lambda2) * 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(phi1) * cos(phi2)
    t_1 = sin(phi1) * sin(phi2)
    if (lambda2 <= (-3.7d-9)) then
        tmp = r * acos((((cos(lambda1) * cos(lambda2)) + (sin(lambda2) * sin(lambda1))) + (phi1 * phi2)))
    else if (lambda2 <= 0.013d0) then
        tmp = r * acos((t_1 + (cos(lambda1) * t_0)))
    else
        tmp = r * acos((t_1 + (cos(lambda2) * 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(phi1) * Math.cos(phi2);
	double t_1 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (lambda2 <= -3.7e-9) {
		tmp = R * Math.acos((((Math.cos(lambda1) * Math.cos(lambda2)) + (Math.sin(lambda2) * Math.sin(lambda1))) + (phi1 * phi2)));
	} else if (lambda2 <= 0.013) {
		tmp = R * Math.acos((t_1 + (Math.cos(lambda1) * t_0)));
	} else {
		tmp = R * Math.acos((t_1 + (Math.cos(lambda2) * t_0)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.cos(phi2)
	t_1 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if lambda2 <= -3.7e-9:
		tmp = R * math.acos((((math.cos(lambda1) * math.cos(lambda2)) + (math.sin(lambda2) * math.sin(lambda1))) + (phi1 * phi2)))
	elif lambda2 <= 0.013:
		tmp = R * math.acos((t_1 + (math.cos(lambda1) * t_0)))
	else:
		tmp = R * math.acos((t_1 + (math.cos(lambda2) * t_0)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda2 <= -3.7e-9)
		tmp = Float64(R * acos(Float64(Float64(Float64(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda2) * sin(lambda1))) + Float64(phi1 * phi2))));
	elseif (lambda2 <= 0.013)
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(lambda1) * t_0))));
	else
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(lambda2) * t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * cos(phi2);
	t_1 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if (lambda2 <= -3.7e-9)
		tmp = R * acos((((cos(lambda1) * cos(lambda2)) + (sin(lambda2) * sin(lambda1))) + (phi1 * phi2)));
	elseif (lambda2 <= 0.013)
		tmp = R * acos((t_1 + (cos(lambda1) * t_0)));
	else
		tmp = R * acos((t_1 + (cos(lambda2) * t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -3.7e-9], N[(R * N[ArcCos[N[(N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 0.013], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[lambda1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[lambda2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -3.7e-9

   1. Initial program 54.4%

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

      1. +-commutative 54.4%

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

      2. associate-*l* 54.4%

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

      3. fma-def 54.4%

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

   3. Simplified 54.4%

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

   4. Taylor expanded in phi1 around 0 30.9%

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

   5. Taylor expanded in phi2 around 0 23.5%

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

   6. Taylor expanded in phi2 around 0 17.6%

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

      1. cos-neg 17.6%

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

      2. sub-neg 17.6%

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

      3. +-commutative 17.6%

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

      4. distribute-neg-in 17.6%

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

      5. remove-double-neg 17.6%

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

      6. sub-neg 17.6%

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

   8. Simplified 17.6%

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

      1. cos-diff 32.0%

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

      2. *-commutative 32.0%

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

   10. Applied egg-rr 32.0%

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

   if -3.7e-9 < lambda2 < 0.0129999999999999994

   1. Initial program 91.6%

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

   2. Taylor expanded in lambda2 around 0 91.6%

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

   if 0.0129999999999999994 < lambda2

   1. Initial program 65.7%

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

   2. Taylor expanded in lambda1 around 0 66.1%

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

      1. cos-neg 66.1%

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

   4. Simplified 66.1%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -3.7 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \phi_2\right)\\ \mathbf{elif}\;\lambda_2 \leq 0.013:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \lambda_2 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \end{array} \]

Alternative 11: 66.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -3.7 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 -3.7e-9)
   (*
    R
    (acos
     (+
      (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda2) (sin lambda1)))
      (* phi1 phi2))))
   (*
    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) {
	double tmp;
	if (lambda2 <= -3.7e-9) {
		tmp = R * acos((((cos(lambda1) * cos(lambda2)) + (sin(lambda2) * sin(lambda1))) + (phi1 * phi2)));
	} else {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= (-3.7d-9)) then
        tmp = r * acos((((cos(lambda1) * cos(lambda2)) + (sin(lambda2) * sin(lambda1))) + (phi1 * phi2)))
    else
        tmp = r * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -3.7e-9) {
		tmp = R * Math.acos((((Math.cos(lambda1) * Math.cos(lambda2)) + (Math.sin(lambda2) * Math.sin(lambda1))) + (phi1 * phi2)));
	} else {
		tmp = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2)))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= -3.7e-9:
		tmp = R * math.acos((((math.cos(lambda1) * math.cos(lambda2)) + (math.sin(lambda2) * math.sin(lambda1))) + (phi1 * phi2)))
	else:
		tmp = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + ((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2)))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= -3.7e-9)
		tmp = Float64(R * acos(Float64(Float64(Float64(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda2) * sin(lambda1))) + Float64(phi1 * phi2))));
	else
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= -3.7e-9)
		tmp = R * acos((((cos(lambda1) * cos(lambda2)) + (sin(lambda2) * sin(lambda1))) + (phi1 * phi2)));
	else
		tmp = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, -3.7e-9], N[(R * N[ArcCos[N[(N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(phi1 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -3.7e-9

   1. Initial program 54.4%

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

      1. +-commutative 54.4%

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

      2. associate-*l* 54.4%

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

      3. fma-def 54.4%

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

   3. Simplified 54.4%

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

   4. Taylor expanded in phi1 around 0 30.9%

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

   5. Taylor expanded in phi2 around 0 23.5%

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

   6. Taylor expanded in phi2 around 0 17.6%

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

      1. cos-neg 17.6%

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

      2. sub-neg 17.6%

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

      3. +-commutative 17.6%

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

      4. distribute-neg-in 17.6%

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

      5. remove-double-neg 17.6%

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

      6. sub-neg 17.6%

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

   8. Simplified 17.6%

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

      1. cos-diff 32.0%

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

      2. *-commutative 32.0%

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

   10. Applied egg-rr 32.0%

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

   if -3.7e-9 < lambda2

   1. Initial program 82.7%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -3.7 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_2 \cdot \sin \lambda_1\right) + \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]

Alternative 12: 56.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_1 \leq -0.14:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{elif}\;\phi_1 \leq 0.205:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \left(\cos \phi_2 \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi1 -0.14)
     (* R (acos (+ (* (sin phi1) (sin phi2)) (* (cos phi1) t_0))))
     (if (<= phi1 0.205)
       (*
        R
        (acos
         (+
          (* phi1 (sin phi2))
          (* (+ 1.0 (* -0.5 (pow phi1 2.0))) (* (cos phi2) t_0)))))
       (* R (acos (fma (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((lambda2 - lambda1));
	double tmp;
	if (phi1 <= -0.14) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * t_0)));
	} else if (phi1 <= 0.205) {
		tmp = R * acos(((phi1 * sin(phi2)) + ((1.0 + (-0.5 * pow(phi1, 2.0))) * (cos(phi2) * t_0))));
	} else {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * cos(phi2))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi1 <= -0.14)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * t_0))));
	elseif (phi1 <= 0.205)
		tmp = Float64(R * acos(Float64(Float64(phi1 * sin(phi2)) + Float64(Float64(1.0 + Float64(-0.5 * (phi1 ^ 2.0))) * Float64(cos(phi2) * t_0)))));
	else
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * cos(phi2)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -0.14], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.205], N[(R * N[ArcCos[N[(N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(-0.5 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.14000000000000001

   1. Initial program 76.0%

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

   2. Taylor expanded in phi2 around 0 46.8%

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

      1. sub-neg 46.8%

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

      2. remove-double-neg 46.8%

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

      3. mul-1-neg 46.8%

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

      4. distribute-neg-in 46.8%

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

      5. +-commutative 46.8%

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

      6. cos-neg 46.8%

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

      7. mul-1-neg 46.8%

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

      8. unsub-neg 46.8%

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

   4. Simplified 46.8%

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

   if -0.14000000000000001 < phi1 < 0.204999999999999988

   1. Initial program 68.4%

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

      1. +-commutative 68.4%

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

      2. associate-*l* 68.4%

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

      3. fma-def 68.4%

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

   3. Simplified 68.4%

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

   4. Taylor expanded in phi1 around 0 67.9%

      \[\leadsto \cos^{-1} \color{blue}{\left(-0.5 \cdot \left({\phi_1}^{2} \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]
   5. Step-by-step derivation

      1. +-commutative 67.9%

         \[\leadsto \cos^{-1} \left(-0.5 \cdot \left({\phi_1}^{2} \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)}\right) \cdot R \]

      2. associate-+r+ 67.9%

         \[\leadsto \cos^{-1} \color{blue}{\left(\left(-0.5 \cdot \left({\phi_1}^{2} \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + \phi_1 \cdot \sin \phi_2\right)} \cdot R \]

   6. Simplified 67.9%

      \[\leadsto \cos^{-1} \color{blue}{\left(\left(-0.5 \cdot {\phi_1}^{2} + 1\right) \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) + \phi_1 \cdot \sin \phi_2\right)} \cdot R \]

   if 0.204999999999999988 < phi1

   1. Initial program 81.4%

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

   3. Simplified 81.4%

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

   4. Taylor expanded in lambda1 around 0 63.0%

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

      1. cos-neg 63.0%

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

   6. Simplified 63.0%

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

   7. Taylor expanded in lambda2 around 0 45.1%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.14:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.205:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \end{array} \]

Alternative 13: 56.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -0.047:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{elif}\;\phi_1 \leq 0.145:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \left(\cos \phi_2 \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))) (t_1 (* (sin phi1) (sin phi2))))
   (if (<= phi1 -0.047)
     (* R (acos (+ t_1 (* (cos phi1) t_0))))
     (if (<= phi1 0.145)
       (*
        R
        (acos
         (+
          (* phi1 (sin phi2))
          (* (+ 1.0 (* -0.5 (pow phi1 2.0))) (* (cos phi2) t_0)))))
       (* R (acos (+ t_1 (* (cos phi1) (cos phi2)))))))))

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 <= -0.047) {
		tmp = R * acos((t_1 + (cos(phi1) * t_0)));
	} else if (phi1 <= 0.145) {
		tmp = R * acos(((phi1 * sin(phi2)) + ((1.0 + (-0.5 * pow(phi1, 2.0))) * (cos(phi2) * t_0))));
	} else {
		tmp = R * acos((t_1 + (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) :: t_1
    real(8) :: tmp
    t_0 = cos((lambda2 - lambda1))
    t_1 = sin(phi1) * sin(phi2)
    if (phi1 <= (-0.047d0)) then
        tmp = r * acos((t_1 + (cos(phi1) * t_0)))
    else if (phi1 <= 0.145d0) then
        tmp = r * acos(((phi1 * sin(phi2)) + ((1.0d0 + ((-0.5d0) * (phi1 ** 2.0d0))) * (cos(phi2) * t_0))))
    else
        tmp = r * acos((t_1 + (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((lambda2 - lambda1));
	double t_1 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (phi1 <= -0.047) {
		tmp = R * Math.acos((t_1 + (Math.cos(phi1) * t_0)));
	} else if (phi1 <= 0.145) {
		tmp = R * Math.acos(((phi1 * Math.sin(phi2)) + ((1.0 + (-0.5 * Math.pow(phi1, 2.0))) * (Math.cos(phi2) * t_0))));
	} else {
		tmp = R * Math.acos((t_1 + (Math.cos(phi1) * Math.cos(phi2))));
	}
	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 <= -0.047:
		tmp = R * math.acos((t_1 + (math.cos(phi1) * t_0)))
	elif phi1 <= 0.145:
		tmp = R * math.acos(((phi1 * math.sin(phi2)) + ((1.0 + (-0.5 * math.pow(phi1, 2.0))) * (math.cos(phi2) * t_0))))
	else:
		tmp = R * math.acos((t_1 + (math.cos(phi1) * math.cos(phi2))))
	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 <= -0.047)
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(phi1) * t_0))));
	elseif (phi1 <= 0.145)
		tmp = Float64(R * acos(Float64(Float64(phi1 * sin(phi2)) + Float64(Float64(1.0 + Float64(-0.5 * (phi1 ^ 2.0))) * Float64(cos(phi2) * t_0)))));
	else
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(phi1) * cos(phi2)))));
	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 <= -0.047)
		tmp = R * acos((t_1 + (cos(phi1) * t_0)));
	elseif (phi1 <= 0.145)
		tmp = R * acos(((phi1 * sin(phi2)) + ((1.0 + (-0.5 * (phi1 ^ 2.0))) * (cos(phi2) * t_0))));
	else
		tmp = R * acos((t_1 + (cos(phi1) * cos(phi2))));
	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, -0.047], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.145], N[(R * N[ArcCos[N[(N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(-0.5 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.047

   1. Initial program 76.0%

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

   2. Taylor expanded in phi2 around 0 46.8%

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

      1. sub-neg 46.8%

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

      2. remove-double-neg 46.8%

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

      3. mul-1-neg 46.8%

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

      4. distribute-neg-in 46.8%

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

      5. +-commutative 46.8%

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

      6. cos-neg 46.8%

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

      7. mul-1-neg 46.8%

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

      8. unsub-neg 46.8%

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

   4. Simplified 46.8%

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

   if -0.047 < phi1 < 0.14499999999999999

   1. Initial program 68.8%

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

      1. +-commutative 68.8%

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

      2. associate-*l* 68.8%

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

      3. fma-def 68.8%

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

   3. Simplified 68.8%

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

   4. Taylor expanded in phi1 around 0 68.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(-0.5 \cdot \left({\phi_1}^{2} \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)} \cdot R \]
   5. Step-by-step derivation

      1. +-commutative 68.3%

         \[\leadsto \cos^{-1} \left(-0.5 \cdot \left({\phi_1}^{2} \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)}\right) \cdot R \]

      2. associate-+r+ 68.3%

         \[\leadsto \cos^{-1} \color{blue}{\left(\left(-0.5 \cdot \left({\phi_1}^{2} \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + \phi_1 \cdot \sin \phi_2\right)} \cdot R \]

   6. Simplified 68.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(\left(-0.5 \cdot {\phi_1}^{2} + 1\right) \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) + \phi_1 \cdot \sin \phi_2\right)} \cdot R \]

   if 0.14499999999999999 < phi1

   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. add-sqr-sqrt 39.4%

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

      2. pow2 39.4%

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

      3. *-commutative 39.4%

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

   3. Applied egg-rr 39.4%

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

   4. Taylor expanded in lambda2 around 0 33.4%

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

   5. Taylor expanded in lambda1 around 0 44.7%

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

4. Final simplification 56.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.047:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.145:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \left(1 + -0.5 \cdot {\phi_1}^{2}\right) \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \]

Alternative 14: 54.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_1 \leq -3.1 \cdot 10^{+139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\phi_1 \leq -0.00089:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \cos \phi_1 \cdot t_1\right)\\ \mathbf{elif}\;\phi_1 \leq 0.06:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_1 + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (* R (acos (+ (* (sin phi1) (sin phi2)) (* (cos phi1) (cos phi2))))))
        (t_1 (cos (- lambda1 lambda2))))
   (if (<= phi1 -3.1e+139)
     t_0
     (if (<= phi1 -0.00089)
       (* R (acos (+ (* (sin phi1) phi2) (* (cos phi1) t_1))))
       (if (<= phi1 0.06)
         (* R (acos (+ (* (cos phi2) t_1) (* phi1 (sin phi2)))))
         t_0)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos(phi2))));
	double t_1 = cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -3.1e+139) {
		tmp = t_0;
	} else if (phi1 <= -0.00089) {
		tmp = R * acos(((sin(phi1) * phi2) + (cos(phi1) * t_1)));
	} else if (phi1 <= 0.06) {
		tmp = R * acos(((cos(phi2) * t_1) + (phi1 * sin(phi2))));
	} else {
		tmp = 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 = r * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos(phi2))))
    t_1 = cos((lambda1 - lambda2))
    if (phi1 <= (-3.1d+139)) then
        tmp = t_0
    else if (phi1 <= (-0.00089d0)) then
        tmp = r * acos(((sin(phi1) * phi2) + (cos(phi1) * t_1)))
    else if (phi1 <= 0.06d0) then
        tmp = r * acos(((cos(phi2) * t_1) + (phi1 * sin(phi2))))
    else
        tmp = 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 = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + (Math.cos(phi1) * Math.cos(phi2))));
	double t_1 = Math.cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -3.1e+139) {
		tmp = t_0;
	} else if (phi1 <= -0.00089) {
		tmp = R * Math.acos(((Math.sin(phi1) * phi2) + (Math.cos(phi1) * t_1)));
	} else if (phi1 <= 0.06) {
		tmp = R * Math.acos(((Math.cos(phi2) * t_1) + (phi1 * Math.sin(phi2))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + (math.cos(phi1) * math.cos(phi2))))
	t_1 = math.cos((lambda1 - lambda2))
	tmp = 0
	if phi1 <= -3.1e+139:
		tmp = t_0
	elif phi1 <= -0.00089:
		tmp = R * math.acos(((math.sin(phi1) * phi2) + (math.cos(phi1) * t_1)))
	elif phi1 <= 0.06:
		tmp = R * math.acos(((math.cos(phi2) * t_1) + (phi1 * math.sin(phi2))))
	else:
		tmp = t_0
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * cos(phi2)))))
	t_1 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -3.1e+139)
		tmp = t_0;
	elseif (phi1 <= -0.00089)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * phi2) + Float64(cos(phi1) * t_1))));
	elseif (phi1 <= 0.06)
		tmp = Float64(R * acos(Float64(Float64(cos(phi2) * t_1) + Float64(phi1 * sin(phi2)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos(phi2))));
	t_1 = cos((lambda1 - lambda2));
	tmp = 0.0;
	if (phi1 <= -3.1e+139)
		tmp = t_0;
	elseif (phi1 <= -0.00089)
		tmp = R * acos(((sin(phi1) * phi2) + (cos(phi1) * t_1)));
	elseif (phi1 <= 0.06)
		tmp = R * acos(((cos(phi2) * t_1) + (phi1 * sin(phi2))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -3.1e+139], t$95$0, If[LessEqual[phi1, -0.00089], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.06], N[(R * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision] + N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.1e139 or 0.059999999999999998 < phi1

   1. Initial program 77.8%

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

      1. add-sqr-sqrt 37.3%

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

      2. pow2 37.3%

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

      3. *-commutative 37.3%

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

   3. Applied egg-rr 37.3%

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

   4. Taylor expanded in lambda2 around 0 31.6%

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

   5. Taylor expanded in lambda1 around 0 43.2%

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

   if -3.1e139 < phi1 < -8.8999999999999995e-4

   1. Initial program 79.7%

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

      1. +-commutative 79.7%

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

      2. associate-*l* 79.7%

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

      3. fma-def 79.7%

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

   3. Simplified 79.7%

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

   4. Taylor expanded in phi2 around 0 47.2%

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

   if -8.8999999999999995e-4 < phi1 < 0.059999999999999998

   1. Initial program 68.8%

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

      1. +-commutative 68.8%

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

      2. associate-*l* 68.8%

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

      3. fma-def 68.8%

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

   3. Simplified 68.8%

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

   4. Taylor expanded in phi1 around 0 67.8%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.1 \cdot 10^{+139}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \mathbf{elif}\;\phi_1 \leq -0.00089:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.06:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \]

Alternative 15: 55.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -4.2 \cdot 10^{+27}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq 0.175:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_1 + \cos \phi_2 \cdot t_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))) (t_1 (* (sin phi1) (sin phi2))))
   (if (<= phi2 -4.2e+27)
     (* R (acos (+ t_1 (* (cos phi1) (cos phi2)))))
     (if (<= phi2 0.175)
       (* 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 (phi2 <= -4.2e+27) {
		tmp = R * acos((t_1 + (cos(phi1) * cos(phi2))));
	} else if (phi2 <= 0.175) {
		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 (phi2 <= (-4.2d+27)) then
        tmp = r * acos((t_1 + (cos(phi1) * cos(phi2))))
    else if (phi2 <= 0.175d0) 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 (phi2 <= -4.2e+27) {
		tmp = R * Math.acos((t_1 + (Math.cos(phi1) * Math.cos(phi2))));
	} else if (phi2 <= 0.175) {
		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 phi2 <= -4.2e+27:
		tmp = R * math.acos((t_1 + (math.cos(phi1) * math.cos(phi2))))
	elif phi2 <= 0.175:
		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 (phi2 <= -4.2e+27)
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(phi1) * cos(phi2)))));
	elseif (phi2 <= 0.175)
		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 (phi2 <= -4.2e+27)
		tmp = R * acos((t_1 + (cos(phi1) * cos(phi2))));
	elseif (phi2 <= 0.175)
		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[phi2, -4.2e+27], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 0.175], 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 3 regimes

2. if phi2 < -4.19999999999999989e27

   1. Initial program 79.5%

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

      1. add-sqr-sqrt 45.8%

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

      2. pow2 45.8%

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

      3. *-commutative 45.8%

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

   3. Applied egg-rr 45.8%

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

   4. Taylor expanded in lambda2 around 0 38.0%

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

   5. Taylor expanded in lambda1 around 0 37.7%

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

   if -4.19999999999999989e27 < phi2 < 0.17499999999999999

   1. Initial program 71.1%

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

   2. Taylor expanded in phi2 around 0 68.7%

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

      1. sub-neg 68.7%

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

      2. remove-double-neg 68.7%

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

      3. mul-1-neg 68.7%

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

      4. distribute-neg-in 68.7%

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

      5. +-commutative 68.7%

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

      6. cos-neg 68.7%

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

      7. mul-1-neg 68.7%

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

      8. unsub-neg 68.7%

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

   4. Simplified 68.7%

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

   if 0.17499999999999999 < phi2

   1. Initial program 75.7%

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

   2. Taylor expanded in phi1 around 0 43.3%

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

      1. sub-neg 43.3%

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

      2. neg-mul-1 43.3%

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

      3. neg-mul-1 43.3%

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

      4. remove-double-neg 43.3%

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

      5. mul-1-neg 43.3%

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

      6. distribute-neg-in 43.3%

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

      7. +-commutative 43.3%

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

      8. cos-neg 43.3%

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

      9. mul-1-neg 43.3%

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

      10. unsub-neg 43.3%

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

   4. Simplified 43.3%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -4.2 \cdot 10^{+27}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq 0.175:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]

Alternative 16: 56.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_1 \leq -0.0034:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.054:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t_0 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= phi1 -0.0034)
     (* R (acos (+ t_0 (* (cos phi1) (cos (- lambda2 lambda1))))))
     (if (<= phi1 0.054)
       (*
        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 <= -0.0034) {
		tmp = R * acos((t_0 + (cos(phi1) * cos((lambda2 - lambda1)))));
	} else if (phi1 <= 0.054) {
		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 <= (-0.0034d0)) then
        tmp = r * acos((t_0 + (cos(phi1) * cos((lambda2 - lambda1)))))
    else if (phi1 <= 0.054d0) 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 <= -0.0034) {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * Math.cos((lambda2 - lambda1)))));
	} else if (phi1 <= 0.054) {
		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 <= -0.0034:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * math.cos((lambda2 - lambda1)))))
	elif phi1 <= 0.054:
		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 <= -0.0034)
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * cos(Float64(lambda2 - lambda1))))));
	elseif (phi1 <= 0.054)
		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 <= -0.0034)
		tmp = R * acos((t_0 + (cos(phi1) * cos((lambda2 - lambda1)))));
	elseif (phi1 <= 0.054)
		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, -0.0034], 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, 0.054], 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 < -0.00339999999999999981

   1. Initial program 76.0%

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

   2. Taylor expanded in phi2 around 0 46.8%

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

      1. sub-neg 46.8%

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

      2. remove-double-neg 46.8%

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

      3. mul-1-neg 46.8%

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

      4. distribute-neg-in 46.8%

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

      5. +-commutative 46.8%

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

      6. cos-neg 46.8%

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

      7. mul-1-neg 46.8%

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

      8. unsub-neg 46.8%

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

   4. Simplified 46.8%

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

   if -0.00339999999999999981 < phi1 < 0.0539999999999999994

   1. Initial program 68.8%

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

      1. +-commutative 68.8%

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

      2. associate-*l* 68.8%

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

      3. fma-def 68.8%

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

   3. Simplified 68.8%

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

   4. Taylor expanded in phi1 around 0 67.8%

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

   if 0.0539999999999999994 < phi1

   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. add-sqr-sqrt 39.4%

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

      2. pow2 39.4%

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

      3. *-commutative 39.4%

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

   3. Applied egg-rr 39.4%

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

   4. Taylor expanded in lambda2 around 0 33.4%

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

   5. Taylor expanded in lambda1 around 0 44.7%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0034:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.054:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \]

Alternative 17: 39.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -205000000:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -205000000.0)
   (* R (acos (cos (- lambda2 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 <= -205000000.0) {
		tmp = R * acos(cos((lambda2 - 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 <= (-205000000.0d0)) then
        tmp = r * acos(cos((lambda2 - 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 <= -205000000.0) {
		tmp = R * Math.acos(Math.cos((lambda2 - 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 <= -205000000.0:
		tmp = R * math.acos(math.cos((lambda2 - 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 <= -205000000.0)
		tmp = Float64(R * acos(cos(Float64(lambda2 - lambda1))));
	else
		tmp = Float64(R * acos(Float64(Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))) + Float64(phi1 * sin(phi2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -205000000.0)
		tmp = R * acos(cos((lambda2 - 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, -205000000.0], N[(R * N[ArcCos[N[Cos[N[(lambda2 - 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 < -2.05e8

   1. Initial program 75.2%

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

      1. +-commutative 75.2%

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

      2. associate-*l* 75.2%

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

      3. fma-def 75.2%

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

   3. Simplified 75.2%

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

   4. Taylor expanded in phi1 around 0 4.1%

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

   5. Taylor expanded in phi2 around 0 4.1%

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

   6. Taylor expanded in phi2 around 0 4.1%

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

      1. cos-neg 4.1%

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

      2. sub-neg 4.1%

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

      3. +-commutative 4.1%

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

      4. distribute-neg-in 4.1%

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

      5. remove-double-neg 4.1%

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

      6. sub-neg 4.1%

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

   8. Simplified 4.1%

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

   9. Taylor expanded in phi1 around 0 15.9%

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

   if -2.05e8 < phi1

   1. Initial program 73.5%

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

      1. +-commutative 73.5%

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

      2. associate-*l* 73.5%

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

      3. fma-def 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in phi1 around 0 43.5%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -205000000:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]

Alternative 18: 44.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq 0.18:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \cos \phi_1 \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t_0 + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= phi2 0.18)
     (* 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 (phi2 <= 0.18) {
		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 (phi2 <= 0.18d0) 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 (phi2 <= 0.18) {
		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 phi2 <= 0.18:
		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 (phi2 <= 0.18)
		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 (phi2 <= 0.18)
		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[phi2, 0.18], 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 phi2 < 0.17999999999999999

   1. Initial program 73.5%

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

      1. +-commutative 73.5%

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

      2. associate-*l* 73.5%

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

      3. fma-def 73.5%

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

   3. Simplified 73.5%

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

   4. Taylor expanded in phi2 around 0 49.6%

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

   if 0.17999999999999999 < phi2

   1. Initial program 75.7%

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

      1. +-commutative 75.7%

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

      2. associate-*l* 75.7%

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

      3. fma-def 75.7%

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

   3. Simplified 75.7%

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

   4. Taylor expanded in phi1 around 0 35.3%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.18:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]

Alternative 19: 30.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 0.013:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 0.013)
   (* R (acos (+ (* phi1 (sin phi2)) (* (cos phi2) (cos lambda1)))))
   (* R (acos (cos (- lambda2 lambda1))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 0.013) {
		tmp = R * acos(((phi1 * sin(phi2)) + (cos(phi2) * cos(lambda1))));
	} else {
		tmp = R * acos(cos((lambda2 - lambda1)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 0.013d0) then
        tmp = r * acos(((phi1 * sin(phi2)) + (cos(phi2) * cos(lambda1))))
    else
        tmp = r * acos(cos((lambda2 - lambda1)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 0.013) {
		tmp = R * Math.acos(((phi1 * Math.sin(phi2)) + (Math.cos(phi2) * Math.cos(lambda1))));
	} else {
		tmp = R * Math.acos(Math.cos((lambda2 - lambda1)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 0.013:
		tmp = R * math.acos(((phi1 * math.sin(phi2)) + (math.cos(phi2) * math.cos(lambda1))))
	else:
		tmp = R * math.acos(math.cos((lambda2 - lambda1)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 0.013)
		tmp = Float64(R * acos(Float64(Float64(phi1 * sin(phi2)) + Float64(cos(phi2) * cos(lambda1)))));
	else
		tmp = Float64(R * acos(cos(Float64(lambda2 - lambda1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 0.013)
		tmp = R * acos(((phi1 * sin(phi2)) + (cos(phi2) * cos(lambda1))));
	else
		tmp = R * acos(cos((lambda2 - lambda1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 0.013], N[(R * N[ArcCos[N[(N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 0.0129999999999999994

   1. Initial program 76.5%

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

      1. +-commutative 76.5%

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

      2. associate-*l* 76.5%

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

      3. fma-def 76.5%

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

   3. Simplified 76.5%

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

   4. Taylor expanded in phi1 around 0 34.2%

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

   5. Taylor expanded in lambda2 around 0 26.1%

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

   if 0.0129999999999999994 < lambda2

   1. Initial program 65.7%

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

      1. +-commutative 65.7%

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

      2. associate-*l* 65.7%

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

      3. fma-def 65.7%

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

   3. Simplified 65.7%

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

   4. Taylor expanded in phi1 around 0 29.9%

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

   5. Taylor expanded in phi2 around 0 23.6%

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

   6. Taylor expanded in phi2 around 0 21.2%

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

      1. cos-neg 21.2%

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

      2. sub-neg 21.2%

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

      3. +-commutative 21.2%

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

      4. distribute-neg-in 21.2%

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

      5. remove-double-neg 21.2%

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

      6. sub-neg 21.2%

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

   8. Simplified 21.2%

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

   9. Taylor expanded in phi1 around 0 31.9%

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

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 0.013:\\ \;\;\;\;R \cdot \cos^{-1} \left(\phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \end{array} \]

Alternative 20: 32.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -7.5 \cdot 10^{-85} \lor \neg \left(\phi_1 \leq 7.2 \cdot 10^{-118}\right):\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi1 -7.5e-85) (not (<= phi1 7.2e-118)))
   (* R (acos (cos (- lambda2 lambda1))))
   (* R (acos (+ (* (cos phi2) (cos (- lambda1 lambda2))) (* phi1 phi2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((phi1 <= -7.5e-85) || !(phi1 <= 7.2e-118)) {
		tmp = R * acos(cos((lambda2 - lambda1)));
	} else {
		tmp = R * acos(((cos(phi2) * cos((lambda1 - lambda2))) + (phi1 * 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 <= (-7.5d-85)) .or. (.not. (phi1 <= 7.2d-118))) then
        tmp = r * acos(cos((lambda2 - lambda1)))
    else
        tmp = r * acos(((cos(phi2) * cos((lambda1 - lambda2))) + (phi1 * 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 <= -7.5e-85) || !(phi1 <= 7.2e-118)) {
		tmp = R * Math.acos(Math.cos((lambda2 - lambda1)));
	} else {
		tmp = R * Math.acos(((Math.cos(phi2) * Math.cos((lambda1 - lambda2))) + (phi1 * phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (phi1 <= -7.5e-85) or not (phi1 <= 7.2e-118):
		tmp = R * math.acos(math.cos((lambda2 - lambda1)))
	else:
		tmp = R * math.acos(((math.cos(phi2) * math.cos((lambda1 - lambda2))) + (phi1 * phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if ((phi1 <= -7.5e-85) || !(phi1 <= 7.2e-118))
		tmp = Float64(R * acos(cos(Float64(lambda2 - lambda1))));
	else
		tmp = Float64(R * acos(Float64(Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))) + Float64(phi1 * phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((phi1 <= -7.5e-85) || ~((phi1 <= 7.2e-118)))
		tmp = R * acos(cos((lambda2 - lambda1)));
	else
		tmp = R * acos(((cos(phi2) * cos((lambda1 - lambda2))) + (phi1 * phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi1, -7.5e-85], N[Not[LessEqual[phi1, 7.2e-118]], $MachinePrecision]], N[(R * N[ArcCos[N[Cos[N[(lambda2 - 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 * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -7.5000000000000003e-85 or 7.2000000000000004e-118 < phi1

   1. Initial program 77.6%

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

      1. +-commutative 77.6%

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

      2. associate-*l* 77.6%

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

      3. fma-def 77.6%

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

   3. Simplified 77.6%

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

   4. Taylor expanded in phi1 around 0 17.5%

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

   5. Taylor expanded in phi2 around 0 12.4%

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

   6. Taylor expanded in phi2 around 0 11.1%

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

      1. cos-neg 11.1%

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

      2. sub-neg 11.1%

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

      3. +-commutative 11.1%

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

      4. distribute-neg-in 11.1%

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

      5. remove-double-neg 11.1%

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

      6. sub-neg 11.1%

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

   8. Simplified 11.1%

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

   9. Taylor expanded in phi1 around 0 21.9%

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

   if -7.5000000000000003e-85 < phi1 < 7.2000000000000004e-118

   1. Initial program 66.3%

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

      1. +-commutative 66.3%

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

      2. associate-*l* 66.3%

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

      3. fma-def 66.3%

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

   3. Simplified 66.3%

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

   4. Taylor expanded in phi1 around 0 66.3%

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

   5. Taylor expanded in phi2 around 0 58.2%

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

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -7.5 \cdot 10^{-85} \lor \neg \left(\phi_1 \leq 7.2 \cdot 10^{-118}\right):\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_1 \cdot \phi_2\right)\\ \end{array} \]

Alternative 21: 26.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

   1. +-commutative 74.0%

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

   2. associate-*l* 74.0%

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

   3. fma-def 74.0%

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

3. Simplified 74.0%

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

4. Taylor expanded in phi1 around 0 33.2%

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

5. Taylor expanded in phi2 around 0 27.1%

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

6. Taylor expanded in phi2 around 0 20.0%

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

   1. cos-neg 20.0%

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

   2. sub-neg 20.0%

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

   3. +-commutative 20.0%

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

   4. distribute-neg-in 20.0%

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

   5. remove-double-neg 20.0%

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

   6. sub-neg 20.0%

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

8. Simplified 20.0%

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

9. Taylor expanded in phi1 around 0 28.0%

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

10. Final simplification 28.0%

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

Alternative 22: 9.5% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 74.0%

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

   1. +-commutative 74.0%

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

   2. associate-*l* 74.0%

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

   3. fma-def 74.0%

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

3. Simplified 74.0%

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

4. Taylor expanded in phi1 around 0 33.2%

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

5. Taylor expanded in phi2 around 0 27.1%

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

6. Taylor expanded in phi2 around 0 20.0%

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

   1. cos-neg 20.0%

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

   2. sub-neg 20.0%

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

   3. +-commutative 20.0%

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

   4. distribute-neg-in 20.0%

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

   5. remove-double-neg 20.0%

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

   6. sub-neg 20.0%

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

8. Simplified 20.0%

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

9. Taylor expanded in phi1 around inf 9.9%

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

10. Final simplification 9.9%

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

Reproduce

herbie shell --seed 2023322 
(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))