Spherical law of cosines

Percentage Accurate: 74.1% → 96.5%

Time: 25.4s

Alternatives: 30

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: 74.1% 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: 96.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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 = (lambda2 - lambda1) * R;
	} else {
		tmp = R * acos(fma(cos(lambda2), (t_1 * cos(lambda1)), (t_0 + (sin(lambda1) * (cos(phi2) * (cos(phi1) * sin(lambda2)))))));
	}
	return tmp;
}

Julia

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 = Float64(Float64(lambda2 - lambda1) * R);
	else
		tmp = Float64(R * acos(fma(cos(lambda2), Float64(t_1 * cos(lambda1)), Float64(t_0 + Float64(sin(lambda1) * Float64(cos(phi2) * Float64(cos(phi1) * sin(lambda2))))))));
	end
	return tmp
end

Wolfram

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[(N[(lambda2 - lambda1), $MachinePrecision] * R), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[lambda2], $MachinePrecision] * N[(t$95$1 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + N[(N[Sin[lambda1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $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 3.4%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 3.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 3.4%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right), R\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)\right), R\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      5. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right)\right), R\right) \]

      6. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)\right), R\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right), R\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      10. --lowering--.f64 3.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   8. Simplified 3.4%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right), R\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right), R\right) \]

      4. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \cos \left(\lambda_2 - \lambda_1\right), R\right) \]

      5. acos-cos-s N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\lambda_2 - \lambda_1\right), R\right) \]

      6. --lowering--.f64 31.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right), R\right) \]

   10. Applied egg-rr 31.4%

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

   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 83.2%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right), R\right) \]

      2. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right), R\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right), R\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      5. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right) \cdot \cos \phi_2 + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      6. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\mathsf{fma}\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1, \cos \phi_2, \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      7. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1\right), \cos \phi_2, \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

   4. Applied egg-rr 98.9%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \lambda_2 \cdot \cos \lambda_1\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \lambda_2 \cdot \left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      4. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right), \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      5. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\cos \lambda_2, \left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right), \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right), \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{*.f64}\left(\cos \lambda_1, \left(\cos \phi_1 \cdot \cos \phi_2\right)\right), \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \left(\cos \phi_1 \cdot \cos \phi_2\right)\right), \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\cos \phi_1, \cos \phi_2\right)\right), \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \phi_2\right)\right), \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right)\right), \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right)\right), \mathsf{+.f64}\left(\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), \left(\sin \phi_1 \cdot \sin \phi_2\right)\right)\right)\right), R\right) \]

   6. Applied egg-rr 99.0%

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

4. Final simplification 95.3%

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

Alternative 2: 96.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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 = (lambda2 - lambda1) * R;
	} else {
		tmp = R * acos(((t_0 + (cos(phi1) * (cos(phi2) * (sin(lambda1) * sin(lambda2))))) + (cos(phi1) * (cos(phi2) * (cos(lambda2) * cos(lambda1))))));
	}
	return tmp;
}

Fortran

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 (acos((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) <= 0.0d0) then
        tmp = (lambda2 - lambda1) * r
    else
        tmp = r * acos(((t_0 + (cos(phi1) * (cos(phi2) * (sin(lambda1) * sin(lambda2))))) + (cos(phi1) * (cos(phi2) * (cos(lambda2) * cos(lambda1))))))
    end if
    code = tmp
end function

Java

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 (Math.acos((t_0 + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2))))) <= 0.0) {
		tmp = (lambda2 - lambda1) * R;
	} else {
		tmp = R * Math.acos(((t_0 + (Math.cos(phi1) * (Math.cos(phi2) * (Math.sin(lambda1) * Math.sin(lambda2))))) + (Math.cos(phi1) * (Math.cos(phi2) * (Math.cos(lambda2) * Math.cos(lambda1))))));
	}
	return tmp;
}

Python

[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 math.acos((t_0 + ((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2))))) <= 0.0:
		tmp = (lambda2 - lambda1) * R
	else:
		tmp = R * math.acos(((t_0 + (math.cos(phi1) * (math.cos(phi2) * (math.sin(lambda1) * math.sin(lambda2))))) + (math.cos(phi1) * (math.cos(phi2) * (math.cos(lambda2) * math.cos(lambda1))))))
	return tmp

Julia

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 = Float64(Float64(lambda2 - lambda1) * R);
	else
		tmp = Float64(R * acos(Float64(Float64(t_0 + Float64(cos(phi1) * Float64(cos(phi2) * Float64(sin(lambda1) * sin(lambda2))))) + Float64(cos(phi1) * Float64(cos(phi2) * Float64(cos(lambda2) * cos(lambda1)))))));
	end
	return tmp
end

MATLAB

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 (acos((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) <= 0.0)
		tmp = (lambda2 - lambda1) * R;
	else
		tmp = R * acos(((t_0 + (cos(phi1) * (cos(phi2) * (sin(lambda1) * sin(lambda2))))) + (cos(phi1) * (cos(phi2) * (cos(lambda2) * 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.
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[(N[(lambda2 - lambda1), $MachinePrecision] * R), $MachinePrecision], N[(R * N[ArcCos[N[(N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $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 3.4%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 3.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 3.4%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right), R\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)\right), R\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      5. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right)\right), R\right) \]

      6. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)\right), R\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right), R\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      10. --lowering--.f64 3.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   8. Simplified 3.4%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right), R\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right), R\right) \]

      4. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \cos \left(\lambda_2 - \lambda_1\right), R\right) \]

      5. acos-cos-s N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\lambda_2 - \lambda_1\right), R\right) \]

      6. --lowering--.f64 31.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right), R\right) \]

   10. Applied egg-rr 31.4%

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

   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 83.2%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right), R\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right), R\right) \]

      4. associate-+r+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + \left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right), R\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right), \left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\right)\right), R\right) \]

   4. Applied egg-rr 98.9%

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

4. Final simplification 95.2%

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

Alternative 3: 96.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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 = (lambda2 - lambda1) * R;
	} else {
		tmp = R * acos((t_0 + (t_1 * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2))))));
	}
	return tmp;
}

Julia

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 = Float64(Float64(lambda2 - lambda1) * R);
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(t_1 * fma(cos(lambda2), cos(lambda1), 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.
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[(N[(lambda2 - lambda1), $MachinePrecision] * R), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(t$95$1 * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 3.4%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 3.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 3.4%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right), R\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)\right), R\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      5. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right)\right), R\right) \]

      6. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)\right), R\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right), R\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      10. --lowering--.f64 3.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   8. Simplified 3.4%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right), R\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right), R\right) \]

      4. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \cos \left(\lambda_2 - \lambda_1\right), R\right) \]

      5. acos-cos-s N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\lambda_2 - \lambda_1\right), R\right) \]

      6. --lowering--.f64 31.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right), R\right) \]

   10. Applied egg-rr 31.4%

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

   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 83.2%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\cos \lambda_2, \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 98.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

   4. Applied egg-rr 98.9%

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

4. Final simplification 95.2%

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

Alternative 4: 96.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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 = (lambda2 - lambda1) * R;
	} else {
		tmp = R * acos((t_0 + (t_1 * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1))))));
	}
	return tmp;
}

Fortran

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 = sin(phi1) * sin(phi2)
    t_1 = cos(phi1) * cos(phi2)
    if (acos((t_0 + (t_1 * cos((lambda1 - lambda2))))) <= 0.0d0) then
        tmp = (lambda2 - lambda1) * r
    else
        tmp = r * acos((t_0 + (t_1 * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1))))))
    end if
    code = tmp
end function

Java

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 t_1 = Math.cos(phi1) * Math.cos(phi2);
	double tmp;
	if (Math.acos((t_0 + (t_1 * Math.cos((lambda1 - lambda2))))) <= 0.0) {
		tmp = (lambda2 - lambda1) * R;
	} else {
		tmp = R * Math.acos((t_0 + (t_1 * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda2) * Math.cos(lambda1))))));
	}
	return tmp;
}

Python

[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)
	t_1 = math.cos(phi1) * math.cos(phi2)
	tmp = 0
	if math.acos((t_0 + (t_1 * math.cos((lambda1 - lambda2))))) <= 0.0:
		tmp = (lambda2 - lambda1) * R
	else:
		tmp = R * math.acos((t_0 + (t_1 * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda2) * math.cos(lambda1))))))
	return tmp

Julia

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 = Float64(Float64(lambda2 - lambda1) * R);
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(t_1 * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda2) * cos(lambda1)))))));
	end
	return tmp
end

MATLAB

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);
	t_1 = cos(phi1) * cos(phi2);
	tmp = 0.0;
	if (acos((t_0 + (t_1 * cos((lambda1 - lambda2))))) <= 0.0)
		tmp = (lambda2 - lambda1) * R;
	else
		tmp = R * acos((t_0 + (t_1 * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * 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.
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[(N[(lambda2 - lambda1), $MachinePrecision] * R), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$0 + N[(t$95$1 * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $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 3.4%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 3.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 3.4%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right), R\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)\right), R\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      5. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right)\right), R\right) \]

      6. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)\right), R\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right), R\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      10. --lowering--.f64 3.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   8. Simplified 3.4%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right), R\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right), R\right) \]

      4. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \cos \left(\lambda_2 - \lambda_1\right), R\right) \]

      5. acos-cos-s N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\lambda_2 - \lambda_1\right), R\right) \]

      6. --lowering--.f64 31.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right), R\right) \]

   10. Applied egg-rr 31.4%

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

   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 83.2%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{+.f64}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\cos \lambda_1, \cos \lambda_2\right)\right)\right)\right)\right), R\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \cos \lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 98.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

   4. Applied egg-rr 98.9%

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

4. Final simplification 95.2%

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

Alternative 5: 82.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{if}\;\phi_2 \leq -9.9 \cdot 10^{-82}:\\ \;\;\;\;R \cdot \frac{\pi}{2} - R \cdot \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t\_0\right)\\ \mathbf{elif}\;\phi_2 \leq 0.052:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))
   (if (<= phi2 -9.9e-82)
     (- (* R (/ PI 2.0)) (* R (asin (+ (* (sin phi1) (sin phi2)) t_0))))
     (if (<= phi2 0.052)
       (*
        R
        (acos
         (+
          (*
           (cos phi1)
           (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda2) (cos lambda1))))
          (* (sin phi1) phi2))))
       (* R (acos (fma (sin phi2) (sin phi1) t_0)))))))

C

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(phi1) * (cos(phi2) * cos((lambda1 - lambda2)));
	double tmp;
	if (phi2 <= -9.9e-82) {
		tmp = (R * (((double) M_PI) / 2.0)) - (R * asin(((sin(phi1) * sin(phi2)) + t_0)));
	} else if (phi2 <= 0.052) {
		tmp = R * acos(((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1)))) + (sin(phi1) * phi2)));
	} else {
		tmp = R * acos(fma(sin(phi2), sin(phi1), t_0));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))
	tmp = 0.0
	if (phi2 <= -9.9e-82)
		tmp = Float64(Float64(R * Float64(pi / 2.0)) - Float64(R * asin(Float64(Float64(sin(phi1) * sin(phi2)) + t_0))));
	elseif (phi2 <= 0.052)
		tmp = Float64(R * acos(Float64(Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda2) * cos(lambda1)))) + Float64(sin(phi1) * phi2))));
	else
		tmp = Float64(R * acos(fma(sin(phi2), sin(phi1), t_0)));
	end
	return tmp
end

Wolfram

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[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -9.9e-82], N[(N[(R * N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision] - N[(R * N[ArcSin[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 0.052], N[(R * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -9.9000000000000001e-82

   1. Initial program 82.1%

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

      1. *-commutative N/A

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

      2. acos-asin N/A

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

      3. sub-neg N/A

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

      4. distribute-rgt-in N/A

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

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot R\right), \color{blue}{\left(\left(\mathsf{neg}\left(\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot R\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), R\right), \left(\color{blue}{\left(\mathsf{neg}\left(\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)} \cdot R\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), R\right), \left(\left(\mathsf{neg}\left(\color{blue}{\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) \cdot R\right)\right) \]

      8. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), R\right), \left(\left(\mathsf{neg}\left(\sin^{-1} \color{blue}{\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) \cdot R\right)\right) \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), R\right), \left(\mathsf{neg}\left(\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\right)\right)\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), R\right), \mathsf{neg.f64}\left(\left(\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\right)\right)\right) \]

   4. Applied egg-rr 82.0%

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

   if -9.9000000000000001e-82 < phi2 < 0.0519999999999999976

   1. Initial program 70.1%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right), R\right) \]

      2. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right), R\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right), R\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right) \cdot \cos \lambda_1 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right) \cdot \cos \lambda_1 + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      8. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\mathsf{fma}\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2, \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      9. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right), \cos \lambda_1, \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

   4. Applied egg-rr 87.7%

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

   5. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\phi_2 \cdot \sin \phi_1 + \left(\cos \lambda_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_1\right) + \cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)}\right), R\right) \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \lambda_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_1\right) + \cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \phi_2 \cdot \sin \phi_1\right)\right), R\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\left(\cos \lambda_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_1\right) + \cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right), \left(\phi_2 \cdot \sin \phi_1\right)\right)\right), R\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1 + \cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right), \left(\phi_2 \cdot \sin \phi_1\right)\right)\right), R\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_1 + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_1\right), \left(\phi_2 \cdot \sin \phi_1\right)\right)\right), R\right) \]

      5. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\left(\cos \phi_1 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right), \left(\phi_2 \cdot \sin \phi_1\right)\right)\right), R\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right), \left(\phi_2 \cdot \sin \phi_1\right)\right)\right), R\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right), \left(\phi_2 \cdot \sin \phi_1\right)\right)\right), R\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), \left(\phi_2 \cdot \sin \phi_1\right)\right)\right), R\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\cos \lambda_1, \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), \left(\phi_2 \cdot \sin \phi_1\right)\right)\right), R\right) \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), \left(\phi_2 \cdot \sin \phi_1\right)\right)\right), R\right) \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), \left(\phi_2 \cdot \sin \phi_1\right)\right)\right), R\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right)\right)\right), \left(\phi_2 \cdot \sin \phi_1\right)\right)\right), R\right) \]

      13. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right)\right)\right), \left(\phi_2 \cdot \sin \phi_1\right)\right)\right), R\right) \]

      14. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right), \left(\phi_2 \cdot \sin \phi_1\right)\right)\right), R\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right), \mathsf{*.f64}\left(\phi_2, \sin \phi_1\right)\right)\right), R\right) \]

   7. Simplified 87.0%

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

   if 0.0519999999999999976 < phi2

   1. Initial program 89.9%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\sin \phi_2 \cdot \sin \phi_1 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right), R\right) \]

      3. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\sin \phi_2, \sin \phi_1, \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right), R\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \sin \phi_1, \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right), R\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right), R\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\cos \phi_1, \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 90.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

   4. Applied egg-rr 90.1%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -9.9 \cdot 10^{-82}:\\ \;\;\;\;R \cdot \frac{\pi}{2} - R \cdot \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 0.052:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \mathbf{if}\;\phi_2 \leq -9.9 \cdot 10^{-82}:\\ \;\;\;\;R \cdot \frac{\pi}{2} - R \cdot \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t\_0\right)\\ \mathbf{elif}\;\phi_2 \leq 2.3 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))
   (if (<= phi2 -9.9e-82)
     (- (* R (/ PI 2.0)) (* R (asin (+ (* (sin phi1) (sin phi2)) t_0))))
     (if (<= phi2 2.3e-10)
       (*
        R
        (acos
         (*
          (cos phi1)
          (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2))))))
       (* R (acos (fma (sin phi2) (sin phi1) t_0)))))))

C

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(phi1) * (cos(phi2) * cos((lambda1 - lambda2)));
	double tmp;
	if (phi2 <= -9.9e-82) {
		tmp = (R * (((double) M_PI) / 2.0)) - (R * asin(((sin(phi1) * sin(phi2)) + t_0)));
	} else if (phi2 <= 2.3e-10) {
		tmp = R * acos((cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2)))));
	} else {
		tmp = R * acos(fma(sin(phi2), sin(phi1), t_0));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))
	tmp = 0.0
	if (phi2 <= -9.9e-82)
		tmp = Float64(Float64(R * Float64(pi / 2.0)) - Float64(R * asin(Float64(Float64(sin(phi1) * sin(phi2)) + t_0))));
	elseif (phi2 <= 2.3e-10)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = Float64(R * acos(fma(sin(phi2), sin(phi1), t_0)));
	end
	return tmp
end

Wolfram

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[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -9.9e-82], N[(N[(R * N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision] - N[(R * N[ArcSin[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 2.3e-10], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -9.9000000000000001e-82

   1. Initial program 82.1%

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

      1. *-commutative N/A

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

      2. acos-asin N/A

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

      3. sub-neg N/A

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

      4. distribute-rgt-in N/A

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

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot R\right), \color{blue}{\left(\left(\mathsf{neg}\left(\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot R\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), R\right), \left(\color{blue}{\left(\mathsf{neg}\left(\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)} \cdot R\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), R\right), \left(\left(\mathsf{neg}\left(\color{blue}{\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) \cdot R\right)\right) \]

      8. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), R\right), \left(\left(\mathsf{neg}\left(\sin^{-1} \color{blue}{\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) \cdot R\right)\right) \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), R\right), \left(\mathsf{neg}\left(\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\right)\right)\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), R\right), \mathsf{neg.f64}\left(\left(\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\right)\right)\right) \]

   4. Applied egg-rr 82.0%

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

   if -9.9000000000000001e-82 < phi2 < 2.30000000000000007e-10

   1. Initial program 70.2%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 70.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 70.2%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      4. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{fma.f64}\left(\cos \lambda_2, \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right)\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 87.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

   7. Applied egg-rr 87.1%

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

   if 2.30000000000000007e-10 < phi2

   1. Initial program 89.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\sin \phi_2 \cdot \sin \phi_1 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right), R\right) \]

      3. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\sin \phi_2, \sin \phi_1, \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right), R\right) \]

      4. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \sin \phi_1, \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right), R\right) \]

      5. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right), R\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\cos \phi_1, \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

      10. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 89.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

   4. Applied egg-rr 89.3%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -9.9 \cdot 10^{-82}:\\ \;\;\;\;R \cdot \frac{\pi}{2} - R \cdot \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.3 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -9.9 \cdot 10^{-82}:\\ \;\;\;\;R \cdot \frac{\pi}{2} - R \cdot \sin^{-1} \left(t\_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(t\_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= phi2 -9.9e-82)
     (-
      (* R (/ PI 2.0))
      (*
       R
       (asin (+ t_0 (* (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))))
     (if (<= phi2 1.9e-10)
       (*
        R
        (acos
         (*
          (cos phi1)
          (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2))))))
       (*
        R
        (-
         (/ PI 2.0)
         (asin
          (+
           t_0
           (* (cos phi2) (* (cos phi1) (cos (- lambda2 lambda1))))))))))))

C

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 (phi2 <= -9.9e-82) {
		tmp = (R * (((double) M_PI) / 2.0)) - (R * asin((t_0 + (cos(phi1) * (cos(phi2) * cos((lambda1 - lambda2)))))));
	} else if (phi2 <= 1.9e-10) {
		tmp = R * acos((cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2)))));
	} else {
		tmp = R * ((((double) M_PI) / 2.0) - asin((t_0 + (cos(phi2) * (cos(phi1) * cos((lambda2 - lambda1)))))));
	}
	return tmp;
}

Julia

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 (phi2 <= -9.9e-82)
		tmp = Float64(Float64(R * Float64(pi / 2.0)) - Float64(R * asin(Float64(t_0 + Float64(cos(phi1) * Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))))));
	elseif (phi2 <= 1.9e-10)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = Float64(R * Float64(Float64(pi / 2.0) - asin(Float64(t_0 + Float64(cos(phi2) * Float64(cos(phi1) * 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.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -9.9e-82], N[(N[(R * N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision] - N[(R * N[ArcSin[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.9e-10], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcSin[N[(t$95$0 + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -9.9000000000000001e-82

   1. Initial program 82.1%

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

      1. *-commutative N/A

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

      2. acos-asin N/A

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

      3. sub-neg N/A

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

      4. distribute-rgt-in N/A

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

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot R\right), \color{blue}{\left(\left(\mathsf{neg}\left(\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot R\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), R\right), \left(\color{blue}{\left(\mathsf{neg}\left(\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)} \cdot R\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), R\right), \left(\left(\mathsf{neg}\left(\color{blue}{\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) \cdot R\right)\right) \]

      8. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), R\right), \left(\left(\mathsf{neg}\left(\sin^{-1} \color{blue}{\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) \cdot R\right)\right) \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), R\right), \left(\mathsf{neg}\left(\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\right)\right)\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), R\right), \mathsf{neg.f64}\left(\left(\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\right)\right)\right) \]

   4. Applied egg-rr 82.0%

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

   if -9.9000000000000001e-82 < phi2 < 1.8999999999999999e-10

   1. Initial program 70.2%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 70.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 70.2%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      3. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      4. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{fma.f64}\left(\cos \lambda_2, \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right)\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 87.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

   7. Applied egg-rr 87.1%

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

   if 1.8999999999999999e-10 < phi2

   1. Initial program 89.1%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right), R\right) \]

      2. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right), R\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right), R\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right) \cdot \cos \lambda_1 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right) \cdot \cos \lambda_1 + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      8. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\mathsf{fma}\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2, \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      9. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right), \cos \lambda_1, \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

   4. Applied egg-rr 98.9%

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

   6. Applied egg-rr 89.2%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -9.9 \cdot 10^{-82}:\\ \;\;\;\;R \cdot \frac{\pi}{2} - R \cdot \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -9.9 \cdot 10^{-82}:\\ \;\;\;\;R \cdot \frac{\pi}{2} - R \cdot \sin^{-1} \left(t\_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 8 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(t\_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= phi2 -9.9e-82)
     (-
      (* R (/ PI 2.0))
      (*
       R
       (asin (+ t_0 (* (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))))
     (if (<= phi2 8e-11)
       (*
        R
        (acos
         (*
          (cos phi1)
          (+
           (* (sin lambda1) (sin lambda2))
           (* (cos lambda2) (cos lambda1))))))
       (*
        R
        (-
         (/ PI 2.0)
         (asin
          (+
           t_0
           (* (cos phi2) (* (cos phi1) (cos (- lambda2 lambda1))))))))))))

C

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 (phi2 <= -9.9e-82) {
		tmp = (R * (((double) M_PI) / 2.0)) - (R * asin((t_0 + (cos(phi1) * (cos(phi2) * cos((lambda1 - lambda2)))))));
	} else if (phi2 <= 8e-11) {
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1)))));
	} else {
		tmp = R * ((((double) M_PI) / 2.0) - asin((t_0 + (cos(phi2) * (cos(phi1) * cos((lambda2 - lambda1)))))));
	}
	return tmp;
}

Java

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 (phi2 <= -9.9e-82) {
		tmp = (R * (Math.PI / 2.0)) - (R * Math.asin((t_0 + (Math.cos(phi1) * (Math.cos(phi2) * Math.cos((lambda1 - lambda2)))))));
	} else if (phi2 <= 8e-11) {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda2) * Math.cos(lambda1)))));
	} else {
		tmp = R * ((Math.PI / 2.0) - Math.asin((t_0 + (Math.cos(phi2) * (Math.cos(phi1) * Math.cos((lambda2 - lambda1)))))));
	}
	return tmp;
}

Python

[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 phi2 <= -9.9e-82:
		tmp = (R * (math.pi / 2.0)) - (R * math.asin((t_0 + (math.cos(phi1) * (math.cos(phi2) * math.cos((lambda1 - lambda2)))))))
	elif phi2 <= 8e-11:
		tmp = R * math.acos((math.cos(phi1) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda2) * math.cos(lambda1)))))
	else:
		tmp = R * ((math.pi / 2.0) - math.asin((t_0 + (math.cos(phi2) * (math.cos(phi1) * math.cos((lambda2 - lambda1)))))))
	return tmp

Julia

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 (phi2 <= -9.9e-82)
		tmp = Float64(Float64(R * Float64(pi / 2.0)) - Float64(R * asin(Float64(t_0 + Float64(cos(phi1) * Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))))));
	elseif (phi2 <= 8e-11)
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda2) * cos(lambda1))))));
	else
		tmp = Float64(R * Float64(Float64(pi / 2.0) - asin(Float64(t_0 + Float64(cos(phi2) * Float64(cos(phi1) * cos(Float64(lambda2 - lambda1))))))));
	end
	return tmp
end

MATLAB

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 (phi2 <= -9.9e-82)
		tmp = (R * (pi / 2.0)) - (R * asin((t_0 + (cos(phi1) * (cos(phi2) * cos((lambda1 - lambda2)))))));
	elseif (phi2 <= 8e-11)
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1)))));
	else
		tmp = R * ((pi / 2.0) - asin((t_0 + (cos(phi2) * (cos(phi1) * 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.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -9.9e-82], N[(N[(R * N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision] - N[(R * N[ArcSin[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 8e-11], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcSin[N[(t$95$0 + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -9.9000000000000001e-82

   1. Initial program 82.1%

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

      1. *-commutative N/A

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

      2. acos-asin N/A

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

      3. sub-neg N/A

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

      4. distribute-rgt-in N/A

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

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2} \cdot R\right), \color{blue}{\left(\left(\mathsf{neg}\left(\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot R\right)}\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), R\right), \left(\color{blue}{\left(\mathsf{neg}\left(\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)} \cdot R\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), R\right), \left(\left(\mathsf{neg}\left(\color{blue}{\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) \cdot R\right)\right) \]

      8. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), R\right), \left(\left(\mathsf{neg}\left(\sin^{-1} \color{blue}{\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right) \cdot R\right)\right) \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), R\right), \left(\mathsf{neg}\left(\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\right)\right)\right) \]

      10. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), R\right), \mathsf{neg.f64}\left(\left(\sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\right)\right)\right) \]

   4. Applied egg-rr 82.0%

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

   if -9.9000000000000001e-82 < phi2 < 7.99999999999999952e-11

   1. Initial program 70.2%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 70.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 70.2%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right), R\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right), R\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right), R\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\cos \lambda_2, \cos \lambda_1\right)\right)\right)\right), R\right) \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1\right)\right)\right)\right), R\right) \]

      12. cos-lowering-cos.f64 87.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right)\right)\right), R\right) \]

   7. Applied egg-rr 87.1%

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

   if 7.99999999999999952e-11 < phi2

   1. Initial program 89.1%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right), R\right) \]

      2. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right), R\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right), R\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right) \cdot \cos \lambda_1 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right) \cdot \cos \lambda_1 + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      8. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\mathsf{fma}\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2, \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      9. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right), \cos \lambda_1, \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

   4. Applied egg-rr 98.9%

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

   6. Applied egg-rr 89.2%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -9.9 \cdot 10^{-82}:\\ \;\;\;\;R \cdot \frac{\pi}{2} - R \cdot \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 8 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -1.42 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(t\_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 9.6 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(t\_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (if (<= phi2 -1.42e-14)
     (*
      R
      (-
       (/ PI 2.0)
       (asin (+ t_0 (* (cos phi1) (* (cos phi2) (cos (- lambda1 lambda2))))))))
     (if (<= phi2 9.6e-11)
       (*
        R
        (acos
         (*
          (cos phi1)
          (+
           (* (sin lambda1) (sin lambda2))
           (* (cos lambda2) (cos lambda1))))))
       (*
        R
        (-
         (/ PI 2.0)
         (asin
          (+
           t_0
           (* (cos phi2) (* (cos phi1) (cos (- lambda2 lambda1))))))))))))

C

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 (phi2 <= -1.42e-14) {
		tmp = R * ((((double) M_PI) / 2.0) - asin((t_0 + (cos(phi1) * (cos(phi2) * cos((lambda1 - lambda2)))))));
	} else if (phi2 <= 9.6e-11) {
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1)))));
	} else {
		tmp = R * ((((double) M_PI) / 2.0) - asin((t_0 + (cos(phi2) * (cos(phi1) * cos((lambda2 - lambda1)))))));
	}
	return tmp;
}

Java

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 (phi2 <= -1.42e-14) {
		tmp = R * ((Math.PI / 2.0) - Math.asin((t_0 + (Math.cos(phi1) * (Math.cos(phi2) * Math.cos((lambda1 - lambda2)))))));
	} else if (phi2 <= 9.6e-11) {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda2) * Math.cos(lambda1)))));
	} else {
		tmp = R * ((Math.PI / 2.0) - Math.asin((t_0 + (Math.cos(phi2) * (Math.cos(phi1) * Math.cos((lambda2 - lambda1)))))));
	}
	return tmp;
}

Python

[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 phi2 <= -1.42e-14:
		tmp = R * ((math.pi / 2.0) - math.asin((t_0 + (math.cos(phi1) * (math.cos(phi2) * math.cos((lambda1 - lambda2)))))))
	elif phi2 <= 9.6e-11:
		tmp = R * math.acos((math.cos(phi1) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda2) * math.cos(lambda1)))))
	else:
		tmp = R * ((math.pi / 2.0) - math.asin((t_0 + (math.cos(phi2) * (math.cos(phi1) * math.cos((lambda2 - lambda1)))))))
	return tmp

Julia

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 (phi2 <= -1.42e-14)
		tmp = Float64(R * Float64(Float64(pi / 2.0) - asin(Float64(t_0 + Float64(cos(phi1) * Float64(cos(phi2) * cos(Float64(lambda1 - lambda2))))))));
	elseif (phi2 <= 9.6e-11)
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda2) * cos(lambda1))))));
	else
		tmp = Float64(R * Float64(Float64(pi / 2.0) - asin(Float64(t_0 + Float64(cos(phi2) * Float64(cos(phi1) * cos(Float64(lambda2 - lambda1))))))));
	end
	return tmp
end

MATLAB

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 (phi2 <= -1.42e-14)
		tmp = R * ((pi / 2.0) - asin((t_0 + (cos(phi1) * (cos(phi2) * cos((lambda1 - lambda2)))))));
	elseif (phi2 <= 9.6e-11)
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1)))));
	else
		tmp = R * ((pi / 2.0) - asin((t_0 + (cos(phi2) * (cos(phi1) * 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.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.42e-14], N[(R * N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcSin[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 9.6e-11], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcSin[N[(t$95$0 + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.42000000000000004e-14

   1. Initial program 85.6%

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

      1. acos-asin N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      4. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      5. asin-lowering-asin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right), R\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2\right), \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\sin \phi_1, \sin \phi_2\right), \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \sin \phi_2\right), \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\cos \phi_1, \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      15. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right)\right), R\right) \]

      16. --lowering--.f64 85.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right)\right)\right)\right)\right), R\right) \]

   4. Applied egg-rr 85.6%

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

   if -1.42000000000000004e-14 < phi2 < 9.6000000000000005e-11

   1. Initial program 68.9%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 68.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 68.9%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right), R\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right), R\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right), R\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\cos \lambda_2, \cos \lambda_1\right)\right)\right)\right), R\right) \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1\right)\right)\right)\right), R\right) \]

      12. cos-lowering-cos.f64 87.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right)\right)\right), R\right) \]

   7. Applied egg-rr 87.2%

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

   if 9.6000000000000005e-11 < phi2

   1. Initial program 89.1%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right), R\right) \]

      2. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right), R\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right), R\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right) \cdot \cos \lambda_1 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right) \cdot \cos \lambda_1 + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      8. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\mathsf{fma}\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2, \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      9. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right), \cos \lambda_1, \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

   4. Applied egg-rr 98.9%

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

   6. Applied egg-rr 89.2%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.42 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 9.6 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ t_1 := \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\\ \mathbf{if}\;\phi_2 \leq -1.06 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 8 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_1\right)\\ \mathbf{elif}\;\phi_2 \leq 3.45 \cdot 10^{+218}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

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
         (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda2) (cos lambda1)))))
   (if (<= phi2 -1.06e-7)
     t_0
     (if (<= phi2 8e-6)
       (* R (acos (* (cos phi1) t_1)))
       (if (<= phi2 3.45e+218) (* R (acos (* (cos phi2) t_1))) t_0)))))

C

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 = (sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1));
	double tmp;
	if (phi2 <= -1.06e-7) {
		tmp = t_0;
	} else if (phi2 <= 8e-6) {
		tmp = R * acos((cos(phi1) * t_1));
	} else if (phi2 <= 3.45e+218) {
		tmp = R * acos((cos(phi2) * t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

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 = (sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1))
    if (phi2 <= (-1.06d-7)) then
        tmp = t_0
    else if (phi2 <= 8d-6) then
        tmp = r * acos((cos(phi1) * t_1))
    else if (phi2 <= 3.45d+218) then
        tmp = r * acos((cos(phi2) * t_1))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda2) * Math.cos(lambda1));
	double tmp;
	if (phi2 <= -1.06e-7) {
		tmp = t_0;
	} else if (phi2 <= 8e-6) {
		tmp = R * Math.acos((Math.cos(phi1) * t_1));
	} else if (phi2 <= 3.45e+218) {
		tmp = R * Math.acos((Math.cos(phi2) * t_1));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[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.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda2) * math.cos(lambda1))
	tmp = 0
	if phi2 <= -1.06e-7:
		tmp = t_0
	elif phi2 <= 8e-6:
		tmp = R * math.acos((math.cos(phi1) * t_1))
	elif phi2 <= 3.45e+218:
		tmp = R * math.acos((math.cos(phi2) * t_1))
	else:
		tmp = t_0
	return tmp

Julia

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(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda2) * cos(lambda1)))
	tmp = 0.0
	if (phi2 <= -1.06e-7)
		tmp = t_0;
	elseif (phi2 <= 8e-6)
		tmp = Float64(R * acos(Float64(cos(phi1) * t_1)));
	elseif (phi2 <= 3.45e+218)
		tmp = Float64(R * acos(Float64(cos(phi2) * t_1)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

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 = (sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1));
	tmp = 0.0;
	if (phi2 <= -1.06e-7)
		tmp = t_0;
	elseif (phi2 <= 8e-6)
		tmp = R * acos((cos(phi1) * t_1));
	elseif (phi2 <= 3.45e+218)
		tmp = R * acos((cos(phi2) * t_1));
	else
		tmp = 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.
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[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.06e-7], t$95$0, If[LessEqual[phi2, 8e-6], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 3.45e+218], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.06e-7 or 3.4500000000000001e218 < phi2

   1. Initial program 87.1%

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

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \color{blue}{\cos \lambda_1}\right)\right)\right), R\right) \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 64.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 64.8%

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

   if -1.06e-7 < phi2 < 7.99999999999999964e-6

   1. Initial program 69.3%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 69.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 69.0%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right), R\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right), R\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right), R\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\cos \lambda_2, \cos \lambda_1\right)\right)\right)\right), R\right) \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1\right)\right)\right)\right), R\right) \]

      12. cos-lowering-cos.f64 87.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right)\right)\right), R\right) \]

   7. Applied egg-rr 87.0%

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

   if 7.99999999999999964e-6 < phi2 < 3.4500000000000001e218

   1. Initial program 87.4%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right), R\right) \]

      2. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right), R\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right), R\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right) \cdot \cos \lambda_1 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right) \cdot \cos \lambda_1 + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      8. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\mathsf{fma}\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2, \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      9. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right), \cos \lambda_1, \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_2\right) + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right), R\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2 + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2 + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2\right)\right), R\right) \]

      3. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_2, \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\cos \lambda_1, \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right)\right)\right)\right), R\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right)\right)\right)\right), R\right) \]

      12. sin-lowering-sin.f64 62.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

   7. Simplified 62.5%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.06 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 8 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 3.45 \cdot 10^{+218}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -2.3 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(t\_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t\_1\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.15 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t\_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

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 (- lambda1 lambda2))))
   (if (<= phi2 -2.3e-14)
     (* R (- (/ PI 2.0) (asin (+ t_0 (* (cos phi1) (* (cos phi2) t_1))))))
     (if (<= phi2 1.15e-10)
       (*
        R
        (acos
         (*
          (cos phi1)
          (+
           (* (sin lambda1) (sin lambda2))
           (* (cos lambda2) (cos lambda1))))))
       (* (acos (+ t_0 (* (* (cos phi1) (cos phi2)) t_1))) R)))))

C

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((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -2.3e-14) {
		tmp = R * ((((double) M_PI) / 2.0) - asin((t_0 + (cos(phi1) * (cos(phi2) * t_1)))));
	} else if (phi2 <= 1.15e-10) {
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1)))));
	} else {
		tmp = acos((t_0 + ((cos(phi1) * cos(phi2)) * t_1))) * R;
	}
	return tmp;
}

Java

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 t_1 = Math.cos((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -2.3e-14) {
		tmp = R * ((Math.PI / 2.0) - Math.asin((t_0 + (Math.cos(phi1) * (Math.cos(phi2) * t_1)))));
	} else if (phi2 <= 1.15e-10) {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda2) * Math.cos(lambda1)))));
	} else {
		tmp = Math.acos((t_0 + ((Math.cos(phi1) * Math.cos(phi2)) * t_1))) * R;
	}
	return tmp;
}

Python

[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)
	t_1 = math.cos((lambda1 - lambda2))
	tmp = 0
	if phi2 <= -2.3e-14:
		tmp = R * ((math.pi / 2.0) - math.asin((t_0 + (math.cos(phi1) * (math.cos(phi2) * t_1)))))
	elif phi2 <= 1.15e-10:
		tmp = R * math.acos((math.cos(phi1) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda2) * math.cos(lambda1)))))
	else:
		tmp = math.acos((t_0 + ((math.cos(phi1) * math.cos(phi2)) * t_1))) * R
	return tmp

Julia

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 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi2 <= -2.3e-14)
		tmp = Float64(R * Float64(Float64(pi / 2.0) - asin(Float64(t_0 + Float64(cos(phi1) * Float64(cos(phi2) * t_1))))));
	elseif (phi2 <= 1.15e-10)
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda2) * cos(lambda1))))));
	else
		tmp = Float64(acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * t_1))) * R);
	end
	return tmp
end

MATLAB

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);
	t_1 = cos((lambda1 - lambda2));
	tmp = 0.0;
	if (phi2 <= -2.3e-14)
		tmp = R * ((pi / 2.0) - asin((t_0 + (cos(phi1) * (cos(phi2) * t_1)))));
	elseif (phi2 <= 1.15e-10)
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1)))));
	else
		tmp = acos((t_0 + ((cos(phi1) * cos(phi2)) * t_1))) * R;
	end
	tmp_2 = tmp;
end

Wolfram

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[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -2.3e-14], N[(R * N[(N[(Pi / 2.0), $MachinePrecision] - N[ArcSin[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.15e-10], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[ArcCos[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -2.29999999999999998e-14

   1. Initial program 85.6%

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

      1. acos-asin N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2} - \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{2}\right), \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), 2\right), \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      4. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      5. asin-lowering-asin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right), R\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(\left(\sin \phi_1 \cdot \sin \phi_2\right), \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\sin \phi_1, \sin \phi_2\right), \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \sin \phi_2\right), \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right), R\right) \]

      10. associate-*l* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\cos \phi_1, \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      14. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      15. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right)\right), R\right) \]

      16. --lowering--.f64 85.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), 2\right), \mathsf{asin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right)\right)\right)\right)\right), R\right) \]

   4. Applied egg-rr 85.6%

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

   if -2.29999999999999998e-14 < phi2 < 1.15000000000000004e-10

   1. Initial program 68.9%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 68.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 68.9%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right), R\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right), R\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right), R\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\cos \lambda_2, \cos \lambda_1\right)\right)\right)\right), R\right) \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1\right)\right)\right)\right), R\right) \]

      12. cos-lowering-cos.f64 87.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right)\right)\right), R\right) \]

   7. Applied egg-rr 87.2%

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

   if 1.15000000000000004e-10 < phi2

   1. Initial program 89.1%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.3 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \left(\frac{\pi}{2} - \sin^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.15 \cdot 10^{-10}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;\phi_2 \leq -2.3 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_1 + \frac{1}{\frac{1}{t\_0}}\right)\\ \mathbf{elif}\;\phi_2 \leq 8 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(t\_0 + t\_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

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)) (cos (- lambda1 lambda2)))))
   (if (<= phi2 -2.3e-14)
     (* R (acos (+ t_1 (/ 1.0 (/ 1.0 t_0)))))
     (if (<= phi2 8e-11)
       (*
        R
        (acos
         (*
          (cos phi1)
          (+
           (* (sin lambda1) (sin lambda2))
           (* (cos lambda2) (cos lambda1))))))
       (* (acos (+ t_0 t_1)) R)))))

C

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)) * cos((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -2.3e-14) {
		tmp = R * acos((t_1 + (1.0 / (1.0 / t_0))));
	} else if (phi2 <= 8e-11) {
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1)))));
	} else {
		tmp = acos((t_0 + t_1)) * R;
	}
	return tmp;
}

Fortran

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 = sin(phi1) * sin(phi2)
    t_1 = (cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))
    if (phi2 <= (-2.3d-14)) then
        tmp = r * acos((t_1 + (1.0d0 / (1.0d0 / t_0))))
    else if (phi2 <= 8d-11) then
        tmp = r * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1)))))
    else
        tmp = acos((t_0 + t_1)) * r
    end if
    code = tmp
end function

Java

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 t_1 = (Math.cos(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2));
	double tmp;
	if (phi2 <= -2.3e-14) {
		tmp = R * Math.acos((t_1 + (1.0 / (1.0 / t_0))));
	} else if (phi2 <= 8e-11) {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda2) * Math.cos(lambda1)))));
	} else {
		tmp = Math.acos((t_0 + t_1)) * R;
	}
	return tmp;
}

Python

[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)
	t_1 = (math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2))
	tmp = 0
	if phi2 <= -2.3e-14:
		tmp = R * math.acos((t_1 + (1.0 / (1.0 / t_0))))
	elif phi2 <= 8e-11:
		tmp = R * math.acos((math.cos(phi1) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda2) * math.cos(lambda1)))))
	else:
		tmp = math.acos((t_0 + t_1)) * R
	return tmp

Julia

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(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2)))
	tmp = 0.0
	if (phi2 <= -2.3e-14)
		tmp = Float64(R * acos(Float64(t_1 + Float64(1.0 / Float64(1.0 / t_0)))));
	elseif (phi2 <= 8e-11)
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda2) * cos(lambda1))))));
	else
		tmp = Float64(acos(Float64(t_0 + t_1)) * R);
	end
	return tmp
end

MATLAB

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);
	t_1 = (cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2));
	tmp = 0.0;
	if (phi2 <= -2.3e-14)
		tmp = R * acos((t_1 + (1.0 / (1.0 / t_0))));
	elseif (phi2 <= 8e-11)
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1)))));
	else
		tmp = acos((t_0 + t_1)) * R;
	end
	tmp_2 = tmp;
end

Wolfram

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[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -2.3e-14], N[(R * N[ArcCos[N[(t$95$1 + N[(1.0 / N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 8e-11], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[ArcCos[N[(t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -2.29999999999999998e-14

   1. Initial program 85.6%

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

      1. sin-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\left(\frac{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)}{2}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right)\right)\right), R\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{\frac{2}{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)}}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right)\right)\right), R\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{2}{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{\frac{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)}{2}}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. sin-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{\sin \phi_1 \cdot \sin \phi_2}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \left(\sin \phi_1 \cdot \sin \phi_2\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\sin \phi_1, \sin \phi_2\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \sin \phi_2\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 85.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right)\right)\right), R\right) \]

   4. Applied egg-rr 85.6%

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

   if -2.29999999999999998e-14 < phi2 < 7.99999999999999952e-11

   1. Initial program 68.9%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 68.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 68.9%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right), R\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right), R\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right), R\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\cos \lambda_2, \cos \lambda_1\right)\right)\right)\right), R\right) \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1\right)\right)\right)\right), R\right) \]

      12. cos-lowering-cos.f64 87.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right)\right)\right), R\right) \]

   7. Applied egg-rr 87.2%

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

   if 7.99999999999999952e-11 < phi2

   1. Initial program 89.1%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.3 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \frac{1}{\frac{1}{\sin \phi_1 \cdot \sin \phi_2}}\right)\\ \mathbf{elif}\;\phi_2 \leq 8 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 83.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -2.3 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 3.8 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

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)))))
          R)))
   (if (<= phi2 -2.3e-14)
     t_0
     (if (<= phi2 3.8e-9)
       (*
        R
        (acos
         (*
          (cos phi1)
          (+
           (* (sin lambda1) (sin lambda2))
           (* (cos lambda2) (cos lambda1))))))
       t_0))))

C

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))))) * R;
	double tmp;
	if (phi2 <= -2.3e-14) {
		tmp = t_0;
	} else if (phi2 <= 3.8e-9) {
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

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 = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * r
    if (phi2 <= (-2.3d-14)) then
        tmp = t_0
    else if (phi2 <= 3.8d-9) then
        tmp = r * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

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.acos(((Math.sin(phi1) * Math.sin(phi2)) + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2))))) * R;
	double tmp;
	if (phi2 <= -2.3e-14) {
		tmp = t_0;
	} else if (phi2 <= 3.8e-9) {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda2) * Math.cos(lambda1)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.acos(((math.sin(phi1) * math.sin(phi2)) + ((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2))))) * R
	tmp = 0
	if phi2 <= -2.3e-14:
		tmp = t_0
	elif phi2 <= 3.8e-9:
		tmp = R * math.acos((math.cos(phi1) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda2) * math.cos(lambda1)))))
	else:
		tmp = t_0
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) * R)
	tmp = 0.0
	if (phi2 <= -2.3e-14)
		tmp = t_0;
	elseif (phi2 <= 3.8e-9)
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda2) * cos(lambda1))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R;
	tmp = 0.0;
	if (phi2 <= -2.3e-14)
		tmp = t_0;
	elseif (phi2 <= 3.8e-9)
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1)))));
	else
		tmp = 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.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = 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]}, If[LessEqual[phi2, -2.3e-14], t$95$0, If[LessEqual[phi2, 3.8e-9], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.29999999999999998e-14 or 3.80000000000000011e-9 < phi2

   1. Initial program 87.2%

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

   if -2.29999999999999998e-14 < phi2 < 3.80000000000000011e-9

   1. Initial program 68.9%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 68.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 68.9%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right), R\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right), R\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right), R\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\cos \lambda_2, \cos \lambda_1\right)\right)\right)\right), R\right) \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1\right)\right)\right)\right), R\right) \]

      12. cos-lowering-cos.f64 87.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right)\right)\right), R\right) \]

   7. Applied egg-rr 87.2%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.3 \cdot 10^{-14}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 3.8 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -4.4 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_1 + t\_0 \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq 1300000000000:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_1 + t\_0 \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2))) (t_1 (* (sin phi1) (sin phi2))))
   (if (<= lambda1 -4.4e-6)
     (* R (acos (+ t_1 (* t_0 (cos lambda1)))))
     (if (<= lambda1 1300000000000.0)
       (* R (acos (+ t_1 (* t_0 (cos lambda2)))))
       (*
        R
        (acos
         (*
          (cos phi2)
          (+
           (* (sin lambda1) (sin lambda2))
           (* (cos lambda2) (cos lambda1))))))))))

C

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(phi1) * cos(phi2);
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if (lambda1 <= -4.4e-6) {
		tmp = R * acos((t_1 + (t_0 * cos(lambda1))));
	} else if (lambda1 <= 1300000000000.0) {
		tmp = R * acos((t_1 + (t_0 * cos(lambda2))));
	} else {
		tmp = R * acos((cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1)))));
	}
	return tmp;
}

Fortran

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 = cos(phi1) * cos(phi2)
    t_1 = sin(phi1) * sin(phi2)
    if (lambda1 <= (-4.4d-6)) then
        tmp = r * acos((t_1 + (t_0 * cos(lambda1))))
    else if (lambda1 <= 1300000000000.0d0) then
        tmp = r * acos((t_1 + (t_0 * cos(lambda2))))
    else
        tmp = r * acos((cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1)))))
    end if
    code = tmp
end function

Java

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(phi1) * Math.cos(phi2);
	double t_1 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (lambda1 <= -4.4e-6) {
		tmp = R * Math.acos((t_1 + (t_0 * Math.cos(lambda1))));
	} else if (lambda1 <= 1300000000000.0) {
		tmp = R * Math.acos((t_1 + (t_0 * Math.cos(lambda2))));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda2) * Math.cos(lambda1)))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.cos(phi2)
	t_1 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if lambda1 <= -4.4e-6:
		tmp = R * math.acos((t_1 + (t_0 * math.cos(lambda1))))
	elif lambda1 <= 1300000000000.0:
		tmp = R * math.acos((t_1 + (t_0 * math.cos(lambda2))))
	else:
		tmp = R * math.acos((math.cos(phi2) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda2) * math.cos(lambda1)))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda1 <= -4.4e-6)
		tmp = Float64(R * acos(Float64(t_1 + Float64(t_0 * cos(lambda1)))));
	elseif (lambda1 <= 1300000000000.0)
		tmp = Float64(R * acos(Float64(t_1 + Float64(t_0 * cos(lambda2)))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda2) * cos(lambda1))))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * cos(phi2);
	t_1 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if (lambda1 <= -4.4e-6)
		tmp = R * acos((t_1 + (t_0 * cos(lambda1))));
	elseif (lambda1 <= 1300000000000.0)
		tmp = R * acos((t_1 + (t_0 * cos(lambda2))));
	else
		tmp = R * acos((cos(phi2) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * 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.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -4.4e-6], N[(R * N[ArcCos[N[(t$95$1 + N[(t$95$0 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, 1300000000000.0], N[(R * N[ArcCos[N[(t$95$1 + N[(t$95$0 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -4.4000000000000002e-6

   1. Initial program 73.9%

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

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \color{blue}{\cos \lambda_1}\right)\right)\right), R\right) \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 73.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 73.6%

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

   if -4.4000000000000002e-6 < lambda1 < 1.3e12

   1. Initial program 88.2%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right), R\right) \]
   4. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \cos \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 86.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\lambda_2\right)\right)\right)\right), R\right) \]

   5. Simplified 86.9%

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

   if 1.3e12 < lambda1

   1. Initial program 63.1%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right), R\right) \]

      2. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right), R\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right), R\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right) \cdot \cos \lambda_1 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right) \cdot \cos \lambda_1 + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      8. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\mathsf{fma}\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2, \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      9. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right), \cos \lambda_1, \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

   4. Applied egg-rr 98.9%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_2\right) + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right), R\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2 + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2 + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2\right)\right), R\right) \]

      3. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_2, \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\cos \lambda_1, \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right)\right)\right)\right), R\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right)\right)\right)\right), R\right) \]

      12. sin-lowering-sin.f64 58.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

   7. Simplified 58.8%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4.4 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq 1300000000000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\\ \mathbf{if}\;\phi_2 \leq -1.7 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq 2.95 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda2) (cos lambda1)))))
   (if (<= phi2 -1.7e-5)
     (* R (acos (+ (* (sin phi1) (sin phi2)) (* (cos phi1) (cos phi2)))))
     (if (<= phi2 2.95e-5)
       (* R (acos (* (cos phi1) t_0)))
       (* R (acos (* (cos phi2) t_0)))))))

C

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(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1));
	double tmp;
	if (phi2 <= -1.7e-5) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos(phi2))));
	} else if (phi2 <= 2.95e-5) {
		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.
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(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1))
    if (phi2 <= (-1.7d-5)) then
        tmp = r * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos(phi2))))
    else if (phi2 <= 2.95d-5) 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;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda2) * Math.cos(lambda1));
	double tmp;
	if (phi2 <= -1.7e-5) {
		tmp = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + (Math.cos(phi1) * Math.cos(phi2))));
	} else if (phi2 <= 2.95e-5) {
		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])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda2) * math.cos(lambda1))
	tmp = 0
	if phi2 <= -1.7e-5:
		tmp = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + (math.cos(phi1) * math.cos(phi2))))
	elif phi2 <= 2.95e-5:
		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])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda2) * cos(lambda1)))
	tmp = 0.0
	if (phi2 <= -1.7e-5)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * cos(phi2)))));
	elseif (phi2 <= 2.95e-5)
		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])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1));
	tmp = 0.0;
	if (phi2 <= -1.7e-5)
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos(phi2))));
	elseif (phi2 <= 2.95e-5)
		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.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.7e-5], 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, 2.95e-5], 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 3 regimes

2. if phi2 < -1.7e-5

   1. Initial program 85.7%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right), R\right) \]
   4. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \cos \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 62.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\lambda_2\right)\right)\right)\right), R\right) \]

   5. Simplified 62.2%

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

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)}\right)\right), R\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\right), R\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\cos \phi_2, \cos \phi_1\right)\right)\right), R\right) \]

      3. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \phi_1\right)\right)\right), R\right) \]

      4. cos-lowering-cos.f64 39.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\phi_1\right)\right)\right)\right), R\right) \]

   8. Simplified 39.5%

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

   if -1.7e-5 < phi2 < 2.9499999999999999e-5

   1. Initial program 69.3%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 69.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 69.0%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right), R\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right), R\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right), R\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\cos \lambda_2, \cos \lambda_1\right)\right)\right)\right), R\right) \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1\right)\right)\right)\right), R\right) \]

      12. cos-lowering-cos.f64 87.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right)\right)\right), R\right) \]

   7. Applied egg-rr 87.0%

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

   if 2.9499999999999999e-5 < phi2

   1. Initial program 89.0%

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

      1. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right), R\right) \]

      2. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right), R\right) \]

      3. distribute-lft-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right), R\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right) + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      6. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right) \cdot \cos \lambda_1 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right) \cdot \cos \lambda_1 + \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      8. fma-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\mathsf{fma}\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2, \cos \lambda_1, \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

      9. fma-lowering-fma.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_2\right), \cos \lambda_1, \left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)\right), R\right) \]

   4. Applied egg-rr 98.9%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \lambda_2 \cdot \cos \phi_2\right) + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)}\right), R\right) \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2 + \cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2 + \left(\sin \lambda_1 \cdot \sin \lambda_2\right) \cdot \cos \phi_2\right)\right), R\right) \]

      3. distribute-rgt-out N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \phi_2 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_2, \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\left(\cos \lambda_1 \cdot \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\cos \lambda_1, \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      8. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \cos \lambda_2\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \left(\sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right)\right), R\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right)\right)\right)\right), R\right) \]

      11. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right)\right)\right)\right), R\right) \]

      12. sin-lowering-sin.f64 56.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

   7. Simplified 56.1%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.7 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq 2.95 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 65.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq 0.052:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -1.5e-5)
   (* R (acos (+ (* (sin phi1) (sin phi2)) (* (cos phi1) (cos phi2)))))
   (if (<= phi2 0.052)
     (*
      R
      (acos
       (*
        (cos phi1)
        (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda2) (cos lambda1))))))
     (* R (acos (* (cos phi2) (cos (- lambda2 lambda1))))))))

C

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 <= -1.5e-5) {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos(phi2))));
	} else if (phi2 <= 0.052) {
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1)))));
	} 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.
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 <= (-1.5d-5)) then
        tmp = r * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos(phi2))))
    else if (phi2 <= 0.052d0) then
        tmp = r * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1)))))
    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;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -1.5e-5) {
		tmp = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + (Math.cos(phi1) * Math.cos(phi2))));
	} else if (phi2 <= 0.052) {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda2) * Math.cos(lambda1)))));
	} 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])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= -1.5e-5:
		tmp = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + (math.cos(phi1) * math.cos(phi2))))
	elif phi2 <= 0.052:
		tmp = R * math.acos((math.cos(phi1) * ((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda2) * math.cos(lambda1)))))
	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])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -1.5e-5)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * cos(phi2)))));
	elseif (phi2 <= 0.052)
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda2) * cos(lambda1))))));
	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])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= -1.5e-5)
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * cos(phi2))));
	elseif (phi2 <= 0.052)
		tmp = R * acos((cos(phi1) * ((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1)))));
	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.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -1.5e-5], 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, 0.052], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $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 < -1.50000000000000004e-5

   1. Initial program 85.7%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right), R\right) \]
   4. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \cos \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 62.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\lambda_2\right)\right)\right)\right), R\right) \]

   5. Simplified 62.2%

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

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)}\right)\right), R\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\right), R\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\cos \phi_2, \cos \phi_1\right)\right)\right), R\right) \]

      3. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \phi_1\right)\right)\right), R\right) \]

      4. cos-lowering-cos.f64 39.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\phi_1\right)\right)\right)\right), R\right) \]

   8. Simplified 39.5%

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

   if -1.50000000000000004e-5 < phi2 < 0.0519999999999999976

   1. Initial program 69.0%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 68.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 68.7%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\right), R\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right), R\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right), R\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\right), R\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\right), R\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\cos \lambda_2, \cos \lambda_1\right)\right)\right)\right), R\right) \]

      11. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1\right)\right)\right)\right), R\right) \]

      12. cos-lowering-cos.f64 86.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right)\right)\right), R\right) \]

   7. Applied egg-rr 86.5%

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

   if 0.0519999999999999976 < phi2

   1. Initial program 89.9%

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

   3. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 50.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 50.4%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.5 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \mathbf{elif}\;\phi_2 \leq 0.052:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 63.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\frac{1}{\frac{4}{0 \cdot 2 + 2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(t\_0 \cdot 2\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + t\_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2))))
   (if (<= (- lambda1 lambda2) -1e-7)
     (*
      R
      (acos
       (/
        1.0
        (/
         4.0
         (+ (* 0.0 2.0) (* 2.0 (* (cos (- lambda1 lambda2)) (* t_0 2.0))))))))
     (* R (acos (+ (* (sin phi1) (sin phi2)) t_0))))))

C

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(phi1) * cos(phi2);
	double tmp;
	if ((lambda1 - lambda2) <= -1e-7) {
		tmp = R * acos((1.0 / (4.0 / ((0.0 * 2.0) + (2.0 * (cos((lambda1 - lambda2)) * (t_0 * 2.0)))))));
	} else {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + t_0));
	}
	return tmp;
}

Fortran

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(phi1) * cos(phi2)
    if ((lambda1 - lambda2) <= (-1d-7)) then
        tmp = r * acos((1.0d0 / (4.0d0 / ((0.0d0 * 2.0d0) + (2.0d0 * (cos((lambda1 - lambda2)) * (t_0 * 2.0d0)))))))
    else
        tmp = r * acos(((sin(phi1) * sin(phi2)) + t_0))
    end if
    code = tmp
end function

Java

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(phi1) * Math.cos(phi2);
	double tmp;
	if ((lambda1 - lambda2) <= -1e-7) {
		tmp = R * Math.acos((1.0 / (4.0 / ((0.0 * 2.0) + (2.0 * (Math.cos((lambda1 - lambda2)) * (t_0 * 2.0)))))));
	} else {
		tmp = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + t_0));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.cos(phi2)
	tmp = 0
	if (lambda1 - lambda2) <= -1e-7:
		tmp = R * math.acos((1.0 / (4.0 / ((0.0 * 2.0) + (2.0 * (math.cos((lambda1 - lambda2)) * (t_0 * 2.0)))))))
	else:
		tmp = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + t_0))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -1e-7)
		tmp = Float64(R * acos(Float64(1.0 / Float64(4.0 / Float64(Float64(0.0 * 2.0) + Float64(2.0 * Float64(cos(Float64(lambda1 - lambda2)) * Float64(t_0 * 2.0))))))));
	else
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + t_0)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * cos(phi2);
	tmp = 0.0;
	if ((lambda1 - lambda2) <= -1e-7)
		tmp = R * acos((1.0 / (4.0 / ((0.0 * 2.0) + (2.0 * (cos((lambda1 - lambda2)) * (t_0 * 2.0)))))));
	else
		tmp = R * acos(((sin(phi1) * sin(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.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -1e-7], N[(R * N[ArcCos[N[(1.0 / N[(4.0 / N[(N[(0.0 * 2.0), $MachinePrecision] + N[(2.0 * N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -9.9999999999999995e-8

   1. Initial program 82.8%

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

      1. sin-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)}{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)}{2} + \frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)}{2} + \frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}{2}\right)\right), R\right) \]

      4. frac-add N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\left(\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)\right) \cdot 2 + 2 \cdot \left(\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{2 \cdot 2}\right)\right), R\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\frac{1}{\frac{2 \cdot 2}{\left(\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)\right) \cdot 2 + 2 \cdot \left(\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right)\right), R\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{2 \cdot 2}{\left(\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)\right) \cdot 2 + 2 \cdot \left(\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)\right), R\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(2 \cdot 2\right), \left(\left(\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)\right) \cdot 2 + 2 \cdot \left(\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(4, \left(\left(\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)\right) \cdot 2 + 2 \cdot \left(\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(4, \mathsf{+.f64}\left(\left(\left(\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)\right) \cdot 2\right), \left(2 \cdot \left(\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right)\right), R\right) \]

   4. Applied egg-rr 67.9%

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{\frac{4}{\left(\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot 2\right)\right)}}\right)} \cdot R \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(4, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\cos \left(\mathsf{neg}\left(\phi_2\right)\right) - \cos \phi_2\right)}, 2\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), 2\right)\right)\right)\right)\right)\right)\right), R\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(4, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\cos \phi_2 - \cos \phi_2\right), 2\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), 2\right)\right)\right)\right)\right)\right)\right), R\right) \]

      2. +-inverses 67.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(4, \mathsf{+.f64}\left(\mathsf{*.f64}\left(0, 2\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), 2\right)\right)\right)\right)\right)\right)\right), R\right) \]

   7. Simplified 67.9%

      \[\leadsto \cos^{-1} \left(\frac{1}{\frac{4}{\color{blue}{0} \cdot 2 + 2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot 2\right)\right)}}\right) \cdot R \]

   if -9.9999999999999995e-8 < (-.f64 lambda1 lambda2)

   1. Initial program 76.5%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right), R\right) \]
   4. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \cos \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 58.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\lambda_2\right)\right)\right)\right), R\right) \]

   5. Simplified 58.7%

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

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)}\right)\right), R\right) \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \left(\cos \phi_2 \cdot \cos \phi_1\right)\right)\right), R\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\cos \phi_2, \cos \phi_1\right)\right)\right), R\right) \]

      3. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \phi_1\right)\right)\right), R\right) \]

      4. cos-lowering-cos.f64 41.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\phi_1\right)\right)\right)\right), R\right) \]

   8. Simplified 41.1%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\frac{1}{\frac{4}{0 \cdot 2 + 2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot 2\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \cos \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 59.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 3.95:\\ \;\;\;\;R \cdot \cos^{-1} \left(\frac{1}{\frac{4}{0 \cdot 2 + 2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot 2\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 3.95)
   (*
    R
    (acos
     (/
      1.0
      (/
       4.0
       (+
        (* 0.0 2.0)
        (*
         2.0
         (* (cos (- lambda1 lambda2)) (* (* (cos phi1) (cos phi2)) 2.0))))))))
   (*
    R
    (acos
     (+ (* (sin lambda1) (sin lambda2)) (* (cos lambda2) (cos lambda1)))))))

C

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 <= 3.95) {
		tmp = R * acos((1.0 / (4.0 / ((0.0 * 2.0) + (2.0 * (cos((lambda1 - lambda2)) * ((cos(phi1) * cos(phi2)) * 2.0)))))));
	} else {
		tmp = R * acos(((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1))));
	}
	return tmp;
}

Fortran

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 <= 3.95d0) then
        tmp = r * acos((1.0d0 / (4.0d0 / ((0.0d0 * 2.0d0) + (2.0d0 * (cos((lambda1 - lambda2)) * ((cos(phi1) * cos(phi2)) * 2.0d0)))))))
    else
        tmp = r * acos(((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * cos(lambda1))))
    end if
    code = tmp
end function

Java

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 <= 3.95) {
		tmp = R * Math.acos((1.0 / (4.0 / ((0.0 * 2.0) + (2.0 * (Math.cos((lambda1 - lambda2)) * ((Math.cos(phi1) * Math.cos(phi2)) * 2.0)))))));
	} else {
		tmp = R * Math.acos(((Math.sin(lambda1) * Math.sin(lambda2)) + (Math.cos(lambda2) * Math.cos(lambda1))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= 3.95:
		tmp = R * math.acos((1.0 / (4.0 / ((0.0 * 2.0) + (2.0 * (math.cos((lambda1 - lambda2)) * ((math.cos(phi1) * math.cos(phi2)) * 2.0)))))))
	else:
		tmp = R * math.acos(((math.sin(lambda1) * math.sin(lambda2)) + (math.cos(lambda2) * math.cos(lambda1))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= 3.95)
		tmp = Float64(R * acos(Float64(1.0 / Float64(4.0 / Float64(Float64(0.0 * 2.0) + Float64(2.0 * Float64(cos(Float64(lambda1 - lambda2)) * Float64(Float64(cos(phi1) * cos(phi2)) * 2.0))))))));
	else
		tmp = Float64(R * acos(Float64(Float64(sin(lambda1) * sin(lambda2)) + Float64(cos(lambda2) * cos(lambda1)))));
	end
	return tmp
end

MATLAB

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 <= 3.95)
		tmp = R * acos((1.0 / (4.0 / ((0.0 * 2.0) + (2.0 * (cos((lambda1 - lambda2)) * ((cos(phi1) * cos(phi2)) * 2.0)))))));
	else
		tmp = R * acos(((sin(lambda1) * sin(lambda2)) + (cos(lambda2) * 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.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, 3.95], N[(R * N[ArcCos[N[(1.0 / N[(4.0 / N[(N[(0.0 * 2.0), $MachinePrecision] + N[(2.0 * N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < 3.9500000000000002

   1. Initial program 84.1%

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

      1. sin-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)}{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)}{2} + \frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)}{2} + \frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}{2}\right)\right), R\right) \]

      4. frac-add N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\left(\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)\right) \cdot 2 + 2 \cdot \left(\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{2 \cdot 2}\right)\right), R\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\frac{1}{\frac{2 \cdot 2}{\left(\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)\right) \cdot 2 + 2 \cdot \left(\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right)\right), R\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{2 \cdot 2}{\left(\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)\right) \cdot 2 + 2 \cdot \left(\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)\right), R\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(2 \cdot 2\right), \left(\left(\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)\right) \cdot 2 + 2 \cdot \left(\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(4, \left(\left(\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)\right) \cdot 2 + 2 \cdot \left(\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(4, \mathsf{+.f64}\left(\left(\left(\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)\right) \cdot 2\right), \left(2 \cdot \left(\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right)\right), R\right) \]

   4. Applied egg-rr 63.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{\frac{4}{\left(\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot 2\right)\right)}}\right)} \cdot R \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(4, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\cos \left(\mathsf{neg}\left(\phi_2\right)\right) - \cos \phi_2\right)}, 2\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), 2\right)\right)\right)\right)\right)\right)\right), R\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(4, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\cos \phi_2 - \cos \phi_2\right), 2\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), 2\right)\right)\right)\right)\right)\right)\right), R\right) \]

      2. +-inverses 63.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(4, \mathsf{+.f64}\left(\mathsf{*.f64}\left(0, 2\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), 2\right)\right)\right)\right)\right)\right)\right), R\right) \]

   7. Simplified 63.4%

      \[\leadsto \cos^{-1} \left(\frac{1}{\frac{4}{\color{blue}{0} \cdot 2 + 2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot 2\right)\right)}}\right) \cdot R \]

   if 3.9500000000000002 < lambda1

   1. Initial program 63.0%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 40.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 40.3%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right), R\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)\right), R\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      5. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right)\right), R\right) \]

      6. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)\right), R\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right), R\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      10. --lowering--.f64 29.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   8. Simplified 29.2%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right), R\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right)\right), R\right) \]

      4. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_2 - \lambda_1\right)\right), R\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      6. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_2 + \left(0 - \lambda_1\right)\right)\right), R\right) \]

      7. cos-PI/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_2 + \left(\cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right) - \lambda_1\right)\right)\right), R\right) \]

      8. associate-+r- N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\left(\lambda_2 + \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) - \lambda_1\right)\right), R\right) \]

      9. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\cos \left(\lambda_2 + \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \lambda_1 + \sin \left(\lambda_2 + \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \lambda_1\right)\right), R\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\left(\cos \left(\lambda_2 + \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \cos \lambda_1\right), \left(\sin \left(\lambda_2 + \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\cos \left(\lambda_2 + \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right), \cos \lambda_1\right), \left(\sin \left(\lambda_2 + \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_2 + \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right), \cos \lambda_1\right), \left(\sin \left(\lambda_2 + \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(\lambda_2, \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right), \cos \lambda_1\right), \left(\sin \left(\lambda_2 + \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      14. cos-PI/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(\lambda_2, 0\right)\right), \cos \lambda_1\right), \left(\sin \left(\lambda_2 + \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      15. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(\lambda_2, 0\right)\right), \mathsf{cos.f64}\left(\lambda_1\right)\right), \left(\sin \left(\lambda_2 + \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right) \cdot \sin \lambda_1\right)\right)\right), R\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{+.f64}\left(\lambda_2, 0\right)\right), \mathsf{cos.f64}\left(\lambda_1\right)\right), \mathsf{*.f64}\left(\sin \left(\lambda_2 + \cos \left(\frac{\mathsf{PI}\left(\right)}{2}\right)\right), \sin \lambda_1\right)\right)\right), R\right) \]

   10. Applied egg-rr 39.4%

       \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 + 0\right) \cdot \cos \lambda_1 + \sin \left(\lambda_2 + 0\right) \cdot \sin \lambda_1\right)} \cdot R \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\sin \left(\lambda_2 + 0\right) \cdot \sin \lambda_1 + \cos \left(\lambda_2 + 0\right) \cdot \cos \lambda_1\right)\right), R\right) \]

       2. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\left(\sin \left(\lambda_2 + 0\right) \cdot \sin \lambda_1\right), \left(\cos \left(\lambda_2 + 0\right) \cdot \cos \lambda_1\right)\right)\right), R\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\left(\sin \lambda_1 \cdot \sin \left(\lambda_2 + 0\right)\right), \left(\cos \left(\lambda_2 + 0\right) \cdot \cos \lambda_1\right)\right)\right), R\right) \]

       4. +-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\left(\sin \lambda_1 \cdot \sin \lambda_2\right), \left(\cos \left(\lambda_2 + 0\right) \cdot \cos \lambda_1\right)\right)\right), R\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\sin \lambda_1, \sin \lambda_2\right), \left(\cos \left(\lambda_2 + 0\right) \cdot \cos \lambda_1\right)\right)\right), R\right) \]

       6. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \sin \lambda_2\right), \left(\cos \left(\lambda_2 + 0\right) \cdot \cos \lambda_1\right)\right)\right), R\right) \]

       7. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \left(\lambda_2 + 0\right) \cdot \cos \lambda_1\right)\right)\right), R\right) \]

       8. +-rgt-identity N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right), R\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\cos \lambda_2, \cos \lambda_1\right)\right)\right), R\right) \]

       10. cos-lowering-cos.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \lambda_1\right)\right)\right), R\right) \]

       11. cos-lowering-cos.f64 39.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\lambda_1\right), \mathsf{sin.f64}\left(\lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right)\right), R\right) \]

   12. Applied egg-rr 39.4%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq 3.95:\\ \;\;\;\;R \cdot \cos^{-1} \left(\frac{1}{\frac{4}{0 \cdot 2 + 2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot 2\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2 \cdot \cos \lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 58.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ R \cdot \cos^{-1} \left(\frac{1}{\frac{4}{0 \cdot 2 + 2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot 2\right)\right)}}\right) \end{array} \]

FPCore

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
   (/
    1.0
    (/
     4.0
     (+
      (* 0.0 2.0)
      (*
       2.0
       (* (cos (- lambda1 lambda2)) (* (* (cos phi1) (cos phi2)) 2.0)))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * acos((1.0 / (4.0 / ((0.0 * 2.0) + (2.0 * (cos((lambda1 - lambda2)) * ((cos(phi1) * cos(phi2)) * 2.0)))))));
}

Fortran

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((1.0d0 / (4.0d0 / ((0.0d0 * 2.0d0) + (2.0d0 * (cos((lambda1 - lambda2)) * ((cos(phi1) * cos(phi2)) * 2.0d0)))))))
end function

Java

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((1.0 / (4.0 / ((0.0 * 2.0) + (2.0 * (Math.cos((lambda1 - lambda2)) * ((Math.cos(phi1) * Math.cos(phi2)) * 2.0)))))));
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.acos((1.0 / (4.0 / ((0.0 * 2.0) + (2.0 * (math.cos((lambda1 - lambda2)) * ((math.cos(phi1) * math.cos(phi2)) * 2.0)))))))

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * acos(Float64(1.0 / Float64(4.0 / Float64(Float64(0.0 * 2.0) + Float64(2.0 * Float64(cos(Float64(lambda1 - lambda2)) * Float64(Float64(cos(phi1) * cos(phi2)) * 2.0))))))))
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * acos((1.0 / (4.0 / ((0.0 * 2.0) + (2.0 * (cos((lambda1 - lambda2)) * ((cos(phi1) * cos(phi2)) * 2.0)))))));
end

Wolfram

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[(1.0 / N[(4.0 / N[(N[(0.0 * 2.0), $MachinePrecision] + N[(2.0 * N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.9%

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

   1. sin-mult N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)}{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

   2. cos-mult N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)}{2} + \frac{\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)}{2} + \frac{\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}{2}\right)\right), R\right) \]

   4. frac-add N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\frac{\left(\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)\right) \cdot 2 + 2 \cdot \left(\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{2 \cdot 2}\right)\right), R\right) \]

   5. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\left(\frac{1}{\frac{2 \cdot 2}{\left(\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)\right) \cdot 2 + 2 \cdot \left(\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}}\right)\right), R\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{2 \cdot 2}{\left(\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)\right) \cdot 2 + 2 \cdot \left(\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right)\right)\right), R\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(2 \cdot 2\right), \left(\left(\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)\right) \cdot 2 + 2 \cdot \left(\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(4, \left(\left(\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)\right) \cdot 2 + 2 \cdot \left(\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(4, \mathsf{+.f64}\left(\left(\left(\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)\right) \cdot 2\right), \left(2 \cdot \left(\left(\cos \left(\phi_1 + \phi_2\right) + \cos \left(\phi_1 - \phi_2\right)\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right)\right), R\right) \]

4. Applied egg-rr 60.4%

   \[\leadsto \cos^{-1} \color{blue}{\left(\frac{1}{\frac{4}{\left(\cos \left(\phi_1 - \phi_2\right) - \cos \left(\phi_1 + \phi_2\right)\right) \cdot 2 + 2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot 2\right)\right)}}\right)} \cdot R \]

5. Taylor expanded in phi1 around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(4, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\cos \left(\mathsf{neg}\left(\phi_2\right)\right) - \cos \phi_2\right)}, 2\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), 2\right)\right)\right)\right)\right)\right)\right), R\right) \]
6. Step-by-step derivation

   1. cos-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(4, \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\cos \phi_2 - \cos \phi_2\right), 2\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), 2\right)\right)\right)\right)\right)\right)\right), R\right) \]

   2. +-inverses 60.4%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(4, \mathsf{+.f64}\left(\mathsf{*.f64}\left(0, 2\right), \mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), 2\right)\right)\right)\right)\right)\right)\right), R\right) \]

7. Simplified 60.4%

   \[\leadsto \cos^{-1} \left(\frac{1}{\frac{4}{\color{blue}{0} \cdot 2 + 2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot 2\right)\right)}}\right) \cdot R \]

8. Final simplification 60.4%

   \[\leadsto R \cdot \cos^{-1} \left(\frac{1}{\frac{4}{0 \cdot 2 + 2 \cdot \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot 2\right)\right)}}\right) \]
9. Add Preprocessing

Alternative 20: 42.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.7 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq 2.9 \cdot 10^{-273}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

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 -1.7e-5)
   (* R (acos (* (cos phi1) (cos lambda1))))
   (if (<= lambda1 2.9e-273)
     (* R (acos (* (cos phi1) (cos lambda2))))
     (* R (acos (* (cos phi2) (cos lambda2)))))))

C

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 <= -1.7e-5) {
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	} else if (lambda1 <= 2.9e-273) {
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	} else {
		tmp = R * acos((cos(phi2) * cos(lambda2)));
	}
	return tmp;
}

Fortran

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 <= (-1.7d-5)) then
        tmp = r * acos((cos(phi1) * cos(lambda1)))
    else if (lambda1 <= 2.9d-273) then
        tmp = r * acos((cos(phi1) * cos(lambda2)))
    else
        tmp = r * acos((cos(phi2) * cos(lambda2)))
    end if
    code = tmp
end function

Java

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 <= -1.7e-5) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else if (lambda1 <= 2.9e-273) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda2)));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos(lambda2)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -1.7e-5:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda1)))
	elif lambda1 <= 2.9e-273:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda2)))
	else:
		tmp = R * math.acos((math.cos(phi2) * math.cos(lambda2)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.7e-5)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda1))));
	elseif (lambda1 <= 2.9e-273)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda2))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(lambda2))));
	end
	return tmp
end

MATLAB

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 <= -1.7e-5)
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	elseif (lambda1 <= 2.9e-273)
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	else
		tmp = R * acos((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.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -1.7e-5], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, 2.9e-273], 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[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -1.7e-5

   1. Initial program 73.9%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 47.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 47.1%

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

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right), R\right) \]
   7. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \lambda_1\right)\right), R\right) \]

      2. cos-lowering-cos.f64 46.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right), R\right) \]

   8. Simplified 46.6%

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

   if -1.7e-5 < lambda1 < 2.89999999999999986e-273

   1. Initial program 92.2%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 41.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 41.7%

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

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \color{blue}{\cos \lambda_2}\right)\right), R\right) \]
   7. Step-by-step derivation

      1. cos-lowering-cos.f64 41.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\lambda_2\right)\right)\right), R\right) \]

   8. Simplified 41.7%

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

   if 2.89999999999999986e-273 < lambda1

   1. Initial program 73.3%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right), R\right) \]
   4. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \cos \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 49.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\lambda_2\right)\right)\right)\right), R\right) \]

   5. Simplified 49.9%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \lambda_2 \cdot \cos \phi_2\right)}\right), R\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \lambda_2, \cos \phi_2\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \phi_2\right)\right), R\right) \]

      3. cos-lowering-cos.f64 29.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\phi_2\right)\right)\right), R\right) \]

   8. Simplified 29.8%

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

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.7 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq 2.9 \cdot 10^{-273}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 50.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 132000000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi2 132000000.0)
     (* R (acos (* (cos phi1) t_0)))
     (* R (acos (* (cos phi2) t_0))))))

C

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 <= 132000000.0) {
		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.
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 <= 132000000.0d0) 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;
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 <= 132000000.0) {
		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])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	tmp = 0
	if phi2 <= 132000000.0:
		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])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi2 <= 132000000.0)
		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])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	tmp = 0.0;
	if (phi2 <= 132000000.0)
		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.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 132000000.0], 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 < 1.32e8

   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. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 49.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 49.4%

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

   if 1.32e8 < phi2

   1. Initial program 89.9%

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

   3. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_2, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 50.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 50.4%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 132000000:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 48.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.052:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

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 0.052)
   (* R (acos (* (cos phi1) (cos (- lambda2 lambda1)))))
   (* R (acos (* (cos phi2) (cos lambda2))))))

C

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 <= 0.052) {
		tmp = R * acos((cos(phi1) * cos((lambda2 - lambda1))));
	} else {
		tmp = R * acos((cos(phi2) * cos(lambda2)));
	}
	return tmp;
}

Fortran

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 <= 0.052d0) then
        tmp = r * acos((cos(phi1) * cos((lambda2 - lambda1))))
    else
        tmp = r * acos((cos(phi2) * cos(lambda2)))
    end if
    code = tmp
end function

Java

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 <= 0.052) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos((lambda2 - lambda1))));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos(lambda2)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 0.052:
		tmp = R * math.acos((math.cos(phi1) * math.cos((lambda2 - lambda1))))
	else:
		tmp = R * math.acos((math.cos(phi2) * math.cos(lambda2)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 0.052)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(Float64(lambda2 - lambda1)))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(lambda2))));
	end
	return tmp
end

MATLAB

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 <= 0.052)
		tmp = R * acos((cos(phi1) * cos((lambda2 - lambda1))));
	else
		tmp = R * acos((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.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 0.052], 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[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.0519999999999999976

   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. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 49.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 49.4%

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

   if 0.0519999999999999976 < phi2

   1. Initial program 89.9%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right), R\right) \]
   4. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \cos \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 70.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\lambda_2\right)\right)\right)\right), R\right) \]

   5. Simplified 70.2%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \lambda_2 \cdot \cos \phi_2\right)}\right), R\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \lambda_2, \cos \phi_2\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \phi_2\right)\right), R\right) \]

      3. cos-lowering-cos.f64 40.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\phi_2\right)\right)\right), R\right) \]

   8. Simplified 40.6%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.052:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 42.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 0.000105:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 0.000105)
   (* R (acos (* (cos phi1) (cos lambda1))))
   (* R (acos (* (cos phi2) (cos lambda2))))))

C

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 <= 0.000105) {
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	} else {
		tmp = R * acos((cos(phi2) * cos(lambda2)));
	}
	return tmp;
}

Fortran

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 (lambda2 <= 0.000105d0) then
        tmp = r * acos((cos(phi1) * cos(lambda1)))
    else
        tmp = r * acos((cos(phi2) * cos(lambda2)))
    end if
    code = tmp
end function

Java

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 <= 0.000105) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos(lambda2)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 0.000105:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R * math.acos((math.cos(phi2) * math.cos(lambda2)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 0.000105)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda1))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(lambda2))));
	end
	return tmp
end

MATLAB

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 (lambda2 <= 0.000105)
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	else
		tmp = R * acos((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.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 0.000105], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.05e-4

   1. Initial program 80.9%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 42.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 42.0%

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

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right), R\right) \]
   7. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \lambda_1\right)\right), R\right) \]

      2. cos-lowering-cos.f64 35.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\lambda_1\right)\right)\right), R\right) \]

   8. Simplified 35.1%

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

   if 1.05e-4 < lambda2

   1. Initial program 71.6%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right), R\right) \]
   4. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \cos \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 71.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\lambda_2\right)\right)\right)\right), R\right) \]

   5. Simplified 71.1%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \lambda_2 \cdot \cos \phi_2\right)}\right), R\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \lambda_2, \cos \phi_2\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \phi_2\right)\right), R\right) \]

      3. cos-lowering-cos.f64 51.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\phi_2\right)\right)\right), R\right) \]

   8. Simplified 51.4%

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

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 0.000105:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 34.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0039:\\ \;\;\;\;R \cdot \cos^{-1} \cos \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -0.0039)
   (* R (acos (cos phi1)))
   (* R (acos (* (cos phi2) (cos lambda2))))))

C

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 <= -0.0039) {
		tmp = R * acos(cos(phi1));
	} else {
		tmp = R * acos((cos(phi2) * cos(lambda2)));
	}
	return tmp;
}

Fortran

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 <= (-0.0039d0)) then
        tmp = r * acos(cos(phi1))
    else
        tmp = r * acos((cos(phi2) * cos(lambda2)))
    end if
    code = tmp
end function

Java

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 <= -0.0039) {
		tmp = R * Math.acos(Math.cos(phi1));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * Math.cos(lambda2)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -0.0039:
		tmp = R * math.acos(math.cos(phi1))
	else:
		tmp = R * math.acos((math.cos(phi2) * math.cos(lambda2)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -0.0039)
		tmp = Float64(R * acos(cos(phi1)));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * cos(lambda2))));
	end
	return tmp
end

MATLAB

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 <= -0.0039)
		tmp = R * acos(cos(phi1));
	else
		tmp = R * acos((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.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -0.0039], N[(R * N[ArcCos[N[Cos[phi1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.0038999999999999998

   1. Initial program 84.8%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 45.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 45.1%

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

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \color{blue}{\left(\cos \left(\mathsf{neg}\left(\lambda_1\right)\right) + -1 \cdot \left(\lambda_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)}\right)\right), R\right) \]
   7. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 + -1 \cdot \left(\lambda_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\cos \lambda_1, \left(-1 \cdot \left(\lambda_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right)\right), R\right) \]

      3. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \left(-1 \cdot \left(\lambda_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right)\right), R\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \left(\mathsf{neg}\left(\lambda_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right)\right), R\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \left(\lambda_2 \cdot \left(\mathsf{neg}\left(\sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right)\right)\right), R\right) \]

      6. sin-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \left(\lambda_2 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin \lambda_1\right)\right)\right)\right)\right)\right)\right)\right), R\right) \]

      7. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \left(\lambda_2 \cdot \sin \lambda_1\right)\right)\right)\right), R\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\lambda_2, \sin \lambda_1\right)\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 25.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\lambda_2, \mathsf{sin.f64}\left(\lambda_1\right)\right)\right)\right)\right), R\right) \]

   8. Simplified 25.0%

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

   9. Taylor expanded in lambda1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \phi_1}\right), R\right) \]
   10. Step-by-step derivation

       1. cos-lowering-cos.f64 25.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\phi_1\right)\right), R\right) \]

   11. Simplified 25.3%

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

   if -0.0038999999999999998 < phi1

   1. Initial program 76.3%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right)\right), R\right) \]
   4. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \cos \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 50.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\phi_1\right), \mathsf{sin.f64}\left(\phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\lambda_2\right)\right)\right)\right), R\right) \]

   5. Simplified 50.2%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \lambda_2 \cdot \cos \phi_2\right)}\right), R\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \lambda_2, \cos \phi_2\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \cos \phi_2\right)\right), R\right) \]

      3. cos-lowering-cos.f64 38.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right), \mathsf{cos.f64}\left(\phi_2\right)\right)\right), R\right) \]

   8. Simplified 38.4%

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

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0039:\\ \;\;\;\;R \cdot \cos^{-1} \cos \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 27.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.0155:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{elif}\;\lambda_1 \leq -8.2 \cdot 10^{-224}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -0.0155)
   (* R (acos (cos lambda1)))
   (if (<= lambda1 -8.2e-224)
     (* R (acos (cos phi1)))
     (* R (acos (cos lambda2))))))

C

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 <= -0.0155) {
		tmp = R * acos(cos(lambda1));
	} else if (lambda1 <= -8.2e-224) {
		tmp = R * acos(cos(phi1));
	} 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.
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 <= (-0.0155d0)) then
        tmp = r * acos(cos(lambda1))
    else if (lambda1 <= (-8.2d-224)) then
        tmp = r * acos(cos(phi1))
    else
        tmp = r * acos(cos(lambda2))
    end if
    code = tmp
end function

Java

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 <= -0.0155) {
		tmp = R * Math.acos(Math.cos(lambda1));
	} else if (lambda1 <= -8.2e-224) {
		tmp = R * Math.acos(Math.cos(phi1));
	} else {
		tmp = R * Math.acos(Math.cos(lambda2));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -0.0155:
		tmp = R * math.acos(math.cos(lambda1))
	elif lambda1 <= -8.2e-224:
		tmp = R * math.acos(math.cos(phi1))
	else:
		tmp = R * math.acos(math.cos(lambda2))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -0.0155)
		tmp = Float64(R * acos(cos(lambda1)));
	elseif (lambda1 <= -8.2e-224)
		tmp = Float64(R * acos(cos(phi1)));
	else
		tmp = Float64(R * acos(cos(lambda2)));
	end
	return tmp
end

MATLAB

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 <= -0.0155)
		tmp = R * acos(cos(lambda1));
	elseif (lambda1 <= -8.2e-224)
		tmp = R * acos(cos(phi1));
	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.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -0.0155], N[(R * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, -8.2e-224], N[(R * N[ArcCos[N[Cos[phi1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[Cos[lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -0.0155

   1. Initial program 73.7%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 48.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 48.5%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right), R\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)\right), R\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      5. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right)\right), R\right) \]

      6. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)\right), R\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right), R\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      10. --lowering--.f64 33.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   8. Simplified 33.2%

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

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \lambda_1}\right), R\right) \]
   10. Step-by-step derivation

       1. cos-lowering-cos.f64 33.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right)\right), R\right) \]

   11. Simplified 33.1%

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

   if -0.0155 < lambda1 < -8.19999999999999972e-224

   1. Initial program 93.1%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 37.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 37.5%

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

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \color{blue}{\left(\cos \left(\mathsf{neg}\left(\lambda_1\right)\right) + -1 \cdot \left(\lambda_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)}\right)\right), R\right) \]
   7. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 + -1 \cdot \left(\lambda_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\cos \lambda_1, \left(-1 \cdot \left(\lambda_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right)\right), R\right) \]

      3. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \left(-1 \cdot \left(\lambda_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right)\right), R\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \left(\mathsf{neg}\left(\lambda_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right)\right), R\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \left(\lambda_2 \cdot \left(\mathsf{neg}\left(\sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right)\right)\right), R\right) \]

      6. sin-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \left(\lambda_2 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin \lambda_1\right)\right)\right)\right)\right)\right)\right)\right), R\right) \]

      7. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \left(\lambda_2 \cdot \sin \lambda_1\right)\right)\right)\right), R\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\lambda_2, \sin \lambda_1\right)\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 16.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\lambda_2, \mathsf{sin.f64}\left(\lambda_1\right)\right)\right)\right)\right), R\right) \]

   8. Simplified 16.6%

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

   9. Taylor expanded in lambda1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \phi_1}\right), R\right) \]
   10. Step-by-step derivation

       1. cos-lowering-cos.f64 20.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\phi_1\right)\right), R\right) \]

   11. Simplified 20.5%

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

   if -8.19999999999999972e-224 < lambda1

   1. Initial program 76.0%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 40.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 40.9%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right), R\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)\right), R\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      5. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right)\right), R\right) \]

      6. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)\right), R\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right), R\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      10. --lowering--.f64 26.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   8. Simplified 26.1%

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

   9. Taylor expanded in lambda1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right), R\right) \]
   10. Step-by-step derivation

       1. cos-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \lambda_2\right), R\right) \]

       2. cos-lowering-cos.f64 18.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right)\right), R\right) \]

   11. Simplified 18.5%

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

4. Final simplification 22.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -0.0155:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{elif}\;\lambda_1 \leq -8.2 \cdot 10^{-224}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 32.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -0.02:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \phi_1\\ \end{array} \end{array} \]

FPCore

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 lambda2) -0.02)
   (* R (acos (cos (- lambda1 lambda2))))
   (* R (acos (cos phi1)))))

C

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 - lambda2) <= -0.02) {
		tmp = R * acos(cos((lambda1 - lambda2)));
	} else {
		tmp = R * acos(cos(phi1));
	}
	return tmp;
}

Fortran

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 - lambda2) <= (-0.02d0)) then
        tmp = r * acos(cos((lambda1 - lambda2)))
    else
        tmp = r * acos(cos(phi1))
    end if
    code = tmp
end function

Java

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 - lambda2) <= -0.02) {
		tmp = R * Math.acos(Math.cos((lambda1 - lambda2)));
	} else {
		tmp = R * Math.acos(Math.cos(phi1));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (lambda1 - lambda2) <= -0.02:
		tmp = R * math.acos(math.cos((lambda1 - lambda2)))
	else:
		tmp = R * math.acos(math.cos(phi1))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -0.02)
		tmp = Float64(R * acos(cos(Float64(lambda1 - lambda2))));
	else
		tmp = Float64(R * acos(cos(phi1)));
	end
	return tmp
end

MATLAB

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 - lambda2) <= -0.02)
		tmp = R * acos(cos((lambda1 - lambda2)));
	else
		tmp = R * acos(cos(phi1));
	end
	tmp_2 = tmp;
end

Wolfram

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[(lambda1 - lambda2), $MachinePrecision], -0.02], N[(R * N[ArcCos[N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[Cos[phi1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -0.0200000000000000004

   1. Initial program 82.3%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 50.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 50.2%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right), R\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)\right), R\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      5. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right)\right), R\right) \]

      6. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)\right), R\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right), R\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      10. --lowering--.f64 36.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   8. Simplified 36.1%

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

   if -0.0200000000000000004 < (-.f64 lambda1 lambda2)

   1. Initial program 76.9%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 37.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 37.2%

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

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \color{blue}{\left(\cos \left(\mathsf{neg}\left(\lambda_1\right)\right) + -1 \cdot \left(\lambda_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)}\right)\right), R\right) \]
   7. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \left(\cos \lambda_1 + -1 \cdot \left(\lambda_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\cos \lambda_1, \left(-1 \cdot \left(\lambda_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right)\right), R\right) \]

      3. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \left(-1 \cdot \left(\lambda_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right)\right), R\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \left(\mathsf{neg}\left(\lambda_2 \cdot \sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right)\right), R\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \left(\lambda_2 \cdot \left(\mathsf{neg}\left(\sin \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right)\right)\right), R\right) \]

      6. sin-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \left(\lambda_2 \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sin \lambda_1\right)\right)\right)\right)\right)\right)\right)\right), R\right) \]

      7. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \left(\lambda_2 \cdot \sin \lambda_1\right)\right)\right)\right), R\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\lambda_2, \sin \lambda_1\right)\right)\right)\right), R\right) \]

      9. sin-lowering-sin.f64 23.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right), \mathsf{*.f64}\left(\lambda_2, \mathsf{sin.f64}\left(\lambda_1\right)\right)\right)\right)\right), R\right) \]

   8. Simplified 23.3%

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

   9. Taylor expanded in lambda1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \phi_1}\right), R\right) \]
   10. Step-by-step derivation

       1. cos-lowering-cos.f64 17.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\phi_1\right)\right), R\right) \]

   11. Simplified 17.7%

       \[\leadsto \cos^{-1} \color{blue}{\cos \phi_1} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -0.02:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \phi_1\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 20.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -100000000:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

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 lambda2) -100000000.0)
   (* R (acos (cos lambda1)))
   (* (- lambda2 lambda1) R)))

C

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 - lambda2) <= -100000000.0) {
		tmp = R * acos(cos(lambda1));
	} else {
		tmp = (lambda2 - lambda1) * R;
	}
	return tmp;
}

Fortran

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 - lambda2) <= (-100000000.0d0)) then
        tmp = r * acos(cos(lambda1))
    else
        tmp = (lambda2 - lambda1) * r
    end if
    code = tmp
end function

Java

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 - lambda2) <= -100000000.0) {
		tmp = R * Math.acos(Math.cos(lambda1));
	} else {
		tmp = (lambda2 - lambda1) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (lambda1 - lambda2) <= -100000000.0:
		tmp = R * math.acos(math.cos(lambda1))
	else:
		tmp = (lambda2 - lambda1) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -100000000.0)
		tmp = Float64(R * acos(cos(lambda1)));
	else
		tmp = Float64(Float64(lambda2 - lambda1) * R);
	end
	return tmp
end

MATLAB

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 - lambda2) <= -100000000.0)
		tmp = R * acos(cos(lambda1));
	else
		tmp = (lambda2 - lambda1) * R;
	end
	tmp_2 = tmp;
end

Wolfram

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[(lambda1 - lambda2), $MachinePrecision], -100000000.0], N[(R * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(lambda2 - lambda1), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -1e8

   1. Initial program 82.1%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 49.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 49.4%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right), R\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)\right), R\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      5. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right)\right), R\right) \]

      6. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)\right), R\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right), R\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      10. --lowering--.f64 34.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   8. Simplified 34.9%

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

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \lambda_1}\right), R\right) \]
   10. Step-by-step derivation

       1. cos-lowering-cos.f64 21.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right)\right), R\right) \]

   11. Simplified 21.5%

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

   if -1e8 < (-.f64 lambda1 lambda2)

   1. Initial program 77.1%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 37.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 37.8%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right), R\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)\right), R\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      5. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right)\right), R\right) \]

      6. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)\right), R\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right), R\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      10. --lowering--.f64 20.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   8. Simplified 20.6%

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

      1. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right), R\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right), R\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right), R\right) \]

      4. cos-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \cos \left(\lambda_2 - \lambda_1\right), R\right) \]

      5. acos-cos-s N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\lambda_2 - \lambda_1\right), R\right) \]

      6. --lowering--.f64 6.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right), R\right) \]

   10. Applied egg-rr 6.3%

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

4. Final simplification 11.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -100000000:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 26.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -2.9e-6)
   (* R (acos (cos lambda1)))
   (* R (acos (cos lambda2)))))

C

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 <= -2.9e-6) {
		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.
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 <= (-2.9d-6)) 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;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -2.9e-6) {
		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])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -2.9e-6:
		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])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -2.9e-6)
		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])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -2.9e-6)
		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.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -2.9e-6], 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 < -2.9000000000000002e-6

   1. Initial program 73.9%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 47.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 47.1%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right), R\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)\right), R\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      5. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right)\right), R\right) \]

      6. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)\right), R\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right), R\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      10. --lowering--.f64 32.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   8. Simplified 32.4%

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

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \lambda_1}\right), R\right) \]
   10. Step-by-step derivation

       1. cos-lowering-cos.f64 32.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\lambda_1\right)\right), R\right) \]

   11. Simplified 32.1%

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

   if -2.9000000000000002e-6 < lambda1

   1. Initial program 80.4%

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

   3. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      4. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

      8. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

      9. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

      11. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 40.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

   5. Simplified 40.4%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right), R\right) \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      2. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

      3. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)\right), R\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

      5. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right)\right), R\right) \]

      6. cos-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)\right), R\right) \]

      7. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right), R\right) \]

      8. neg-mul-1 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

      10. --lowering--.f64 23.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

   8. Simplified 23.7%

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

   9. Taylor expanded in lambda1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)}\right), R\right) \]
   10. Step-by-step derivation

       1. cos-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \lambda_2\right), R\right) \]

       2. cos-lowering-cos.f64 18.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\lambda_2\right)\right), R\right) \]

   11. Simplified 18.0%

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

4. Final simplification 21.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.9 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 29: 9.1% accurate, 122.6× speedup

Math

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

FPCore

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 lambda1) R))

C

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

Fortran

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 - lambda1) * r
end function

Java

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 - lambda1) * R;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Taylor expanded in phi2 around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

   2. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

   4. remove-double-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

   6. distribute-neg-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

   8. cos-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

   9. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

   10. mul-1-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

   11. unsub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

   12. --lowering--.f64 41.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

5. Simplified 41.9%

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

6. Taylor expanded in phi1 around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right), R\right) \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

   2. remove-double-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

   3. distribute-neg-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)\right), R\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

   5. neg-mul-1 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right)\right), R\right) \]

   6. cos-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)\right), R\right) \]

   7. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right), R\right) \]

   8. neg-mul-1 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

   10. --lowering--.f64 25.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

8. Simplified 25.7%

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

   1. cos-diff N/A

      \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right), R\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right), R\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \sin \lambda_1\right), R\right) \]

   4. cos-diff N/A

      \[\leadsto \mathsf{*.f64}\left(\cos^{-1} \cos \left(\lambda_2 - \lambda_1\right), R\right) \]

   5. acos-cos-s N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\lambda_2 - \lambda_1\right), R\right) \]

   6. --lowering--.f64 5.9%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right), R\right) \]

10. Applied egg-rr 5.9%

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

Alternative 30: 3.5% accurate, 204.3× speedup

Math

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

FPCore

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 (* lambda1 R))

C

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

Fortran

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 = lambda1 * r
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

3. Taylor expanded in phi2 around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}\right), R\right) \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

   2. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

   4. remove-double-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\left(\mathsf{neg}\left(-1 \cdot \lambda_1\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

   6. distribute-neg-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)\right)\right), R\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right)\right), R\right) \]

   8. cos-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right), R\right) \]

   9. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)\right), R\right) \]

   10. mul-1-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)\right)\right), R\right) \]

   11. unsub-neg N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_2 - \lambda_1\right)\right)\right)\right), R\right) \]

   12. --lowering--.f64 41.9%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_2, \lambda_1\right)\right)\right)\right), R\right) \]

5. Simplified 41.9%

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

6. Taylor expanded in phi1 around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\color{blue}{\cos \left(\lambda_2 - \lambda_1\right)}\right), R\right) \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

   2. remove-double-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right), R\right) \]

   3. distribute-neg-in N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)\right), R\right) \]

   4. +-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

   5. neg-mul-1 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\mathsf{neg}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right)\right), R\right) \]

   6. cos-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)\right), R\right) \]

   7. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + -1 \cdot \lambda_2\right)\right)\right), R\right) \]

   8. neg-mul-1 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)\right), R\right) \]

   9. sub-neg N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\left(\lambda_1 - \lambda_2\right)\right)\right), R\right) \]

   10. --lowering--.f64 25.7%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right), R\right) \]

8. Simplified 25.7%

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

9. Taylor expanded in lambda1 around inf

   \[\leadsto \color{blue}{R \cdot \lambda_1} \]
10. Step-by-step derivation

    1. *-commutative N/A

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

    2. *-lowering-*.f64 5.3%

       \[\leadsto \mathsf{*.f64}\left(\lambda_1, \color{blue}{R}\right) \]

11. Simplified 5.3%

    \[\leadsto \color{blue}{\lambda_1 \cdot R} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024164 
(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))