Spherical law of cosines

Percentage Accurate: 73.8% → 96.5%

Time: 24.5s

Alternatives: 26

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 73.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.5% accurate, 0.3× speedup

Math

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

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (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
        (* t_1 (cos lambda2))
        (cos lambda1)
        (+
         t_0
         (* (cos phi1) (* (sin lambda1) (* (cos phi2) (sin lambda2)))))))))))

C

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

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	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(Float64(t_1 * cos(lambda2)), cos(lambda1), Float64(t_0 + Float64(cos(phi1) * Float64(sin(lambda1) * Float64(cos(phi2) * sin(lambda2))))))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, 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[(t$95$1 * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[(N[Cos[phi2], $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 10.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 10.4%

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

   5. Simplified 10.4%

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

   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 10.4%

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

   8. Simplified 10.4%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
   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 29.0%

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

   10. Applied egg-rr 29.0%

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

   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 77.4%

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

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

4. Final simplification 95.1%

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

Alternative 2: 96.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double 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(((cos(phi1) * (sin(lambda1) * (cos(phi2) * sin(lambda2)))) + (t_0 + (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.
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(((cos(phi1) * (sin(lambda1) * (cos(phi2) * sin(lambda2)))) + (t_0 + (cos(phi1) * (cos(phi2) * (cos(lambda2) * cos(lambda1)))))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double 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(((Math.cos(phi1) * (Math.sin(lambda1) * (Math.cos(phi2) * Math.sin(lambda2)))) + (t_0 + (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])
[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(((math.cos(phi1) * (math.sin(lambda1) * (math.cos(phi2) * math.sin(lambda2)))) + (t_0 + (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])
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(cos(phi1) * Float64(sin(lambda1) * Float64(cos(phi2) * sin(lambda2)))) + Float64(t_0 + 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])){:}
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(((cos(phi1) * (sin(lambda1) * (cos(phi2) * sin(lambda2)))) + (t_0 + (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.
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[(N[Cos[phi1], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 + 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]), $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 10.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 10.4%

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

   5. Simplified 10.4%

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

   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 10.4%

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

   8. Simplified 10.4%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
   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 29.0%

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

   10. Applied egg-rr 29.0%

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

   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 77.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. 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. distribute-rgt-in N/A

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

      3. associate-+r+ N/A

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

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

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

   4. Applied egg-rr 98.9%

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

4. Final simplification 95.0%

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

Alternative 3: 96.5% accurate, 0.4× speedup

Math

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

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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);
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])
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.
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 10.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 10.4%

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

   5. Simplified 10.4%

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

   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 10.4%

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

   8. Simplified 10.4%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
   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 29.0%

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

   10. Applied egg-rr 29.0%

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

   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 77.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. 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 95.0%

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

Alternative 4: 96.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: 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 * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2))))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(phi1) * Math.sin(phi2);
	double 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.cos(lambda2) * Math.cos(lambda1)) + (Math.sin(lambda1) * Math.sin(lambda2))))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * math.sin(phi2)
	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.cos(lambda2) * math.cos(lambda1)) + (math.sin(lambda1) * math.sin(lambda2))))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	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(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda1) * sin(lambda2)))))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(phi1) * sin(phi2);
	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 * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2))))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, 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[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $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 10.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 10.4%

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

   5. Simplified 10.4%

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

   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 10.4%

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

   8. Simplified 10.4%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
   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 29.0%

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

   10. Applied egg-rr 29.0%

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

   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 77.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. 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 95.0%

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

Alternative 5: 84.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \lambda_2 \cdot \cos \lambda_1\\ \mathbf{if}\;\phi_2 \leq -6.4 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\sin \phi_1 \cdot \left(2 + \frac{\frac{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \left(\cos \phi_2 \cdot 2\right)}{\sin \phi_2}}{\sin \phi_1}\right)\right) \cdot \frac{\sin \phi_2}{2}\right)\\ \mathbf{elif}\;\phi_2 \leq 1.6 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(t\_0 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos lambda2) (cos lambda1))))
   (if (<= phi2 -6.4e-8)
     (*
      R
      (acos
       (*
        (*
         (sin phi1)
         (+
          2.0
          (/
           (/
            (* (* (cos phi1) (cos (- lambda2 lambda1))) (* (cos phi2) 2.0))
            (sin phi2))
           (sin phi1))))
        (/ (sin phi2) 2.0))))
     (if (<= phi2 1.6e-6)
       (*
        R
        (acos
         (+
          (* (cos phi1) (+ t_0 (* (sin lambda1) (sin lambda2))))
          (* (sin phi1) phi2))))
       (*
        R
        (acos
         (+ (* (sin phi1) (sin phi2)) (* (cos phi1) (* (cos phi2) t_0)))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(lambda2) * cos(lambda1);
	double tmp;
	if (phi2 <= -6.4e-8) {
		tmp = R * acos(((sin(phi1) * (2.0 + ((((cos(phi1) * cos((lambda2 - lambda1))) * (cos(phi2) * 2.0)) / sin(phi2)) / sin(phi1)))) * (sin(phi2) / 2.0)));
	} else if (phi2 <= 1.6e-6) {
		tmp = R * acos(((cos(phi1) * (t_0 + (sin(lambda1) * sin(lambda2)))) + (sin(phi1) * phi2)));
	} else {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * (cos(phi2) * t_0))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(lambda2) * cos(lambda1)
    if (phi2 <= (-6.4d-8)) then
        tmp = r * acos(((sin(phi1) * (2.0d0 + ((((cos(phi1) * cos((lambda2 - lambda1))) * (cos(phi2) * 2.0d0)) / sin(phi2)) / sin(phi1)))) * (sin(phi2) / 2.0d0)))
    else if (phi2 <= 1.6d-6) then
        tmp = r * acos(((cos(phi1) * (t_0 + (sin(lambda1) * sin(lambda2)))) + (sin(phi1) * phi2)))
    else
        tmp = r * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * (cos(phi2) * t_0))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(lambda2) * Math.cos(lambda1);
	double tmp;
	if (phi2 <= -6.4e-8) {
		tmp = R * Math.acos(((Math.sin(phi1) * (2.0 + ((((Math.cos(phi1) * Math.cos((lambda2 - lambda1))) * (Math.cos(phi2) * 2.0)) / Math.sin(phi2)) / Math.sin(phi1)))) * (Math.sin(phi2) / 2.0)));
	} else if (phi2 <= 1.6e-6) {
		tmp = R * Math.acos(((Math.cos(phi1) * (t_0 + (Math.sin(lambda1) * Math.sin(lambda2)))) + (Math.sin(phi1) * phi2)));
	} else {
		tmp = R * Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + (Math.cos(phi1) * (Math.cos(phi2) * t_0))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(lambda2) * math.cos(lambda1)
	tmp = 0
	if phi2 <= -6.4e-8:
		tmp = R * math.acos(((math.sin(phi1) * (2.0 + ((((math.cos(phi1) * math.cos((lambda2 - lambda1))) * (math.cos(phi2) * 2.0)) / math.sin(phi2)) / math.sin(phi1)))) * (math.sin(phi2) / 2.0)))
	elif phi2 <= 1.6e-6:
		tmp = R * math.acos(((math.cos(phi1) * (t_0 + (math.sin(lambda1) * math.sin(lambda2)))) + (math.sin(phi1) * phi2)))
	else:
		tmp = R * math.acos(((math.sin(phi1) * math.sin(phi2)) + (math.cos(phi1) * (math.cos(phi2) * t_0))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(lambda2) * cos(lambda1))
	tmp = 0.0
	if (phi2 <= -6.4e-8)
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * Float64(2.0 + Float64(Float64(Float64(Float64(cos(phi1) * cos(Float64(lambda2 - lambda1))) * Float64(cos(phi2) * 2.0)) / sin(phi2)) / sin(phi1)))) * Float64(sin(phi2) / 2.0))));
	elseif (phi2 <= 1.6e-6)
		tmp = Float64(R * acos(Float64(Float64(cos(phi1) * Float64(t_0 + Float64(sin(lambda1) * sin(lambda2)))) + Float64(sin(phi1) * phi2))));
	else
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * Float64(cos(phi2) * t_0)))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(lambda2) * cos(lambda1);
	tmp = 0.0;
	if (phi2 <= -6.4e-8)
		tmp = R * acos(((sin(phi1) * (2.0 + ((((cos(phi1) * cos((lambda2 - lambda1))) * (cos(phi2) * 2.0)) / sin(phi2)) / sin(phi1)))) * (sin(phi2) / 2.0)));
	elseif (phi2 <= 1.6e-6)
		tmp = R * acos(((cos(phi1) * (t_0 + (sin(lambda1) * sin(lambda2)))) + (sin(phi1) * phi2)));
	else
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * (cos(phi2) * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -6.4e-8], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[(2.0 + N[(N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[Sin[phi2], $MachinePrecision]), $MachinePrecision] / N[Sin[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sin[phi2], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.6e-6], N[(R * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[(t$95$0 + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -6.4000000000000004e-8

   1. Initial program 77.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. 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-mult N/A

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

      3. associate-*l/ N/A

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

      4. sin-mult N/A

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

      5. clear-num N/A

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

      6. frac-add N/A

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

      7. div-inv N/A

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

      8. clear-num N/A

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

      9. sin-mult N/A

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

   4. Applied egg-rr 77.6%

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

   6. Applied egg-rr 77.7%

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

   if -6.4000000000000004e-8 < phi2 < 1.5999999999999999e-6

   1. Initial program 67.9%

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

      1. +-commutative N/A

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

      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 88.1%

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

   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

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

   7. Simplified 88.1%

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

   if 1.5999999999999999e-6 < phi2

   1. Initial program 80.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. 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 98.8%

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

   5. Taylor expanded in lambda1 around 0

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

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

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

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

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

      3. sin-lowering-sin.f64 80.2%

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

   7. Simplified 80.2%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -6.4 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\sin \phi_1 \cdot \left(2 + \frac{\frac{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot \left(\cos \phi_2 \cdot 2\right)}{\sin \phi_2}}{\sin \phi_1}\right)\right) \cdot \frac{\sin \phi_2}{2}\right)\\ \mathbf{elif}\;\phi_2 \leq 1.6 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \lambda_2 \cdot \cos \lambda_1\\ \mathbf{if}\;\phi_2 \leq -6.4 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \frac{\cos \phi_2}{\frac{1}{\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(t\_0 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos lambda2) (cos lambda1))))
   (if (<= phi2 -6.4e-8)
     (*
      R
      (acos
       (fma
        (sin phi2)
        (sin phi1)
        (/ (cos phi2) (/ 1.0 (* (cos phi1) (cos (- lambda1 lambda2))))))))
     (if (<= phi2 1.35e-6)
       (*
        R
        (acos
         (+
          (* (cos phi1) (+ t_0 (* (sin lambda1) (sin lambda2))))
          (* (sin phi1) phi2))))
       (*
        R
        (acos
         (+ (* (sin phi1) (sin phi2)) (* (cos phi1) (* (cos phi2) t_0)))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(lambda2) * cos(lambda1);
	double tmp;
	if (phi2 <= -6.4e-8) {
		tmp = R * acos(fma(sin(phi2), sin(phi1), (cos(phi2) / (1.0 / (cos(phi1) * cos((lambda1 - lambda2)))))));
	} else if (phi2 <= 1.35e-6) {
		tmp = R * acos(((cos(phi1) * (t_0 + (sin(lambda1) * sin(lambda2)))) + (sin(phi1) * phi2)));
	} else {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * (cos(phi2) * t_0))));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(lambda2) * cos(lambda1))
	tmp = 0.0
	if (phi2 <= -6.4e-8)
		tmp = Float64(R * acos(fma(sin(phi2), sin(phi1), Float64(cos(phi2) / Float64(1.0 / Float64(cos(phi1) * cos(Float64(lambda1 - lambda2))))))));
	elseif (phi2 <= 1.35e-6)
		tmp = Float64(R * acos(Float64(Float64(cos(phi1) * Float64(t_0 + Float64(sin(lambda1) * sin(lambda2)))) + Float64(sin(phi1) * phi2))));
	else
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * Float64(cos(phi2) * t_0)))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -6.4e-8], N[(R * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] / N[(1.0 / N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.35e-6], N[(R * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[(t$95$0 + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -6.4000000000000004e-8

   1. Initial program 77.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. 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. distribute-rgt-in N/A

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

      3. associate-+r+ N/A

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

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

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

   4. Applied egg-rr 99.2%

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

      1. associate-+l+ N/A

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

      2. *-commutative N/A

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

      3. fma-define N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-in N/A

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

   6. Applied egg-rr 77.7%

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

      1. cos-mult N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. clear-num N/A

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

      7. cos-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \left(\frac{1}{\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{/.f64}\left(1, \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 77.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right)\right)\right)\right)\right), R\right) \]

   8. Applied egg-rr 77.7%

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

   if -6.4000000000000004e-8 < phi2 < 1.34999999999999999e-6

   1. Initial program 67.9%

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

      1. +-commutative N/A

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

      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 88.1%

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

   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

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

   7. Simplified 88.1%

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

   if 1.34999999999999999e-6 < phi2

   1. Initial program 80.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. 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 98.8%

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

   5. Taylor expanded in lambda1 around 0

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

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

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

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

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

      3. sin-lowering-sin.f64 80.2%

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

   7. Simplified 80.2%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -6.4 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \frac{\cos \phi_2}{\frac{1}{\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right) + \sin \phi_1 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -5.5 \cdot 10^{-12}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \frac{\cos \phi_2}{\frac{1}{\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.5 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 -5.5e-12)
   (*
    R
    (acos
     (fma
      (sin phi2)
      (sin phi1)
      (/ (cos phi2) (/ 1.0 (* (cos phi1) (cos (- lambda1 lambda2))))))))
   (if (<= phi2 2.5e-7)
     (*
      R
      (acos
       (*
        (cos phi1)
        (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2))))))
     (*
      R
      (acos
       (+
        (* (sin phi1) (sin phi2))
        (* (cos phi1) (* (cos phi2) (* (cos lambda2) (cos lambda1))))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -5.5e-12) {
		tmp = R * acos(fma(sin(phi2), sin(phi1), (cos(phi2) / (1.0 / (cos(phi1) * cos((lambda1 - lambda2)))))));
	} else if (phi2 <= 2.5e-7) {
		tmp = R * acos((cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2)))));
	} else {
		tmp = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * (cos(phi2) * (cos(lambda2) * cos(lambda1))))));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -5.5e-12)
		tmp = Float64(R * acos(fma(sin(phi2), sin(phi1), Float64(cos(phi2) / Float64(1.0 / Float64(cos(phi1) * cos(Float64(lambda1 - lambda2))))))));
	elseif (phi2 <= 2.5e-7)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = Float64(R * acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(cos(phi1) * Float64(cos(phi2) * Float64(cos(lambda2) * cos(lambda1)))))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -5.5e-12], N[(R * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] / N[(1.0 / N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 2.5e-7], 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[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $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 3 regimes

2. if phi2 < -5.5000000000000004e-12

   1. Initial program 77.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. 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. distribute-rgt-in N/A

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

      3. associate-+r+ N/A

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

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

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

   4. Applied egg-rr 99.2%

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

      1. associate-+l+ N/A

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

      2. *-commutative N/A

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

      3. fma-define N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. associate-*r* N/A

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

      9. distribute-lft-in N/A

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

   6. Applied egg-rr 77.7%

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

      1. cos-mult N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. clear-num N/A

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

      7. cos-mult N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \left(\frac{1}{\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{/.f64}\left(1, \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\cos \phi_1, \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right), R\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right)\right)\right), R\right) \]

      12. --lowering--.f64 77.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{fma.f64}\left(\mathsf{sin.f64}\left(\phi_2\right), \mathsf{sin.f64}\left(\phi_1\right), \mathsf{/.f64}\left(\mathsf{cos.f64}\left(\phi_2\right), \mathsf{/.f64}\left(1, \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right)\right)\right)\right)\right), R\right) \]

   8. Applied egg-rr 77.7%

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

   if -5.5000000000000004e-12 < phi2 < 2.49999999999999989e-7

   1. Initial program 67.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 67.8%

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

   5. Simplified 67.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
   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. fma-define N/A

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

      3. *-commutative N/A

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

      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 88.0%

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

   7. Applied egg-rr 88.0%

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

   if 2.49999999999999989e-7 < phi2

   1. Initial program 80.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. 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 98.8%

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

   5. Taylor expanded in lambda1 around 0

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

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

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

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

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

      3. sin-lowering-sin.f64 80.2%

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

   7. Simplified 80.2%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -5.5 \cdot 10^{-12}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \frac{\cos \phi_2}{\frac{1}{\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)}}\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.5 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -2.6 \cdot 10^{-9}:\\ \;\;\;\;\cos^{-1} \left(t\_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 -2.6e-9)
     (*
      (acos (+ t_0 (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
      R)
     (if (<= phi2 6.8e-7)
       (*
        R
        (acos
         (*
          (cos phi1)
          (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2))))))
       (*
        R
        (acos
         (+
          t_0
          (* (cos phi1) (* (cos phi2) (* (cos lambda2) (cos lambda1)))))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi2 <= -2.6e-9) {
		tmp = acos((t_0 + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R;
	} else if (phi2 <= 6.8e-7) {
		tmp = R * acos((cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2)))));
	} else {
		tmp = R * acos((t_0 + (cos(phi1) * (cos(phi2) * (cos(lambda2) * cos(lambda1))))));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi2 <= -2.6e-9)
		tmp = Float64(acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) * R);
	elseif (phi2 <= 6.8e-7)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * Float64(cos(phi2) * Float64(cos(lambda2) * cos(lambda1)))))));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -2.6e-9], N[(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] * R), $MachinePrecision], If[LessEqual[phi2, 6.8e-7], 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[(t$95$0 + 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 3 regimes

2. if phi2 < -2.6000000000000001e-9

   1. Initial program 77.7%

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

   if -2.6000000000000001e-9 < phi2 < 6.79999999999999948e-7

   1. Initial program 67.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 67.8%

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

   5. Simplified 67.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
   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. fma-define N/A

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

      3. *-commutative N/A

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

      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 88.0%

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

   7. Applied egg-rr 88.0%

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

   if 6.79999999999999948e-7 < phi2

   1. Initial program 80.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. 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 98.8%

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

   5. Taylor expanded in lambda1 around 0

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

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

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

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

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

      3. sin-lowering-sin.f64 80.2%

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

   7. Simplified 80.2%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.6 \cdot 10^{-9}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -1.08 \cdot 10^{-10}:\\ \;\;\;\;\cos^{-1} \left(t\_1 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 2.9 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_1 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))) (t_1 (* (sin phi1) (sin phi2))))
   (if (<= phi2 -1.08e-10)
     (* (acos (+ t_1 (* (* (cos phi1) (cos phi2)) t_0))) R)
     (if (<= phi2 2.9e-7)
       (*
        R
        (acos
         (*
          (cos phi1)
          (fma (cos lambda2) (cos lambda1) (* (sin lambda1) (sin lambda2))))))
       (* R (acos (+ t_1 (* (cos phi2) (* (cos phi1) t_0)))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi2 <= -1.08e-10) {
		tmp = acos((t_1 + ((cos(phi1) * cos(phi2)) * t_0))) * R;
	} else if (phi2 <= 2.9e-7) {
		tmp = R * acos((cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda1) * sin(lambda2)))));
	} else {
		tmp = R * acos((t_1 + (cos(phi2) * (cos(phi1) * t_0))));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi2 <= -1.08e-10)
		tmp = Float64(acos(Float64(t_1 + Float64(Float64(cos(phi1) * cos(phi2)) * t_0))) * R);
	elseif (phi2 <= 2.9e-7)
		tmp = Float64(R * acos(Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(phi2) * Float64(cos(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.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -1.08e-10], N[(N[ArcCos[N[(t$95$1 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 2.9e-7], 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[(t$95$1 + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.08000000000000002e-10

   1. Initial program 77.7%

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

   if -1.08000000000000002e-10 < phi2 < 2.8999999999999998e-7

   1. Initial program 67.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 67.8%

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

   5. Simplified 67.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]
   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. fma-define N/A

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

      3. *-commutative N/A

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

      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 88.0%

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

   7. Applied egg-rr 88.0%

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

   if 2.8999999999999998e-7 < phi2

   1. Initial program 80.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      9. cos-lowering-cos.f64 80.1%

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

   4. Applied egg-rr 80.1%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.08 \cdot 10^{-10}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 2.9 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right)\\ t_1 := \cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\\ \mathbf{if}\;\phi_1 \leq -7 \cdot 10^{+65}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_1\right)\\ \mathbf{elif}\;\phi_1 \leq -3.2 \cdot 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq 0.0011:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 lambda2)))))))
        (t_1
         (+ (* (cos lambda2) (cos lambda1)) (* (sin lambda1) (sin lambda2)))))
   (if (<= phi1 -7e+65)
     (* R (acos (* (cos phi1) t_1)))
     (if (<= phi1 -3.2e-13)
       t_0
       (if (<= phi1 0.0011) (* R (acos (* (cos phi2) t_1))) t_0)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * acos(((sin(phi1) * sin(phi2)) + (cos(phi1) * (cos(phi2) * cos(lambda2)))));
	double t_1 = (cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2));
	double tmp;
	if (phi1 <= -7e+65) {
		tmp = R * acos((cos(phi1) * t_1));
	} else if (phi1 <= -3.2e-13) {
		tmp = t_0;
	} else if (phi1 <= 0.0011) {
		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.
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(lambda2)))))
    t_1 = (cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2))
    if (phi1 <= (-7d+65)) then
        tmp = r * acos((cos(phi1) * t_1))
    else if (phi1 <= (-3.2d-13)) then
        tmp = t_0
    else if (phi1 <= 0.0011d0) 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;
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(lambda2)))));
	double t_1 = (Math.cos(lambda2) * Math.cos(lambda1)) + (Math.sin(lambda1) * Math.sin(lambda2));
	double tmp;
	if (phi1 <= -7e+65) {
		tmp = R * Math.acos((Math.cos(phi1) * t_1));
	} else if (phi1 <= -3.2e-13) {
		tmp = t_0;
	} else if (phi1 <= 0.0011) {
		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])
[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(lambda2)))))
	t_1 = (math.cos(lambda2) * math.cos(lambda1)) + (math.sin(lambda1) * math.sin(lambda2))
	tmp = 0
	if phi1 <= -7e+65:
		tmp = R * math.acos((math.cos(phi1) * t_1))
	elif phi1 <= -3.2e-13:
		tmp = t_0
	elif phi1 <= 0.0011:
		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])
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(cos(phi1) * Float64(cos(phi2) * cos(lambda2))))))
	t_1 = Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda1) * sin(lambda2)))
	tmp = 0.0
	if (phi1 <= -7e+65)
		tmp = Float64(R * acos(Float64(cos(phi1) * t_1)));
	elseif (phi1 <= -3.2e-13)
		tmp = t_0;
	elseif (phi1 <= 0.0011)
		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])){:}
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(lambda2)))));
	t_1 = (cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2));
	tmp = 0.0;
	if (phi1 <= -7e+65)
		tmp = R * acos((cos(phi1) * t_1));
	elseif (phi1 <= -3.2e-13)
		tmp = t_0;
	elseif (phi1 <= 0.0011)
		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.
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[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -7e+65], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, -3.2e-13], t$95$0, If[LessEqual[phi1, 0.0011], 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 phi1 < -7.0000000000000002e65

   1. Initial program 82.2%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 56.5%

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

   5. Simplified 56.5%

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

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

      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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

      10. cos-lowering-cos.f64 61.1%

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

   7. Applied egg-rr 61.1%

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

   if -7.0000000000000002e65 < phi1 < -3.2e-13 or 0.00110000000000000007 < phi1

   1. Initial program 80.5%

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

   3. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. cos-neg N/A

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

      11. cos-lowering-cos.f64 55.9%

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

   5. Simplified 55.9%

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

   if -3.2e-13 < phi1 < 0.00110000000000000007

   1. Initial program 65.8%

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

      1. +-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 88.8%

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

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. cos-lowering-cos.f64 87.8%

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

   7. Simplified 87.8%

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

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -7 \cdot 10^{+65}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -3.2 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.0011:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\phi_2 \leq -2.75 \cdot 10^{-9}:\\ \;\;\;\;\cos^{-1} \left(t\_1 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_1 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))) (t_1 (* (sin phi1) (sin phi2))))
   (if (<= phi2 -2.75e-9)
     (* (acos (+ t_1 (* (* (cos phi1) (cos phi2)) t_0))) R)
     (if (<= phi2 2.2e-7)
       (*
        R
        (acos
         (*
          (cos phi1)
          (+
           (* (cos lambda2) (cos lambda1))
           (* (sin lambda1) (sin lambda2))))))
       (* R (acos (+ t_1 (* (cos phi2) (* (cos phi1) t_0)))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi2 <= -2.75e-9) {
		tmp = acos((t_1 + ((cos(phi1) * cos(phi2)) * t_0))) * R;
	} else if (phi2 <= 2.2e-7) {
		tmp = R * acos((cos(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))));
	} else {
		tmp = R * acos((t_1 + (cos(phi2) * (cos(phi1) * t_0))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos((lambda1 - lambda2))
    t_1 = sin(phi1) * sin(phi2)
    if (phi2 <= (-2.75d-9)) then
        tmp = acos((t_1 + ((cos(phi1) * cos(phi2)) * t_0))) * r
    else if (phi2 <= 2.2d-7) then
        tmp = r * acos((cos(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))))
    else
        tmp = r * acos((t_1 + (cos(phi2) * (cos(phi1) * t_0))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double t_1 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (phi2 <= -2.75e-9) {
		tmp = Math.acos((t_1 + ((Math.cos(phi1) * Math.cos(phi2)) * t_0))) * R;
	} else if (phi2 <= 2.2e-7) {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.cos(lambda2) * Math.cos(lambda1)) + (Math.sin(lambda1) * Math.sin(lambda2)))));
	} else {
		tmp = R * Math.acos((t_1 + (Math.cos(phi2) * (Math.cos(phi1) * t_0))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	t_1 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if phi2 <= -2.75e-9:
		tmp = math.acos((t_1 + ((math.cos(phi1) * math.cos(phi2)) * t_0))) * R
	elif phi2 <= 2.2e-7:
		tmp = R * math.acos((math.cos(phi1) * ((math.cos(lambda2) * math.cos(lambda1)) + (math.sin(lambda1) * math.sin(lambda2)))))
	else:
		tmp = R * math.acos((t_1 + (math.cos(phi2) * (math.cos(phi1) * t_0))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi2 <= -2.75e-9)
		tmp = Float64(acos(Float64(t_1 + Float64(Float64(cos(phi1) * cos(phi2)) * t_0))) * R);
	elseif (phi2 <= 2.2e-7)
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = Float64(R * acos(Float64(t_1 + Float64(cos(phi2) * Float64(cos(phi1) * t_0)))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	t_1 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if (phi2 <= -2.75e-9)
		tmp = acos((t_1 + ((cos(phi1) * cos(phi2)) * t_0))) * R;
	elseif (phi2 <= 2.2e-7)
		tmp = R * acos((cos(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))));
	else
		tmp = R * acos((t_1 + (cos(phi2) * (cos(phi1) * t_0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -2.75e-9], N[(N[ArcCos[N[(t$95$1 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi2, 2.2e-7], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(t$95$1 + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -2.7499999999999998e-9

   1. Initial program 77.7%

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

   if -2.7499999999999998e-9 < phi2 < 2.2000000000000001e-7

   1. Initial program 67.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 67.8%

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

   5. Simplified 67.8%

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

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

      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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

      10. cos-lowering-cos.f64 87.9%

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

   7. Applied egg-rr 87.9%

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

   if 2.2000000000000001e-7 < phi2

   1. Initial program 80.1%

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

      1. associate-*l* N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      9. cos-lowering-cos.f64 80.1%

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

   4. Applied egg-rr 80.1%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.75 \cdot 10^{-9}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 83.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -2.6 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 -2.6e-9)
     t_0
     (if (<= phi2 2.2e-7)
       (*
        R
        (acos
         (*
          (cos phi1)
          (+
           (* (cos lambda2) (cos lambda1))
           (* (sin lambda1) (sin lambda2))))))
       t_0))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R;
	double tmp;
	if (phi2 <= -2.6e-9) {
		tmp = t_0;
	} else if (phi2 <= 2.2e-7) {
		tmp = R * acos((cos(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * r
    if (phi2 <= (-2.6d-9)) then
        tmp = t_0
    else if (phi2 <= 2.2d-7) then
        tmp = r * acos((cos(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.acos(((Math.sin(phi1) * Math.sin(phi2)) + ((Math.cos(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2))))) * R;
	double tmp;
	if (phi2 <= -2.6e-9) {
		tmp = t_0;
	} else if (phi2 <= 2.2e-7) {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.cos(lambda2) * Math.cos(lambda1)) + (Math.sin(lambda1) * Math.sin(lambda2)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.acos(((math.sin(phi1) * math.sin(phi2)) + ((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2))))) * R
	tmp = 0
	if phi2 <= -2.6e-9:
		tmp = t_0
	elif phi2 <= 2.2e-7:
		tmp = R * math.acos((math.cos(phi1) * ((math.cos(lambda2) * math.cos(lambda1)) + (math.sin(lambda1) * math.sin(lambda2)))))
	else:
		tmp = t_0
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(acos(Float64(Float64(sin(phi1) * sin(phi2)) + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) * R)
	tmp = 0.0
	if (phi2 <= -2.6e-9)
		tmp = t_0;
	elseif (phi2 <= 2.2e-7)
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = acos(((sin(phi1) * sin(phi2)) + ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))) * R;
	tmp = 0.0;
	if (phi2 <= -2.6e-9)
		tmp = t_0;
	elseif (phi2 <= 2.2e-7)
		tmp = R * acos((cos(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))));
	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.
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, -2.6e-9], t$95$0, If[LessEqual[phi2, 2.2e-7], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.6000000000000001e-9 or 2.2000000000000001e-7 < phi2

   1. Initial program 78.7%

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

   if -2.6000000000000001e-9 < phi2 < 2.2000000000000001e-7

   1. Initial program 67.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 67.8%

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

   5. Simplified 67.8%

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

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

      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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

      10. cos-lowering-cos.f64 87.9%

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

   7. Applied egg-rr 87.9%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.6 \cdot 10^{-9}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ \mathbf{if}\;\lambda_1 \leq -4.6 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 5.4 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(t\_0 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 (<= lambda1 -4.6e-7)
     (* R (acos (+ t_0 (* (cos phi2) (* (cos phi1) (cos lambda1))))))
     (if (<= lambda1 5.4e-5)
       (* R (acos (+ t_0 (* (cos phi1) (* (cos phi2) (cos lambda2))))))
       (*
        R
        (acos
         (*
          (cos phi2)
          (+
           (* (cos lambda2) (cos lambda1))
           (* (sin lambda1) (sin lambda2))))))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(phi1) * sin(phi2);
	double tmp;
	if (lambda1 <= -4.6e-7) {
		tmp = R * acos((t_0 + (cos(phi2) * (cos(phi1) * cos(lambda1)))));
	} else if (lambda1 <= 5.4e-5) {
		tmp = R * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda2)))));
	} else {
		tmp = R * acos((cos(phi2) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(phi1) * sin(phi2)
    if (lambda1 <= (-4.6d-7)) then
        tmp = r * acos((t_0 + (cos(phi2) * (cos(phi1) * cos(lambda1)))))
    else if (lambda1 <= 5.4d-5) then
        tmp = r * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda2)))))
    else
        tmp = r * acos((cos(phi2) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(phi1) * Math.sin(phi2);
	double tmp;
	if (lambda1 <= -4.6e-7) {
		tmp = R * Math.acos((t_0 + (Math.cos(phi2) * (Math.cos(phi1) * Math.cos(lambda1)))));
	} else if (lambda1 <= 5.4e-5) {
		tmp = R * Math.acos((t_0 + (Math.cos(phi1) * (Math.cos(phi2) * Math.cos(lambda2)))));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * ((Math.cos(lambda2) * Math.cos(lambda1)) + (Math.sin(lambda1) * Math.sin(lambda2)))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(phi1) * math.sin(phi2)
	tmp = 0
	if lambda1 <= -4.6e-7:
		tmp = R * math.acos((t_0 + (math.cos(phi2) * (math.cos(phi1) * math.cos(lambda1)))))
	elif lambda1 <= 5.4e-5:
		tmp = R * math.acos((t_0 + (math.cos(phi1) * (math.cos(phi2) * math.cos(lambda2)))))
	else:
		tmp = R * math.acos((math.cos(phi2) * ((math.cos(lambda2) * math.cos(lambda1)) + (math.sin(lambda1) * math.sin(lambda2)))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (lambda1 <= -4.6e-7)
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi2) * Float64(cos(phi1) * cos(lambda1))))));
	elseif (lambda1 <= 5.4e-5)
		tmp = Float64(R * acos(Float64(t_0 + Float64(cos(phi1) * Float64(cos(phi2) * cos(lambda2))))));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda1) * sin(lambda2))))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(phi1) * sin(phi2);
	tmp = 0.0;
	if (lambda1 <= -4.6e-7)
		tmp = R * acos((t_0 + (cos(phi2) * (cos(phi1) * cos(lambda1)))));
	elseif (lambda1 <= 5.4e-5)
		tmp = R * acos((t_0 + (cos(phi1) * (cos(phi2) * cos(lambda2)))));
	else
		tmp = R * acos((cos(phi2) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -4.6e-7], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, 5.4e-5], N[(R * N[ArcCos[N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -4.5999999999999999e-7

   1. Initial program 62.0%

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

      1. 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 99.0%

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

   5. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

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

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

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

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

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

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

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

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

      6. associate-*r* N/A

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

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

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

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

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

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

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

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

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

      11. cos-lowering-cos.f64 61.9%

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

   7. Simplified 61.9%

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

   if -4.5999999999999999e-7 < lambda1 < 5.3999999999999998e-5

   1. Initial program 88.6%

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

   3. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      10. cos-neg N/A

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

      11. cos-lowering-cos.f64 88.6%

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

   5. Simplified 88.6%

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

   if 5.3999999999999998e-5 < lambda1

   1. Initial program 56.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. 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 99.3%

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

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. cos-lowering-cos.f64 48.5%

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

   7. Simplified 48.5%

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

4. Final simplification 71.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4.6 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot \cos \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_1 \leq 5.4 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\\ \mathbf{if}\;\phi_2 \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (+ (* (cos lambda2) (cos lambda1)) (* (sin lambda1) (sin lambda2)))))
   (if (<= phi2 1.5e-6)
     (* R (acos (* (cos phi1) t_0)))
     (* R (acos (* (cos phi2) t_0))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2));
	double tmp;
	if (phi2 <= 1.5e-6) {
		tmp = R * acos((cos(phi1) * t_0));
	} else {
		tmp = R * acos((cos(phi2) * t_0));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2))
    if (phi2 <= 1.5d-6) then
        tmp = r * acos((cos(phi1) * t_0))
    else
        tmp = r * acos((cos(phi2) * t_0))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (Math.cos(lambda2) * Math.cos(lambda1)) + (Math.sin(lambda1) * Math.sin(lambda2));
	double tmp;
	if (phi2 <= 1.5e-6) {
		tmp = R * Math.acos((Math.cos(phi1) * t_0));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * t_0));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (math.cos(lambda2) * math.cos(lambda1)) + (math.sin(lambda1) * math.sin(lambda2))
	tmp = 0
	if phi2 <= 1.5e-6:
		tmp = R * math.acos((math.cos(phi1) * t_0))
	else:
		tmp = R * math.acos((math.cos(phi2) * t_0))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda1) * sin(lambda2)))
	tmp = 0.0
	if (phi2 <= 1.5e-6)
		tmp = Float64(R * acos(Float64(cos(phi1) * t_0)));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * t_0)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2));
	tmp = 0.0;
	if (phi2 <= 1.5e-6)
		tmp = R * acos((cos(phi1) * t_0));
	else
		tmp = R * acos((cos(phi2) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, 1.5e-6], 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 < 1.5e-6

   1. Initial program 71.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 48.0%

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

   5. Simplified 48.0%

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

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

      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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

      10. cos-lowering-cos.f64 60.0%

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

   7. Applied egg-rr 60.0%

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

   if 1.5e-6 < phi2

   1. Initial program 80.1%

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

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

   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. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      13. cos-lowering-cos.f64 62.2%

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

   7. Simplified 62.2%

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

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.5 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 69.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 3e-6)
   (*
    R
    (acos
     (*
      (cos phi1)
      (+ (* (cos lambda2) (cos lambda1)) (* (sin lambda1) (sin lambda2))))))
   (* R (acos (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3e-6) {
		tmp = R * acos((cos(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))));
	} else {
		tmp = R * acos(((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 3d-6) then
        tmp = r * acos((cos(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))))
    else
        tmp = r * acos(((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3e-6) {
		tmp = R * Math.acos((Math.cos(phi1) * ((Math.cos(lambda2) * Math.cos(lambda1)) + (Math.sin(lambda1) * Math.sin(lambda2)))));
	} else {
		tmp = R * Math.acos(((Math.cos(phi1) * Math.cos(phi2)) * Math.cos((lambda1 - lambda2))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 3e-6:
		tmp = R * math.acos((math.cos(phi1) * ((math.cos(lambda2) * math.cos(lambda1)) + (math.sin(lambda1) * math.sin(lambda2)))))
	else:
		tmp = R * math.acos(((math.cos(phi1) * math.cos(phi2)) * math.cos((lambda1 - lambda2))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 3e-6)
		tmp = Float64(R * acos(Float64(cos(phi1) * Float64(Float64(cos(lambda2) * cos(lambda1)) + Float64(sin(lambda1) * sin(lambda2))))));
	else
		tmp = Float64(R * acos(Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2)))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 3e-6)
		tmp = R * acos((cos(phi1) * ((cos(lambda2) * cos(lambda1)) + (sin(lambda1) * sin(lambda2)))));
	else
		tmp = R * acos(((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 3e-6], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.0000000000000001e-6

   1. Initial program 71.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 48.0%

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

   5. Simplified 48.0%

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

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

      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. +-lowering-+.f64 N/A

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

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

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

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

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

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

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

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

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

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

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

      10. cos-lowering-cos.f64 60.0%

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

   7. Applied egg-rr 60.0%

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

   if 3.0000000000000001e-6 < phi2

   1. Initial program 80.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. 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. /-lowering-/.f64 N/A

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

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

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

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

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

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

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

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

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

      7. +-lowering-+.f64 55.6%

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

   4. Applied egg-rr 55.6%

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

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \left(\cos \left(\mathsf{neg}\left(\phi_2\right)\right) - \cos \phi_2\right)\right)}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right)\right)\right), R\right) \]
   6. Step-by-step derivation

      1. cos-neg N/A

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

      2. +-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2} \cdot 0\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right)\right)\right), R\right) \]

      3. metadata-eval 55.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right)\right)\right), R\right) \]

   7. Simplified 55.9%

      \[\leadsto \cos^{-1} \left(\color{blue}{0} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 59.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ R \cdot \cos^{-1} \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.8%

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

   1. sin-mult N/A

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

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

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

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

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

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

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

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

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

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

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

   7. +-lowering-+.f64 57.4%

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

4. Applied egg-rr 57.4%

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

5. Taylor expanded in phi1 around 0

   \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \left(\cos \left(\mathsf{neg}\left(\phi_2\right)\right) - \cos \phi_2\right)\right)}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right)\right)\right), R\right) \]
6. Step-by-step derivation

   1. cos-neg N/A

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

   2. +-inverses N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2} \cdot 0\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right)\right)\right), R\right) \]

   3. metadata-eval 57.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{+.f64}\left(0, \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \mathsf{cos.f64}\left(\phi_2\right)\right), \mathsf{cos.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right)\right)\right)\right)\right), R\right) \]

7. Simplified 57.2%

   \[\leadsto \cos^{-1} \left(\color{blue}{0} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

8. Final simplification 57.2%

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

Alternative 17: 29.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;t\_0 \leq 0.986:\\ \;\;\;\;R \cdot \cos^{-1} t\_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \pi\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= t_0 0.986)
     (* R (acos t_0))
     (* R (fabs (remainder (- lambda1 lambda2) (* 2.0 PI)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (t_0 <= 0.986) {
		tmp = R * acos(t_0);
	} else {
		tmp = R * fabs(remainder((lambda1 - lambda2), (2.0 * ((double) M_PI))));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double tmp;
	if (t_0 <= 0.986) {
		tmp = R * Math.acos(t_0);
	} else {
		tmp = R * Math.abs(Math.IEEEremainder((lambda1 - lambda2), (2.0 * Math.PI)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	tmp = 0
	if t_0 <= 0.986:
		tmp = R * math.acos(t_0)
	else:
		tmp = R * math.fabs(math.remainder((lambda1 - lambda2), (2.0 * math.pi)))
	return tmp

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.986], N[(R * N[ArcCos[t$95$0], $MachinePrecision]), $MachinePrecision], N[(R * N[Abs[N[With[{TMP1 = N[(lambda1 - lambda2), $MachinePrecision], TMP2 = N[(2.0 * Pi), $MachinePrecision]}, TMP1 - Round[TMP1 / TMP2] * TMP2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 lambda1 lambda2)) < 0.98599999999999999

   1. Initial program 73.4%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 45.7%

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

   5. Simplified 45.7%

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

   6. Taylor expanded in 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 32.3%

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

   8. Simplified 32.3%

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

   if 0.98599999999999999 < (cos.f64 (-.f64 lambda1 lambda2))

   1. Initial program 74.5%

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

   3. Taylor expanded in 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 30.9%

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

   5. Simplified 30.9%

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

   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 7.4%

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

   8. Simplified 7.4%

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

      1. cos-acos N/A

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

      2. acos-cos N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\left|\left(\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right|\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\left(\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{remainder.f64}\left(\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right), R\right) \]

      5. acos-cos-s N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{remainder.f64}\left(\left(\lambda_1 - \lambda_2\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{remainder.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right), R\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{remainder.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{*.f64}\left(2, \mathsf{PI}\left(\right)\right)\right)\right), R\right) \]

      8. PI-lowering-PI.f64 19.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{fabs.f64}\left(\mathsf{remainder.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{*.f64}\left(2, \mathsf{PI.f64}\left(\right)\right)\right)\right), R\right) \]

   10. Applied egg-rr 19.4%

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

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\lambda_1 - \lambda_2\right) \leq 0.986:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left|\left(\left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \pi\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 58.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ \mathbf{if}\;\phi_2 \leq 3 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (if (<= phi2 3e-6)
     (* R (acos (* (cos phi1) t_0)))
     (* R (acos (* (cos phi2) t_0))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 3e-6) {
		tmp = R * acos((cos(phi1) * t_0));
	} else {
		tmp = R * acos((cos(phi2) * t_0));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos((lambda2 - lambda1))
    if (phi2 <= 3d-6) then
        tmp = r * acos((cos(phi1) * t_0))
    else
        tmp = r * acos((cos(phi2) * t_0))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 3e-6) {
		tmp = R * Math.acos((Math.cos(phi1) * t_0));
	} else {
		tmp = R * Math.acos((Math.cos(phi2) * t_0));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	tmp = 0
	if phi2 <= 3e-6:
		tmp = R * math.acos((math.cos(phi1) * t_0))
	else:
		tmp = R * math.acos((math.cos(phi2) * t_0))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi2 <= 3e-6)
		tmp = Float64(R * acos(Float64(cos(phi1) * t_0)));
	else
		tmp = Float64(R * acos(Float64(cos(phi2) * t_0)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	tmp = 0.0;
	if (phi2 <= 3e-6)
		tmp = R * acos((cos(phi1) * t_0));
	else
		tmp = R * acos((cos(phi2) * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 3e-6], 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 < 3.0000000000000001e-6

   1. Initial program 71.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 48.0%

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

   5. Simplified 48.0%

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

   if 3.0000000000000001e-6 < phi2

   1. Initial program 80.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. 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 55.2%

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

   5. Simplified 55.2%

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

4. Final simplification 49.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 29.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{if}\;t\_0 \leq 0.999999999996:\\ \;\;\;\;R \cdot \cos^{-1} t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (if (<= t_0 0.999999999996) (* R (acos t_0)) (* (- lambda2 lambda1) R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (t_0 <= 0.999999999996) {
		tmp = R * acos(t_0);
	} 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.
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((lambda1 - lambda2))
    if (t_0 <= 0.999999999996d0) then
        tmp = r * acos(t_0)
    else
        tmp = (lambda2 - lambda1) * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda1 - lambda2));
	double tmp;
	if (t_0 <= 0.999999999996) {
		tmp = R * Math.acos(t_0);
	} else {
		tmp = (lambda2 - lambda1) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda1 - lambda2))
	tmp = 0
	if t_0 <= 0.999999999996:
		tmp = R * math.acos(t_0)
	else:
		tmp = (lambda2 - lambda1) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (t_0 <= 0.999999999996)
		tmp = Float64(R * acos(t_0));
	else
		tmp = Float64(Float64(lambda2 - lambda1) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda1 - lambda2));
	tmp = 0.0;
	if (t_0 <= 0.999999999996)
		tmp = R * acos(t_0);
	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.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 0.999999999996], N[(R * N[ArcCos[t$95$0], $MachinePrecision]), $MachinePrecision], N[(N[(lambda2 - lambda1), $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 lambda1 lambda2)) < 0.999999999995999977

   1. Initial program 72.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. 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 44.9%

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

   5. Simplified 44.9%

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

   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 30.8%

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

   8. Simplified 30.8%

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

   if 0.999999999995999977 < (cos.f64 (-.f64 lambda1 lambda2))

   1. Initial program 78.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 29.4%

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

   5. Simplified 29.4%

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

   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 5.3%

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

   8. Simplified 5.3%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
   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 10.3%

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

   10. Applied egg-rr 10.3%

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

4. Final simplification 25.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\lambda_1 - \lambda_2\right) \leq 0.999999999996:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 42.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 5.1 \cdot 10^{-16}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 5.1e-16)
   (* R (acos (* (cos phi1) (cos lambda1))))
   (* R (acos (* (cos phi1) (cos lambda2))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 5.1e-16) {
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	} else {
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 5.1d-16) then
        tmp = r * acos((cos(phi1) * cos(lambda1)))
    else
        tmp = r * acos((cos(phi1) * cos(lambda2)))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 5.1e-16) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda2)));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 5.1e-16:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda2)))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 5.1e-16)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda1))));
	else
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda2))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 5.1e-16)
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	else
		tmp = R * acos((cos(phi1) * cos(lambda2)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 5.1e-16], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 5.1e-16

   1. Initial program 80.2%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 43.3%

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

   5. Simplified 43.3%

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

   6. Taylor expanded in 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 34.7%

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

   8. Simplified 34.7%

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

   if 5.1e-16 < lambda2

   1. Initial program 57.6%

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

   3. Taylor expanded in 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 35.4%

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

   5. Simplified 35.4%

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

   6. Taylor expanded in 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 35.1%

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

   8. Simplified 35.1%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 5.1 \cdot 10^{-16}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 37.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.8 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 \cdot \left(\frac{\lambda_1}{\lambda_2} + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 2.8e-6)
   (* R (acos (* (cos phi1) (cos lambda1))))
   (* R (acos (cos (* lambda2 (+ (/ lambda1 lambda2) -1.0)))))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 2.8e-6) {
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	} else {
		tmp = R * acos(cos((lambda2 * ((lambda1 / lambda2) + -1.0))));
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 2.8d-6) then
        tmp = r * acos((cos(phi1) * cos(lambda1)))
    else
        tmp = r * acos(cos((lambda2 * ((lambda1 / lambda2) + (-1.0d0)))))
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 2.8e-6) {
		tmp = R * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R * Math.acos(Math.cos((lambda2 * ((lambda1 / lambda2) + -1.0))));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 2.8e-6:
		tmp = R * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R * math.acos(math.cos((lambda2 * ((lambda1 / lambda2) + -1.0))))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 2.8e-6)
		tmp = Float64(R * acos(Float64(cos(phi1) * cos(lambda1))));
	else
		tmp = Float64(R * acos(cos(Float64(lambda2 * Float64(Float64(lambda1 / lambda2) + -1.0)))));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 2.8e-6)
		tmp = R * acos((cos(phi1) * cos(lambda1)));
	else
		tmp = R * acos(cos((lambda2 * ((lambda1 / lambda2) + -1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 2.8e-6], N[(R * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[ArcCos[N[Cos[N[(lambda2 * N[(N[(lambda1 / lambda2), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 2.79999999999999987e-6

   1. Initial program 80.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 43.2%

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

   5. Simplified 43.2%

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

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{*.f64}\left(\mathsf{cos.f64}\left(\phi_1\right), \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right), R\right) \]
   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 34.6%

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

   8. Simplified 34.6%

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

   if 2.79999999999999987e-6 < lambda2

   1. Initial program 57.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 35.6%

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

   5. Simplified 35.6%

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

   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 27.0%

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

   8. Simplified 27.0%

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

   9. Taylor expanded in lambda2 around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{acos.f64}\left(\mathsf{cos.f64}\left(\color{blue}{\left(\lambda_2 \cdot \left(\frac{\lambda_1}{\lambda_2} - 1\right)\right)}\right)\right), R\right) \]
   10. Step-by-step derivation

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

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

       2. sub-neg N/A

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

       3. metadata-eval N/A

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

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

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

       5. /-lowering-/.f64 27.1%

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

   11. Simplified 27.1%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.8 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_2 \cdot \left(\frac{\lambda_1}{\lambda_2} + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 42.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.8%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
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 41.1%

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

5. Simplified 41.1%

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

6. Final simplification 41.1%

   \[\leadsto R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \]
7. Add Preprocessing

Alternative 23: 26.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -8 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -8e-8) (* R (acos (cos lambda1))) (* R (acos (cos lambda2)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -8e-8) {
		tmp = R * acos(cos(lambda1));
	} else {
		tmp = R * acos(cos(lambda2));
	}
	return tmp;
}

Fortran

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

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -8e-8) {
		tmp = R * Math.acos(Math.cos(lambda1));
	} else {
		tmp = R * Math.acos(Math.cos(lambda2));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -8e-8:
		tmp = R * math.acos(math.cos(lambda1))
	else:
		tmp = R * math.acos(math.cos(lambda2))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -8e-8)
		tmp = Float64(R * acos(cos(lambda1)));
	else
		tmp = Float64(R * acos(cos(lambda2)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -8e-8)
		tmp = R * acos(cos(lambda1));
	else
		tmp = R * acos(cos(lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -8e-8], 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 < -8.0000000000000002e-8

   1. Initial program 62.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 41.2%

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

   5. Simplified 41.2%

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

   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 27.7%

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

   8. Simplified 27.7%

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

   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 27.9%

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

   11. Simplified 27.9%

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

   if -8.0000000000000002e-8 < lambda1

   1. Initial program 78.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 41.0%

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

   5. Simplified 41.0%

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

   6. Taylor expanded in 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 23.3%

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

   8. Simplified 23.3%

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

   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 2 regimes into one program.

4. Final simplification 21.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -8 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 20.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.7 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

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

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.7e-8) {
		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.
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-8)) 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;
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-8) {
		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])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -1.7e-8:
		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])
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-8)
		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])){:}
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-8)
		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.
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-8], 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 lambda1 < -1.7e-8

   1. Initial program 62.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 41.2%

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

   5. Simplified 41.2%

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

   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 27.7%

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

   8. Simplified 27.7%

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

   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 27.9%

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

   11. Simplified 27.9%

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

   if -1.7e-8 < lambda1

   1. Initial program 78.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   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 41.0%

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

   5. Simplified 41.0%

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

   6. Taylor expanded in 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 23.3%

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

   8. Simplified 23.3%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
   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 5.4%

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

   10. Applied egg-rr 5.4%

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

4. Final simplification 11.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.7 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;\left(\lambda_2 - \lambda_1\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 9.0% accurate, 122.6× speedup

Math

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

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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);
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.
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;
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])
[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])
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])){:}
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.
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 73.8%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
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 41.1%

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

5. Simplified 41.1%

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

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 24.5%

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

8. Simplified 24.5%

   \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
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 5.5%

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

10. Applied egg-rr 5.5%

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

Alternative 26: 3.6% accurate, 204.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 73.8%

   \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
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 41.1%

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

5. Simplified 41.1%

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

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 24.5%

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

8. Simplified 24.5%

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

9. Taylor expanded in lambda1 around inf

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

11. Simplified 5.5%

    \[\leadsto \color{blue}{\lambda_1} \cdot R \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024159 
(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))