Spherical law of cosines

Percentage Accurate: 73.7% → 96.6%

Time: 39.0s

Alternatives: 25

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.7% 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: 96.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\lambda_2, R, \lambda_1 \cdot \left(-R\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<=
      (acos
       (+
        (* (sin phi1) (sin phi2))
        (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
      0.0)
   (fma lambda2 R (* lambda1 (- R)))
   (*
    R
    (acos
     (fma
      (sin phi1)
      (sin phi2)
      (*
       (cos phi1)
       (*
        (cos phi2)
        (+
         (* (cos lambda1) (cos lambda2))
         (* (sin lambda1) (sin lambda2))))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) <= 0.0) {
		tmp = fma(lambda2, R, (lambda1 * -R));
	} else {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * (cos(phi2) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda1) * sin(lambda2)))))));
	}
	return tmp;
}

Julia

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

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[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], 0.0], N[(lambda2 * R + N[(lambda1 * (-R)), $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))) < 0.0

   1. Initial program 9.2%

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

   3. Simplified 9.2%

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

   5. Taylor expanded in phi2 around 0 9.2%

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

   6. Taylor expanded in phi1 around 0 9.2%

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

   7. Taylor expanded in lambda1 around inf 41.0%

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

   8. Taylor expanded in lambda1 around 0 41.0%

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

      1. +-commutative 41.0%

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

      2. *-commutative 41.0%

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

      3. fma-define 41.1%

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

      4. mul-1-neg 41.1%

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

      5. distribute-rgt-neg-in 41.1%

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

   10. Simplified 41.1%

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

   if 0.0 < (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))))

   1. Initial program 81.5%

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

   3. Simplified 81.6%

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

      1. cos-diff 99.1%

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

      2. distribute-lft-in 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. distribute-lft-out 99.1%

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

      2. *-commutative 99.1%

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

      3. fma-undefine 99.1%

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

      4. *-commutative 99.1%

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

   8. Simplified 99.1%

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

   9. Taylor expanded in phi2 around inf 99.1%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\lambda_2, R, \lambda_1 \cdot \left(-R\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\cos^{-1} \left(t\_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\lambda_2, R, \lambda_1 \cdot \left(-R\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<=
        (acos (+ t_0 (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
        0.0)
     (fma lambda2 R (* lambda1 (- R)))
     (*
      R
      (acos
       (+
        t_0
        (*
         (cos phi1)
         (*
          (cos phi2)
          (+
           (* (cos lambda1) (cos lambda2))
           (* (sin lambda1) (sin lambda2)))))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (acos((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) <= 0.0) {
		tmp = fma(lambda2, R, (lambda1 * -R));
	} else {
		tmp = R * acos((t_0 + (cos(phi1) * (cos(phi2) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda1) * sin(lambda2)))))));
	}
	return tmp;
}

Julia

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

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[ArcCos[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(lambda2 * R + N[(lambda1 * (-R)), $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))) < 0.0

   1. Initial program 9.2%

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

   3. Simplified 9.2%

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

   5. Taylor expanded in phi2 around 0 9.2%

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

   6. Taylor expanded in phi1 around 0 9.2%

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

   7. Taylor expanded in lambda1 around inf 41.0%

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

   8. Taylor expanded in lambda1 around 0 41.0%

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

      1. +-commutative 41.0%

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

      2. *-commutative 41.0%

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

      3. fma-define 41.1%

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

      4. mul-1-neg 41.1%

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

      5. distribute-rgt-neg-in 41.1%

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

   10. Simplified 41.1%

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

   if 0.0 < (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))))

   1. Initial program 81.5%

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

   3. Simplified 81.6%

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

      1. cos-diff 99.1%

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

      2. distribute-lft-in 99.1%

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

   6. Applied egg-rr 99.1%

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

      1. distribute-lft-out 99.1%

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

      2. *-commutative 99.1%

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

      3. fma-undefine 99.1%

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

      4. *-commutative 99.1%

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

   8. Simplified 99.1%

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

   9. Taylor expanded in phi1 around 0 99.1%

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

4. Final simplification 94.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\lambda_2, R, \lambda_1 \cdot \left(-R\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 76.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\lambda_2, R, \lambda_1 \cdot \left(-R\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t\_0\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<=
        (acos (+ (* (sin phi1) (sin phi2)) (* (* (cos phi1) (cos phi2)) t_0)))
        0.0)
     (fma lambda2 R (* lambda1 (- R)))
     (*
      R
      (acos (fma (sin phi1) (sin phi2) (* (cos phi1) (* (cos phi2) t_0))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * t_0))) <= 0.0) {
		tmp = fma(lambda2, R, (lambda1 * -R));
	} else {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * (cos(phi2) * t_0))));
	}
	return tmp;
}

Julia

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

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(lambda2 * R + N[(lambda1 * (-R)), $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))) < 0.0

   1. Initial program 9.2%

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

   3. Simplified 9.2%

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

   5. Taylor expanded in phi2 around 0 9.2%

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

   6. Taylor expanded in phi1 around 0 9.2%

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

   7. Taylor expanded in lambda1 around inf 41.0%

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

   8. Taylor expanded in lambda1 around 0 41.0%

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

      1. +-commutative 41.0%

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

      2. *-commutative 41.0%

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

      3. fma-define 41.1%

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

      4. mul-1-neg 41.1%

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

      5. distribute-rgt-neg-in 41.1%

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

   10. Simplified 41.1%

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

   if 0.0 < (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))))

   1. Initial program 81.5%

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

   3. Simplified 81.6%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\lambda_2, R, \lambda_1 \cdot \left(-R\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\cos^{-1} \left(t\_0 + t\_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\lambda_2, R, \lambda_1 \cdot \left(-R\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(t\_1, \cos \left(\lambda_2 - \lambda_1\right), t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))) (t_1 (* (cos phi1) (cos phi2))))
   (if (<= (acos (+ t_0 (* t_1 (cos (- lambda1 lambda2))))) 0.0)
     (fma lambda2 R (* lambda1 (- R)))
     (* R (acos (fma t_1 (cos (- lambda2 lambda1)) t_0))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double t_1 = cos(phi1) * cos(phi2);
	double tmp;
	if (acos((t_0 + (t_1 * cos((lambda1 - lambda2))))) <= 0.0) {
		tmp = fma(lambda2, R, (lambda1 * -R));
	} else {
		tmp = R * acos(fma(t_1, cos((lambda2 - lambda1)), t_0));
	}
	return tmp;
}

Julia

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

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[ArcCos[N[(t$95$0 + N[(t$95$1 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 0.0], N[(lambda2 * R + N[(lambda1 * (-R)), $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$1 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))) < 0.0

   1. Initial program 9.2%

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

   3. Simplified 9.2%

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

   5. Taylor expanded in phi2 around 0 9.2%

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

   6. Taylor expanded in phi1 around 0 9.2%

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

   7. Taylor expanded in lambda1 around inf 41.0%

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

   8. Taylor expanded in lambda1 around 0 41.0%

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

      1. +-commutative 41.0%

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

      2. *-commutative 41.0%

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

      3. fma-define 41.1%

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

      4. mul-1-neg 41.1%

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

      5. distribute-rgt-neg-in 41.1%

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

   10. Simplified 41.1%

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

   if 0.0 < (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))))

   1. Initial program 81.5%

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

   3. Simplified 81.6%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\lambda_2, R, \lambda_1 \cdot \left(-R\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\lambda_2, R, \lambda_1 \cdot \left(-R\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (acos
          (+
           (* (sin phi1) (sin phi2))
           (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))))
   (if (<= t_0 0.0) (fma lambda2 R (* lambda1 (- R))) (* t_0 R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	double tmp;
	if (t_0 <= 0.0) {
		tmp = fma(lambda2, R, (lambda1 * -R));
	} else {
		tmp = t_0 * R;
	}
	return tmp;
}

Julia

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

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[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]}, If[LessEqual[t$95$0, 0.0], N[(lambda2 * R + N[(lambda1 * (-R)), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * R), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2))))) < 0.0

   1. Initial program 9.2%

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

   3. Simplified 9.2%

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

   5. Taylor expanded in phi2 around 0 9.2%

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

   6. Taylor expanded in phi1 around 0 9.2%

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

   7. Taylor expanded in lambda1 around inf 41.0%

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

   8. Taylor expanded in lambda1 around 0 41.0%

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

      1. +-commutative 41.0%

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

      2. *-commutative 41.0%

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

      3. fma-define 41.1%

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

      4. mul-1-neg 41.1%

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

      5. distribute-rgt-neg-in 41.1%

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

   10. Simplified 41.1%

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

   if 0.0 < (acos.f64 (+.f64 (*.f64 (sin.f64 phi1) (sin.f64 phi2)) (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (cos.f64 (-.f64 lambda1 lambda2)))))

   1. Initial program 81.5%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\lambda_2, R, \lambda_1 \cdot \left(-R\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0054:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.36 \cdot 10^{-61}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -0.0054)
   (*
    R
    (acos
     (fma
      (sin phi1)
      (sin phi2)
      (* (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2)))))))
   (if (<= phi1 1.36e-61)
     (*
      R
      (acos
       (fma
        (sin phi1)
        (sin phi2)
        (*
         (cos phi2)
         (+
          (* (cos lambda1) (cos lambda2))
          (* (sin lambda1) (sin lambda2)))))))
     (*
      R
      (log
       (exp
        (acos
         (fma
          (cos phi1)
          (* (cos phi2) (cos (- lambda2 lambda1)))
          (* (sin phi1) (sin phi2))))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.0054) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * (cos(phi2) * cos((lambda1 - lambda2))))));
	} else if (phi1 <= 1.36e-61) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi2) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda1) * sin(lambda2))))));
	} else {
		tmp = R * log(exp(acos(fma(cos(phi1), (cos(phi2) * cos((lambda2 - lambda1))), (sin(phi1) * sin(phi2))))));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -0.0054)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * Float64(cos(phi2) * cos(Float64(lambda1 - lambda2)))))));
	elseif (phi1 <= 1.36e-61)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi2) * Float64(Float64(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda1) * sin(lambda2)))))));
	else
		tmp = Float64(R * log(exp(acos(fma(cos(phi1), Float64(cos(phi2) * cos(Float64(lambda2 - lambda1))), Float64(sin(phi1) * sin(phi2)))))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -0.0054], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 1.36e-61], N[(R * N[ArcCos[N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[Log[N[Exp[N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.0054000000000000003

   1. Initial program 81.9%

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

   3. Simplified 82.0%

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

   if -0.0054000000000000003 < phi1 < 1.35999999999999995e-61

   1. Initial program 66.4%

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

   3. Simplified 66.4%

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

      1. cos-diff 83.9%

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

      2. distribute-lft-in 83.9%

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

   6. Applied egg-rr 83.9%

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

      1. distribute-lft-out 83.9%

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

      2. *-commutative 83.9%

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

      3. fma-undefine 83.9%

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

      4. *-commutative 83.9%

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

   8. Simplified 83.9%

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

   9. Taylor expanded in phi2 around inf 83.9%

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

   10. Taylor expanded in phi1 around 0 83.9%

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

   if 1.35999999999999995e-61 < phi1

   1. Initial program 81.2%

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

   3. Simplified 81.3%

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

      1. add-log-exp 81.3%

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

      2. cos-diff 95.6%

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

      3. *-commutative 95.6%

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

      4. *-commutative 95.6%

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

      5. cos-diff 81.3%

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

      6. fma-define 81.3%

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

      7. associate-*r* 81.3%

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

      8. fma-define 81.3%

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

      9. cos-diff 95.6%

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

      10. *-commutative 95.6%

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

      11. *-commutative 95.6%

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

      12. cos-diff 81.3%

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

   6. Applied egg-rr 81.3%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0054:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 1.36 \cdot 10^{-61}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \log \left(e^{\cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ t_1 := \cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq -0.066:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 8.5 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 23500000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 4.65 \cdot 10^{+27}:\\ \;\;\;\;R \cdot \cos^{-1} t\_1\\ \mathbf{elif}\;\phi_2 \leq 7.8 \cdot 10^{+94} \lor \neg \left(\phi_2 \leq 2.8 \cdot 10^{+118}\right):\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_1 + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          R
          (acos
           (+
            (* (sin phi1) (sin phi2))
            (* (* (cos phi1) (cos phi2)) (cos lambda1))))))
        (t_1 (* (cos phi2) (cos (- lambda2 lambda1)))))
   (if (<= phi2 -0.066)
     t_0
     (if (<= phi2 8.5e-10)
       (*
        R
        (acos
         (*
          (cos phi1)
          (+
           (* (cos lambda1) (cos lambda2))
           (* (sin lambda1) (sin lambda2))))))
       (if (<= phi2 23500000.0)
         t_0
         (if (<= phi2 4.65e+27)
           (* R (acos t_1))
           (if (or (<= phi2 7.8e+94) (not (<= phi2 2.8e+118)))
             t_0
             (* R (acos (+ t_1 (* phi1 (sin phi2))))))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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)) * cos(lambda1))));
	double t_1 = cos(phi2) * cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= -0.066) {
		tmp = t_0;
	} else if (phi2 <= 8.5e-10) {
		tmp = R * acos((cos(phi1) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda1) * sin(lambda2)))));
	} else if (phi2 <= 23500000.0) {
		tmp = t_0;
	} else if (phi2 <= 4.65e+27) {
		tmp = R * acos(t_1);
	} else if ((phi2 <= 7.8e+94) || !(phi2 <= 2.8e+118)) {
		tmp = t_0;
	} else {
		tmp = R * acos((t_1 + (phi1 * sin(phi2))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = r * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos(lambda1))))
    t_1 = cos(phi2) * cos((lambda2 - lambda1))
    if (phi2 <= (-0.066d0)) then
        tmp = t_0
    else if (phi2 <= 8.5d-10) then
        tmp = r * acos((cos(phi1) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda1) * sin(lambda2)))))
    else if (phi2 <= 23500000.0d0) then
        tmp = t_0
    else if (phi2 <= 4.65d+27) then
        tmp = r * acos(t_1)
    else if ((phi2 <= 7.8d+94) .or. (.not. (phi2 <= 2.8d+118))) then
        tmp = t_0
    else
        tmp = r * acos((t_1 + (phi1 * sin(phi2))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos(lambda1))));
	double t_1 = Math.cos(phi2) * Math.cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= -0.066) {
		tmp = t_0;
	} else if (phi2 <= 8.5e-10) {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.cos(lambda1) * Math.cos(lambda2)) + (Math.sin(lambda1) * Math.sin(lambda2)))));
	} else if (phi2 <= 23500000.0) {
		tmp = t_0;
	} else if (phi2 <= 4.65e+27) {
		tmp = R * Math.acos(t_1);
	} else if ((phi2 <= 7.8e+94) || !(phi2 <= 2.8e+118)) {
		tmp = t_0;
	} else {
		tmp = R * Math.acos((t_1 + (phi1 * Math.sin(phi2))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + ((math.cos(phi1) * math.cos(phi2)) * math.cos(lambda1))))
	t_1 = math.cos(phi2) * math.cos((lambda2 - lambda1))
	tmp = 0
	if phi2 <= -0.066:
		tmp = t_0
	elif phi2 <= 8.5e-10:
		tmp = R * math.acos((math.cos(phi1) * ((math.cos(lambda1) * math.cos(lambda2)) + (math.sin(lambda1) * math.sin(lambda2)))))
	elif phi2 <= 23500000.0:
		tmp = t_0
	elif phi2 <= 4.65e+27:
		tmp = R * math.acos(t_1)
	elif (phi2 <= 7.8e+94) or not (phi2 <= 2.8e+118):
		tmp = t_0
	else:
		tmp = R * math.acos((t_1 + (phi1 * math.sin(phi2))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(lambda1)))))
	t_1 = Float64(cos(phi2) * cos(Float64(lambda2 - lambda1)))
	tmp = 0.0
	if (phi2 <= -0.066)
		tmp = t_0;
	elseif (phi2 <= 8.5e-10)
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda1) * sin(lambda2))))));
	elseif (phi2 <= 23500000.0)
		tmp = t_0;
	elseif (phi2 <= 4.65e+27)
		tmp = Float64(R * acos(t_1));
	elseif ((phi2 <= 7.8e+94) || !(phi2 <= 2.8e+118))
		tmp = t_0;
	else
		tmp = Float64(R * acos(Float64(t_1 + Float64(phi1 * sin(phi2)))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = R * acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos(lambda1))));
	t_1 = cos(phi2) * cos((lambda2 - lambda1));
	tmp = 0.0;
	if (phi2 <= -0.066)
		tmp = t_0;
	elseif (phi2 <= 8.5e-10)
		tmp = R * acos((cos(phi1) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda1) * sin(lambda2)))));
	elseif (phi2 <= 23500000.0)
		tmp = t_0;
	elseif (phi2 <= 4.65e+27)
		tmp = R * acos(t_1);
	elseif ((phi2 <= 7.8e+94) || ~((phi2 <= 2.8e+118)))
		tmp = t_0;
	else
		tmp = R * acos((t_1 + (phi1 * sin(phi2))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.066], t$95$0, If[LessEqual[phi2, 8.5e-10], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 23500000.0], t$95$0, If[LessEqual[phi2, 4.65e+27], N[(R * N[ArcCos[t$95$1], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[phi2, 7.8e+94], N[Not[LessEqual[phi2, 2.8e+118]], $MachinePrecision]], t$95$0, N[(R * N[ArcCos[N[(t$95$1 + N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if phi2 < -0.066000000000000003 or 8.4999999999999996e-10 < phi2 < 2.35e7 or 4.65000000000000021e27 < phi2 < 7.79999999999999971e94 or 2.79999999999999986e118 < phi2

   1. Initial program 82.0%

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

   3. Taylor expanded in lambda2 around 0 57.0%

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

   if -0.066000000000000003 < phi2 < 8.4999999999999996e-10

   1. Initial program 68.7%

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

   3. Simplified 68.7%

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

   5. Taylor expanded in phi2 around 0 67.1%

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

      1. cos-diff 82.9%

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

      2. +-commutative 82.9%

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

      3. *-commutative 82.9%

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

      4. *-commutative 82.9%

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

   7. Applied egg-rr 82.9%

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

   if 2.35e7 < phi2 < 4.65000000000000021e27

   1. Initial program 98.8%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in phi1 around 0 82.0%

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

   if 7.79999999999999971e94 < phi2 < 2.79999999999999986e118

   1. Initial program 80.1%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in phi1 around 0 74.2%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.066:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 8.5 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 23500000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 4.65 \cdot 10^{+27}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 7.8 \cdot 10^{+94} \lor \neg \left(\phi_2 \leq 2.8 \cdot 10^{+118}\right):\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right) + \phi_1 \cdot \sin \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -4.1 \cdot 10^{+54}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.28 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= lambda2 -4.1e+54)
     (*
      R
      (acos
       (*
        (cos phi1)
        (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))))))
     (if (<= lambda2 1.28e-18)
       (* R (acos (+ t_0 (* (cos phi1) (* (cos phi2) (cos lambda1))))))
       (* R (acos (+ t_0 (* (cos phi1) (* (cos phi2) (cos lambda2))))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (lambda2 <= -4.1e+54) {
		tmp = R * acos((cos(phi1) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda1) * sin(lambda2)))));
	} else if (lambda2 <= 1.28e-18) {
		tmp = R * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda1)))));
	} else {
		tmp = R * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda2)))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(phi1) * sin(phi2)
    if (lambda2 <= (-4.1d+54)) then
        tmp = r * acos((cos(phi1) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda1) * sin(lambda2)))))
    else if (lambda2 <= 1.28d-18) then
        tmp = r * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda1)))))
    else
        tmp = r * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda2)))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (lambda2 <= -4.1e+54) {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.cos(lambda1) * Math.cos(lambda2)) + (Math.sin(lambda1) * Math.sin(lambda2)))));
	} else if (lambda2 <= 1.28e-18) {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * (Math.cos(phi2) * Math.cos(lambda1)))));
	} else {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * (Math.cos(phi2) * Math.cos(lambda2)))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if lambda2 <= -4.1e+54:
		tmp = R * math.acos((math.cos(phi1) * ((math.cos(lambda1) * math.cos(lambda2)) + (math.sin(lambda1) * math.sin(lambda2)))))
	elif lambda2 <= 1.28e-18:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * (math.cos(phi2) * math.cos(lambda1)))))
	else:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * (math.cos(phi2) * math.cos(lambda2)))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda2 <= -4.1e+54)
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda1) * sin(lambda2))))));
	elseif (lambda2 <= 1.28e-18)
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * Float64(cos(phi2) * cos(lambda1))))));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * Float64(cos(phi2) * cos(lambda2))))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if (lambda2 <= -4.1e+54)
		tmp = R * acos((cos(phi1) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda1) * sin(lambda2)))));
	elseif (lambda2 <= 1.28e-18)
		tmp = R * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda1)))));
	else
		tmp = R * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda2)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -4.1e+54], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 1.28e-18], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -4.09999999999999967e54

   1. Initial program 64.7%

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

   3. Simplified 64.7%

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

   5. Taylor expanded in phi2 around 0 41.7%

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

      1. cos-diff 57.6%

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

      2. +-commutative 57.6%

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

      3. *-commutative 57.6%

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

      4. *-commutative 57.6%

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

   7. Applied egg-rr 57.6%

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

   if -4.09999999999999967e54 < lambda2 < 1.27999999999999993e-18

   1. Initial program 81.7%

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

   3. Taylor expanded in lambda2 around 0 79.2%

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

      1. *-commutative 79.2%

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

      2. associate-*r* 79.2%

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

   5. Simplified 79.2%

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

   if 1.27999999999999993e-18 < lambda2

   1. Initial program 69.4%

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

   3. Taylor expanded in lambda1 around 0 69.3%

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

      1. cos-neg 69.3%

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

      2. *-commutative 69.3%

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

   5. Simplified 69.3%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -4.1 \cdot 10^{+54}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.28 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_2 \leq -4.1 \cdot 10^{+54}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.28 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= lambda2 -4.1e+54)
     (*
      R
      (acos
       (*
        (cos phi1)
        (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))))))
     (if (<= lambda2 1.28e-18)
       (* R (acos (+ t_0 (* (* (cos phi1) (cos phi2)) (cos lambda1)))))
       (* R (acos (+ t_0 (* (cos phi1) (* (cos phi2) (cos lambda2))))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (lambda2 <= -4.1e+54) {
		tmp = R * acos((cos(phi1) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda1) * sin(lambda2)))));
	} else if (lambda2 <= 1.28e-18) {
		tmp = R * acos((t_0 + ((cos(phi1) * cos(phi2)) * cos(lambda1))));
	} else {
		tmp = R * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda2)))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(phi1) * sin(phi2)
    if (lambda2 <= (-4.1d+54)) then
        tmp = r * acos((cos(phi1) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda1) * sin(lambda2)))))
    else if (lambda2 <= 1.28d-18) then
        tmp = r * acos((t_0 + ((cos(phi1) * cos(phi2)) * cos(lambda1))))
    else
        tmp = r * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda2)))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (lambda2 <= -4.1e+54) {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.cos(lambda1) * Math.cos(lambda2)) + (Math.sin(lambda1) * Math.sin(lambda2)))));
	} else if (lambda2 <= 1.28e-18) {
		tmp = R * Math.acos((t_0 + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos(lambda1))));
	} else {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * (Math.cos(phi2) * Math.cos(lambda2)))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if lambda2 <= -4.1e+54:
		tmp = R * math.acos((math.cos(phi1) * ((math.cos(lambda1) * math.cos(lambda2)) + (math.sin(lambda1) * math.sin(lambda2)))))
	elif lambda2 <= 1.28e-18:
		tmp = R * math.acos((t_0 + ((math.cos(phi1) * math.cos(phi2)) * math.cos(lambda1))))
	else:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * (math.cos(phi2) * math.cos(lambda2)))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda2 <= -4.1e+54)
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda1) * sin(lambda2))))));
	elseif (lambda2 <= 1.28e-18)
		tmp = Float64(R * acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(lambda1)))));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * Float64(cos(phi2) * cos(lambda2))))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if (lambda2 <= -4.1e+54)
		tmp = R * acos((cos(phi1) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda1) * sin(lambda2)))));
	elseif (lambda2 <= 1.28e-18)
		tmp = R * acos((t_0 + ((cos(phi1) * cos(phi2)) * cos(lambda1))));
	else
		tmp = R * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda2)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -4.1e+54], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 1.28e-18], N[(R * N[ArcCos[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -4.09999999999999967e54

   1. Initial program 64.7%

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

   3. Simplified 64.7%

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

   5. Taylor expanded in phi2 around 0 41.7%

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

      1. cos-diff 57.6%

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

      2. +-commutative 57.6%

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

      3. *-commutative 57.6%

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

      4. *-commutative 57.6%

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

   7. Applied egg-rr 57.6%

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

   if -4.09999999999999967e54 < lambda2 < 1.27999999999999993e-18

   1. Initial program 81.7%

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

   3. Taylor expanded in lambda2 around 0 79.2%

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

   if 1.27999999999999993e-18 < lambda2

   1. Initial program 69.4%

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

   3. Taylor expanded in lambda1 around 0 69.3%

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

      1. cos-neg 69.3%

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

      2. *-commutative 69.3%

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

   5. Simplified 69.3%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -4.1 \cdot 10^{+54}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.28 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 10^{-13}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1e-13)
   (*
    R
    (acos
     (*
      (cos phi1)
      (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2))))))
   (* R (acos (* (cos phi2) (cos (- lambda2 lambda1)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1e-13) {
		tmp = R * acos((cos(phi1) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda1) * sin(lambda2)))));
	} else {
		tmp = R * acos((cos(phi2) * cos((lambda2 - lambda1))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 1d-13) then
        tmp = r * acos((cos(phi1) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda1) * sin(lambda2)))))
    else
        tmp = r * acos((cos(phi2) * cos((lambda2 - lambda1))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1e-13) {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.cos(lambda1) * Math.cos(lambda2)) + (Math.sin(lambda1) * Math.sin(lambda2)))));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos((lambda2 - lambda1))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1e-13:
		tmp = R * math.acos((math.cos(phi1) * ((math.cos(lambda1) * math.cos(lambda2)) + (math.sin(lambda1) * math.sin(lambda2)))))
	else:
		tmp = R * math.acos((math.cos(phi2) * math.cos((lambda2 - lambda1))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1e-13)
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1e-13)
		tmp = R * acos((cos(phi1) * ((cos(lambda1) * cos(lambda2)) + (sin(lambda1) * sin(lambda2)))));
	else
		tmp = R * acos((cos(phi2) * cos((lambda2 - lambda1))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1e-13], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1e-13

   1. Initial program 74.1%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in phi2 around 0 50.9%

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

      1. cos-diff 61.6%

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

      2. +-commutative 61.6%

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

      3. *-commutative 61.6%

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

      4. *-commutative 61.6%

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

   7. Applied egg-rr 61.6%

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

   if 1e-13 < phi2

   1. Initial program 80.1%

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

   3. Simplified 80.1%

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

   5. Taylor expanded in phi1 around 0 48.5%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 10^{-13}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\cos \left(\lambda_1 - \lambda_2\right) \leq 0.99998:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (cos (- lambda1 lambda2)) 0.99998)
   (* R (acos (* (cos phi2) (cos (- lambda2 lambda1)))))
   (* R (acos (+ (* (sin phi1) (sin phi2)) (* (cos phi1) (cos phi2)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (cos((lambda1 - lambda2)) <= 0.99998) {
		tmp = R * acos((cos(phi2) * cos((lambda2 - lambda1))));
	} else {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos(phi2))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (cos((lambda1 - lambda2)) <= 0.99998d0) then
        tmp = r * acos((cos(phi2) * cos((lambda2 - lambda1))))
    else
        tmp = r * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos(phi2))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (Math.cos((lambda1 - lambda2)) <= 0.99998) {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos((lambda2 - lambda1))));
	} else {
		tmp = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + (Math.cos(phi1) * Math.cos(phi2))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if math.cos((lambda1 - lambda2)) <= 0.99998:
		tmp = R * math.acos((math.cos(phi2) * math.cos((lambda2 - lambda1))))
	else:
		tmp = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + (math.cos(phi1) * math.cos(phi2))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (cos(Float64(lambda1 - lambda2)) <= 0.99998)
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(Float64(lambda2 - lambda1)))));
	else
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * cos(phi2)))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (cos((lambda1 - lambda2)) <= 0.99998)
		tmp = R * acos((cos(phi2) * cos((lambda2 - lambda1))));
	else
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos(phi2))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision], 0.99998], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 lambda1 lambda2)) < 0.99997999999999998

   1. Initial program 77.7%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in phi1 around 0 46.3%

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

   if 0.99997999999999998 < (cos.f64 (-.f64 lambda1 lambda2))

   1. Initial program 70.0%

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

   3. Simplified 70.0%

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

   5. Taylor expanded in lambda2 around 0 70.0%

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

      1. *-commutative 70.0%

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

   7. Simplified 70.0%

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

   8. Taylor expanded in lambda1 around 0 69.1%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\lambda_1 - \lambda_2\right) \leq 0.99998:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.062:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 0.095:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \left(-0.5 \cdot {\phi_2}^{2} + 1\right) \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -0.062)
   (* R (acos (fma (sin phi1) (sin phi2) (* (cos phi1) (cos phi2)))))
   (if (<= phi2 0.095)
     (*
      R
      (acos
       (+
        (* (sin phi1) phi2)
        (*
         (+ (* -0.5 (pow phi2 2.0)) 1.0)
         (* (cos phi1) (cos (- lambda1 lambda2)))))))
     (* R (acos (* (cos phi2) (cos (- lambda2 lambda1))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -0.062) {
		tmp = R * acos(fma(sin(phi1), sin(phi2), (cos(phi1) * cos(phi2))));
	} else if (phi2 <= 0.095) {
		tmp = R * acos(((sin(phi1) * phi2) + (((-0.5 * pow(phi2, 2.0)) + 1.0) * (cos(phi1) * cos((lambda1 - lambda2))))));
	} else {
		tmp = R * acos((cos(phi2) * cos((lambda2 - lambda1))));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -0.062)
		tmp = Float64(R * acos(fma(sin(phi1), sin(phi2), Float64(cos(phi1) * cos(phi2)))));
	elseif (phi2 <= 0.095)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * phi2) + Float64(Float64(Float64(-0.5 * (phi2 ^ 2.0)) + 1.0) * Float64(cos(phi1) * cos(Float64(lambda1 - lambda2)))))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -0.062], 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], If[LessEqual[phi2, 0.095], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision] + N[(N[(N[(-0.5 * N[Power[phi2, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.062

   1. Initial program 85.6%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in lambda2 around 0 62.8%

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

      1. *-commutative 62.8%

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

   7. Simplified 62.8%

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

   8. Taylor expanded in lambda1 around 0 38.8%

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

   if -0.062 < phi2 < 0.095000000000000001

   1. Initial program 68.3%

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

   3. Simplified 68.3%

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

   5. Taylor expanded in phi2 around 0 67.3%

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

      1. distribute-rgt-in 67.3%

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

      2. *-commutative 67.3%

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

      3. associate-+l+ 67.3%

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

   7. Simplified 67.3%

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

   if 0.095000000000000001 < phi2

   1. Initial program 80.5%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in phi1 around 0 48.5%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.062:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_1, \sin \phi_2, \cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 0.095:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \left(-0.5 \cdot {\phi_2}^{2} + 1\right) \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.056:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq 0.095:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \left(-0.5 \cdot {\phi_2}^{2} + 1\right) \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -0.056)
   (* R (acos (+ (* (sin phi1) (sin phi2)) (* (cos phi1) (cos phi2)))))
   (if (<= phi2 0.095)
     (*
      R
      (acos
       (+
        (* (sin phi1) phi2)
        (*
         (+ (* -0.5 (pow phi2 2.0)) 1.0)
         (* (cos phi1) (cos (- lambda1 lambda2)))))))
     (* R (acos (* (cos phi2) (cos (- lambda2 lambda1))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -0.056) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos(phi2))));
	} else if (phi2 <= 0.095) {
		tmp = R * acos(((sin(phi1) * phi2) + (((-0.5 * pow(phi2, 2.0)) + 1.0) * (cos(phi1) * cos((lambda1 - lambda2))))));
	} else {
		tmp = R * acos((cos(phi2) * cos((lambda2 - lambda1))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= (-0.056d0)) then
        tmp = r * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos(phi2))))
    else if (phi2 <= 0.095d0) then
        tmp = r * acos(((sin(phi1) * phi2) + ((((-0.5d0) * (phi2 ** 2.0d0)) + 1.0d0) * (cos(phi1) * cos((lambda1 - lambda2))))))
    else
        tmp = r * acos((cos(phi2) * cos((lambda2 - lambda1))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -0.056) {
		tmp = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + (Math.cos(phi1) * Math.cos(phi2))));
	} else if (phi2 <= 0.095) {
		tmp = R * Math.acos(((Math.sin(phi1) * phi2) + (((-0.5 * Math.pow(phi2, 2.0)) + 1.0) * (Math.cos(phi1) * Math.cos((lambda1 - lambda2))))));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos((lambda2 - lambda1))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= -0.056:
		tmp = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + (math.cos(phi1) * math.cos(phi2))))
	elif phi2 <= 0.095:
		tmp = R * math.acos(((math.sin(phi1) * phi2) + (((-0.5 * math.pow(phi2, 2.0)) + 1.0) * (math.cos(phi1) * math.cos((lambda1 - lambda2))))))
	else:
		tmp = R * math.acos((math.cos(phi2) * math.cos((lambda2 - lambda1))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -0.056)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * cos(phi2)))));
	elseif (phi2 <= 0.095)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * phi2) + Float64(Float64(Float64(-0.5 * (phi2 ^ 2.0)) + 1.0) * Float64(cos(phi1) * cos(Float64(lambda1 - lambda2)))))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(Float64(lambda2 - lambda1)))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= -0.056)
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos(phi2))));
	elseif (phi2 <= 0.095)
		tmp = R * acos(((sin(phi1) * phi2) + (((-0.5 * (phi2 ^ 2.0)) + 1.0) * (cos(phi1) * cos((lambda1 - lambda2))))));
	else
		tmp = R * acos((cos(phi2) * cos((lambda2 - lambda1))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -0.056], 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], If[LessEqual[phi2, 0.095], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision] + N[(N[(N[(-0.5 * N[Power[phi2, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.0560000000000000012

   1. Initial program 85.6%

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

   3. Simplified 85.7%

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

   5. Taylor expanded in lambda2 around 0 62.8%

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

      1. *-commutative 62.8%

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

   7. Simplified 62.8%

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

   8. Taylor expanded in lambda1 around 0 38.7%

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

   if -0.0560000000000000012 < phi2 < 0.095000000000000001

   1. Initial program 68.3%

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

   3. Simplified 68.3%

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

   5. Taylor expanded in phi2 around 0 67.3%

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

      1. distribute-rgt-in 67.3%

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

      2. *-commutative 67.3%

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

      3. associate-+l+ 67.3%

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

   7. Simplified 67.3%

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

   if 0.095000000000000001 < phi2

   1. Initial program 80.5%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in phi1 around 0 48.5%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.056:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq 0.095:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \phi_2 + \left(-0.5 \cdot {\phi_2}^{2} + 1\right) \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 42.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.32 \cdot 10^{+91}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.38 \cdot 10^{+20}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.32e+91)
   (* R (acos (* (cos phi1) (cos lambda1))))
   (if (<= phi1 -1.38e+20)
     (* R (acos (* (cos phi1) (cos lambda2))))
     (* R (acos (* (cos phi2) (cos lambda1)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.32e+91) {
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	} else if (phi1 <= -1.38e+20) {
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	} else {
		tmp = R * acos((cos(phi2) * cos(lambda1)));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-1.32d+91)) then
        tmp = r * acos((cos(phi1) * cos(lambda1)))
    else if (phi1 <= (-1.38d+20)) then
        tmp = r * acos((cos(phi1) * cos(lambda2)))
    else
        tmp = r * acos((cos(phi2) * cos(lambda1)))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.32e+91) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else if (phi1 <= -1.38e+20) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda2)));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos(lambda1)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1.32e+91:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda1)))
	elif phi1 <= -1.38e+20:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda2)))
	else:
		tmp = R * math.acos((math.cos(phi2) * math.cos(lambda1)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.32e+91)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda1))));
	elseif (phi1 <= -1.38e+20)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda2))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(lambda1))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1.32e+91)
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	elseif (phi1 <= -1.38e+20)
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	else
		tmp = R * acos((cos(phi2) * cos(lambda1)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.32e+91], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, -1.38e+20], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.32000000000000003e91

   1. Initial program 81.0%

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

   3. Simplified 81.0%

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

   5. Taylor expanded in lambda2 around 0 67.1%

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

      1. *-commutative 67.1%

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

   7. Simplified 67.1%

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

   8. Taylor expanded in phi2 around 0 40.3%

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

   if -1.32000000000000003e91 < phi1 < -1.38e20

   1. Initial program 93.5%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in phi2 around 0 41.1%

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

   6. Taylor expanded in lambda1 around 0 26.4%

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

      1. *-commutative 26.4%

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

   8. Simplified 26.4%

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

   if -1.38e20 < 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. Add Preprocessing

   3. Taylor expanded in lambda2 around 0 51.1%

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

   4. Taylor expanded in phi1 around 0 30.9%

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

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.32 \cdot 10^{+91}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.38 \cdot 10^{+20}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 41.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 8.8 \cdot 10^{-76}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.02 \cdot 10^{-44}:\\ \;\;\;\;R \cdot \left|\left(\left(\lambda_2 - \lambda_1\right) \mathsf{rem} \left(2 \cdot \pi\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 8.8e-76)
   (* R (acos (* (cos phi1) (cos lambda1))))
   (if (<= lambda2 1.02e-44)
     (* R (fabs (remainder (- lambda2 lambda1) (* 2.0 PI))))
     (* R (acos (* (cos phi1) (cos lambda2)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 8.8e-76) {
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	} else if (lambda2 <= 1.02e-44) {
		tmp = R * fabs(remainder((lambda2 - lambda1), (2.0 * ((double) M_PI))));
	} else {
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 8.8e-76) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else if (lambda2 <= 1.02e-44) {
		tmp = R * Math.abs(Math.IEEEremainder((lambda2 - lambda1), (2.0 * Math.PI)));
	} else {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda2)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 8.8e-76:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda1)))
	elif lambda2 <= 1.02e-44:
		tmp = R * math.fabs(math.remainder((lambda2 - lambda1), (2.0 * math.pi)))
	else:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda2)))
	return tmp

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 8.8e-76], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 1.02e-44], N[(R * N[Abs[N[With[{TMP1 = N[(lambda2 - lambda1), $MachinePrecision], TMP2 = N[(2.0 * Pi), $MachinePrecision]}, TMP1 - Round[TMP1 / TMP2] * TMP2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < 8.79999999999999997e-76

   1. Initial program 77.6%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in lambda2 around 0 63.5%

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

      1. *-commutative 63.5%

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

   7. Simplified 63.5%

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

   8. Taylor expanded in phi2 around 0 39.1%

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

   if 8.79999999999999997e-76 < lambda2 < 1.0199999999999999e-44

   1. Initial program 56.8%

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

   3. Simplified 56.8%

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

   5. Taylor expanded in phi2 around 0 30.3%

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

   6. Taylor expanded in phi1 around 0 9.3%

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

      1. acos-cos 52.1%

         \[\leadsto R \cdot \color{blue}{\left|\left(\left(\lambda_2 - \lambda_1\right) \mathsf{rem} \left(2 \cdot \pi\right)\right)\right|} \]

   8. Applied egg-rr 52.1%

      \[\leadsto R \cdot \color{blue}{\left|\left(\left(\lambda_2 - \lambda_1\right) \mathsf{rem} \left(2 \cdot \pi\right)\right)\right|} \]

   if 1.0199999999999999e-44 < lambda2

   1. Initial program 72.3%

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

   3. Simplified 72.3%

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

   5. Taylor expanded in phi2 around 0 35.4%

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

   6. Taylor expanded in lambda1 around 0 31.6%

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

      1. *-commutative 31.6%

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

   8. Simplified 31.6%

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

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 8.8 \cdot 10^{-76}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.02 \cdot 10^{-44}:\\ \;\;\;\;R \cdot \left|\left(\left(\lambda_2 - \lambda_1\right) \mathsf{rem} \left(2 \cdot \pi\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 4.5 \cdot 10^{-56}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi2 4.5e-56)
     (* R (acos (* (cos phi1) t_0)))
     (* R (acos (* (cos phi2) t_0))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 4.5e-56) {
		tmp = R * acos((cos(phi1) * t_0));
	} else {
		tmp = R * acos((cos(phi2) * t_0));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((lambda2 - lambda1))
    if (phi2 <= 4.5d-56) then
        tmp = r * acos((cos(phi1) * t_0))
    else
        tmp = r * acos((cos(phi2) * t_0))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 4.5e-56) {
		tmp = R * Math.acos((Math.cos(phi1) * t_0));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * t_0));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	tmp = 0
	if phi2 <= 4.5e-56:
		tmp = R * math.acos((math.cos(phi1) * t_0))
	else:
		tmp = R * math.acos((math.cos(phi2) * t_0))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi2 <= 4.5e-56)
		tmp = Float64(R * acos(Float64(cos(phi1) * t_0)));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * t_0)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	tmp = 0.0;
	if (phi2 <= 4.5e-56)
		tmp = R * acos((cos(phi1) * t_0));
	else
		tmp = R * acos((cos(phi2) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 4.5e-56], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.5000000000000001e-56

   1. Initial program 75.4%

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

   3. Simplified 75.4%

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

   5. Taylor expanded in phi2 around 0 50.8%

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

   if 4.5000000000000001e-56 < phi2

   1. Initial program 76.1%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in phi1 around 0 43.5%

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

Alternative 17: 53.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.6 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.6e-9)
   (* R (acos (* (cos phi1) (cos (- lambda2 lambda1)))))
   (* R (acos (* (cos phi2) (cos lambda1))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.6e-9) {
		tmp = R * acos((cos(phi1) * cos((lambda2 - lambda1))));
	} else {
		tmp = R * acos((cos(phi2) * cos(lambda1)));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 1.6d-9) then
        tmp = r * acos((cos(phi1) * cos((lambda2 - lambda1))))
    else
        tmp = r * acos((cos(phi2) * cos(lambda1)))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.6e-9) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos((lambda2 - lambda1))));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos(lambda1)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.6e-9:
		tmp = R * math.acos((math.cos(phi1) * math.cos((lambda2 - lambda1))))
	else:
		tmp = R * math.acos((math.cos(phi2) * math.cos(lambda1)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.6e-9)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(Float64(lambda2 - lambda1)))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(lambda1))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.6e-9)
		tmp = R * acos((cos(phi1) * cos((lambda2 - lambda1))));
	else
		tmp = R * acos((cos(phi2) * cos(lambda1)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.6e-9], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.60000000000000006e-9

   1. Initial program 74.2%

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

   3. Simplified 74.2%

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

   5. Taylor expanded in phi2 around 0 51.1%

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

   if 1.60000000000000006e-9 < phi2

   1. Initial program 79.8%

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

   3. Taylor expanded in lambda2 around 0 51.6%

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

   4. Taylor expanded in phi1 around 0 29.1%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.6 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 36.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.02 \cdot 10^{+68}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -1.02e+68)
   (* R (acos (cos lambda1)))
   (* R (acos (* (cos phi1) (cos lambda2))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.02e+68) {
		tmp = R * acos(cos(lambda1));
	} else {
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-1.02d+68)) then
        tmp = r * acos(cos(lambda1))
    else
        tmp = r * acos((cos(phi1) * cos(lambda2)))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.02e+68) {
		tmp = R * Math.acos(Math.cos(lambda1));
	} else {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda2)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -1.02e+68:
		tmp = R * math.acos(math.cos(lambda1))
	else:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda2)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.02e+68)
		tmp = Float64(R * acos(cos(lambda1)));
	else
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda2))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -1.02e+68)
		tmp = R * acos(cos(lambda1));
	else
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -1.02e+68], N[(R * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.02e68

   1. Initial program 65.5%

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

   3. Simplified 65.5%

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

   5. Taylor expanded in phi2 around 0 43.3%

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

   6. Taylor expanded in phi1 around 0 33.8%

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

   7. Taylor expanded in lambda2 around 0 33.9%

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

      1. cos-neg 33.9%

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

   9. Simplified 33.9%

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

   if -1.02e68 < lambda1

   1. Initial program 77.9%

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

   3. Simplified 77.9%

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

   5. Taylor expanded in phi2 around 0 43.7%

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

   6. Taylor expanded in lambda1 around 0 34.6%

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

      1. *-commutative 34.6%

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

   8. Simplified 34.6%

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

Alternative 19: 23.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -5.8 \cdot 10^{-97}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left|\left(\left(\lambda_2 - \lambda_1\right) \mathsf{rem} \left(2 \cdot \pi\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -5.8e-97)
   (* R (acos (cos (- lambda2 lambda1))))
   (* R (fabs (remainder (- lambda2 lambda1) (* 2.0 PI))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -5.8e-97) {
		tmp = R * acos(cos((lambda2 - lambda1)));
	} else {
		tmp = R * fabs(remainder((lambda2 - lambda1), (2.0 * ((double) M_PI))));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -5.8e-97) {
		tmp = R * Math.acos(Math.cos((lambda2 - lambda1)));
	} else {
		tmp = R * Math.abs(Math.IEEEremainder((lambda2 - lambda1), (2.0 * Math.PI)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -5.8e-97:
		tmp = R * math.acos(math.cos((lambda2 - lambda1)))
	else:
		tmp = R * math.fabs(math.remainder((lambda2 - lambda1), (2.0 * math.pi)))
	return tmp

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -5.8e-97], N[(R * N[ArcCos[N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[Abs[N[With[{TMP1 = N[(lambda2 - lambda1), $MachinePrecision], TMP2 = N[(2.0 * Pi), $MachinePrecision]}, TMP1 - Round[TMP1 / TMP2] * TMP2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -5.7999999999999999e-97

   1. Initial program 68.7%

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

   3. Simplified 68.7%

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

   5. Taylor expanded in phi2 around 0 46.3%

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

   6. Taylor expanded in phi1 around 0 29.7%

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

   if -5.7999999999999999e-97 < lambda1

   1. Initial program 79.0%

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

   3. Simplified 79.0%

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

   5. Taylor expanded in phi2 around 0 42.2%

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

   6. Taylor expanded in phi1 around 0 21.5%

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

      1. acos-cos 20.6%

         \[\leadsto R \cdot \color{blue}{\left|\left(\left(\lambda_2 - \lambda_1\right) \mathsf{rem} \left(2 \cdot \pi\right)\right)\right|} \]

   8. Applied egg-rr 20.6%

      \[\leadsto R \cdot \color{blue}{\left|\left(\left(\lambda_2 - \lambda_1\right) \mathsf{rem} \left(2 \cdot \pi\right)\right)\right|} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 26.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.038:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -0.038)
   (* R (acos (cos lambda1)))
   (* R (acos (cos lambda2)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -0.038) {
		tmp = R * acos(cos(lambda1));
	} else {
		tmp = R * acos(cos(lambda2));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-0.038d0)) then
        tmp = r * acos(cos(lambda1))
    else
        tmp = r * acos(cos(lambda2))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -0.038) {
		tmp = R * Math.acos(Math.cos(lambda1));
	} else {
		tmp = R * Math.acos(Math.cos(lambda2));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -0.038:
		tmp = R * math.acos(math.cos(lambda1))
	else:
		tmp = R * math.acos(math.cos(lambda2))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -0.038)
		tmp = Float64(R * acos(cos(lambda1)));
	else
		tmp = Float64(R * acos(cos(lambda2)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -0.038)
		tmp = R * acos(cos(lambda1));
	else
		tmp = R * acos(cos(lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -0.038], N[(R * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[Cos[lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -0.0379999999999999991

   1. Initial program 64.4%

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

   3. Simplified 64.4%

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

   5. Taylor expanded in phi2 around 0 43.9%

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

   6. Taylor expanded in phi1 around 0 32.4%

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

   7. Taylor expanded in lambda2 around 0 32.5%

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

      1. cos-neg 32.5%

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

   9. Simplified 32.5%

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

   if -0.0379999999999999991 < lambda1

   1. Initial program 79.0%

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

   3. Simplified 79.0%

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

   5. Taylor expanded in phi2 around 0 43.5%

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

   6. Taylor expanded in phi1 around 0 21.7%

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

   7. Taylor expanded in lambda1 around 0 15.4%

      \[\leadsto R \cdot \cos^{-1} \color{blue}{\cos \lambda_2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 19.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2 \cdot 10^{+37}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 - \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -2e+37) (* R (acos (cos lambda1))) (* R (- lambda2 lambda1))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -2e+37) {
		tmp = R * acos(cos(lambda1));
	} else {
		tmp = R * (lambda2 - lambda1);
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-2d+37)) then
        tmp = r * acos(cos(lambda1))
    else
        tmp = r * (lambda2 - lambda1)
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -2e+37) {
		tmp = R * Math.acos(Math.cos(lambda1));
	} else {
		tmp = R * (lambda2 - lambda1);
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -2e+37:
		tmp = R * math.acos(math.cos(lambda1))
	else:
		tmp = R * (lambda2 - lambda1)
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -2e+37)
		tmp = Float64(R * acos(cos(lambda1)));
	else
		tmp = Float64(R * Float64(lambda2 - lambda1));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -2e+37)
		tmp = R * acos(cos(lambda1));
	else
		tmp = R * (lambda2 - lambda1);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -2e+37], N[(R * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.99999999999999991e37

   1. Initial program 66.6%

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

   3. Simplified 66.6%

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

   5. Taylor expanded in phi2 around 0 44.5%

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

   6. Taylor expanded in phi1 around 0 33.0%

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

   7. Taylor expanded in lambda2 around 0 33.1%

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

      1. cos-neg 33.1%

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

   9. Simplified 33.1%

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

   if -1.99999999999999991e37 < lambda1

   1. Initial program 77.9%

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

   3. Simplified 77.9%

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

   5. Taylor expanded in phi2 around 0 43.4%

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

   6. Taylor expanded in phi1 around 0 22.0%

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

   7. Taylor expanded in lambda2 around 0 7.6%

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

Alternative 22: 26.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

3. Simplified 75.6%

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

5. Taylor expanded in phi2 around 0 43.6%

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

6. Taylor expanded in phi1 around 0 24.2%

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

Alternative 23: 8.8% accurate, 68.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -6.6 \cdot 10^{-132}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -6.6e-132) (* lambda1 (- R)) (* lambda2 R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -6.6e-132) {
		tmp = lambda1 * -R;
	} else {
		tmp = lambda2 * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-6.6d-132)) then
        tmp = lambda1 * -r
    else
        tmp = lambda2 * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -6.6e-132) {
		tmp = lambda1 * -R;
	} else {
		tmp = lambda2 * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -6.6e-132:
		tmp = lambda1 * -R
	else:
		tmp = lambda2 * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -6.6e-132)
		tmp = Float64(lambda1 * Float64(-R));
	else
		tmp = Float64(lambda2 * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -6.6e-132)
		tmp = lambda1 * -R;
	else
		tmp = lambda2 * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -6.6e-132], N[(lambda1 * (-R)), $MachinePrecision], N[(lambda2 * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -6.5999999999999994e-132

   1. Initial program 70.4%

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

   3. Simplified 70.5%

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

   5. Taylor expanded in phi2 around 0 46.6%

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

   6. Taylor expanded in phi1 around 0 28.7%

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

   7. Taylor expanded in lambda2 around 0 11.4%

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

      1. associate-*r* 11.4%

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

      2. mul-1-neg 11.4%

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

   9. Simplified 11.4%

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

   if -6.5999999999999994e-132 < lambda1

   1. Initial program 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in phi2 around 0 42.0%

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

   6. Taylor expanded in phi1 around 0 21.8%

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

   7. Taylor expanded in lambda2 around inf 6.1%

      \[\leadsto \color{blue}{R \cdot \lambda_2} \]
   8. Step-by-step derivation

      1. *-commutative 6.1%

         \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

   9. Simplified 6.1%

      \[\leadsto \color{blue}{\lambda_2 \cdot R} \]
3. Recombined 2 regimes into one program.

4. Final simplification 7.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -6.6 \cdot 10^{-132}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 9.0% accurate, 122.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

3. Simplified 75.6%

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

5. Taylor expanded in phi2 around 0 43.6%

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

6. Taylor expanded in phi1 around 0 24.2%

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

7. Taylor expanded in lambda2 around 0 7.0%

   \[\leadsto R \cdot \color{blue}{\left(\lambda_2 - \lambda_1\right)} \]
8. Add Preprocessing

Alternative 25: 6.8% accurate, 204.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \lambda_2 \cdot R \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return lambda2 * R

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.6%

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

3. Simplified 75.6%

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

5. Taylor expanded in phi2 around 0 43.6%

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

6. Taylor expanded in phi1 around 0 24.2%

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

7. Taylor expanded in lambda2 around inf 5.7%

   \[\leadsto \color{blue}{R \cdot \lambda_2} \]
8. Step-by-step derivation

   1. *-commutative 5.7%

      \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

9. Simplified 5.7%

   \[\leadsto \color{blue}{\lambda_2 \cdot R} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(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))