Spherical law of cosines

Percentage Accurate: 73.8% → 96.2%

Time: 24.3s

Alternatives: 32

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: 96.2% accurate, 0.4× speedup

Math

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

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))))
   (*
    R_s
    (if (<=
         (*
          (acos
           (+ t_0 (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2)))))
          R_m)
         0.0)
      (* lambda1 (- (/ (* lambda2 R_m) lambda1) R_m))
      (*
       R_m
       (acos
        (fma
         (*
          (cos phi2)
          (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2))))
         (cos phi1)
         t_0)))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, 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))))) * R_m) <= 0.0) {
		tmp = lambda1 * (((lambda2 * R_m) / lambda1) - R_m);
	} else {
		tmp = R_m * acos(fma((cos(phi2) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2)))), cos(phi1), t_0));
	}
	return R_s * tmp;
}

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (Float64(acos(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))) * R_m) <= 0.0)
		tmp = Float64(lambda1 * Float64(Float64(Float64(lambda2 * R_m) / lambda1) - R_m));
	else
		tmp = Float64(R_m * acos(fma(Float64(cos(phi2) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2)))), cos(phi1), t_0)));
	end
	return Float64(R_s * tmp)
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[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$95$m), $MachinePrecision], 0.0], N[(lambda1 * N[(N[(N[(lambda2 * R$95$m), $MachinePrecision] / lambda1), $MachinePrecision] - R$95$m), $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 70.2%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 39.9

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

   5. Simplified 39.9%

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

   6. Taylor expanded in phi1 around 0

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

      1. sub-neg N/A

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

      2. remove-double-neg N/A

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

      3. distribute-neg-in N/A

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

      4. +-commutative N/A

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

      5. neg-mul-1 N/A

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

      6. cos-neg N/A

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

      7. lower-cos.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. sub-neg N/A

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

      10. lower--.f64 25.6

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

   8. Simplified 25.6%

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

      1. flip-- N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. difference-of-squares N/A

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

      8. lift--.f64 N/A

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

      9. acos-cos-s N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-acos.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 25.6

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

      15. lift-acos.f64 N/A

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

      16. lift-cos.f64 N/A

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

      17. lift--.f64 N/A

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

      18. cos-diff N/A

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

      19. lift-cos.f64 N/A

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

      20. lift-cos.f64 N/A

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

      21. *-commutative N/A

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

      22. lift-cos.f64 N/A

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

      23. lift-cos.f64 N/A

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

      24. lift-sin.f64 N/A

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

      25. lift-sin.f64 N/A

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

      26. *-commutative N/A

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

      27. lift-sin.f64 N/A

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

      28. lift-sin.f64 N/A

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

   10. Applied egg-rr 13.6%

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

   11. Taylor expanded in lambda1 around inf

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. mul-1-neg N/A

          \[\leadsto \lambda_1 \cdot \left(\frac{R \cdot \lambda_2}{\lambda_1} + \color{blue}{\left(\mathsf{neg}\left(R\right)\right)}\right) \]

       4. unsub-neg N/A

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

       5. lower--.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 7.9

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

   13. Simplified 7.9%

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

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

   1. Initial program 83.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 83.9

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

   4. Applied egg-rr 83.9%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 98.7

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

   6. Applied egg-rr 98.7%

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

4. Final simplification 55.8%

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

Alternative 2: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := \sin \phi_1 \cdot \sin \phi_2\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R\_m, \sin^{-1} \left(\mathsf{fma}\left(t\_0, t\_1, t\_2\right)\right) \cdot \left(-R\_m\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.057:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, t\_1 \cdot t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi1) (cos phi2)))
        (t_2 (* (sin phi1) (sin phi2))))
   (*
    R_s
    (if (<= phi1 -2.6e-21)
      (fma (* PI 0.5) R_m (* (asin (fma t_0 t_1 t_2)) (- R_m)))
      (if (<= phi1 0.057)
        (*
         R_m
         (acos
          (fma
           (*
            (cos phi2)
            (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2))))
           (fma
            (* phi1 phi1)
            (fma
             (* phi1 phi1)
             (fma (* phi1 phi1) -0.001388888888888889 0.041666666666666664)
             -0.5)
            1.0)
           t_2)))
        (* R_m (acos (fma (sin phi2) (sin phi1) (* t_1 t_0)))))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = cos(phi1) * cos(phi2);
	double t_2 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi1 <= -2.6e-21) {
		tmp = fma((((double) M_PI) * 0.5), R_m, (asin(fma(t_0, t_1, t_2)) * -R_m));
	} else if (phi1 <= 0.057) {
		tmp = R_m * acos(fma((cos(phi2) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2)))), fma((phi1 * phi1), fma((phi1 * phi1), fma((phi1 * phi1), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0), t_2));
	} else {
		tmp = R_m * acos(fma(sin(phi2), sin(phi1), (t_1 * t_0)));
	}
	return R_s * tmp;
}

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi1) * cos(phi2))
	t_2 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi1 <= -2.6e-21)
		tmp = fma(Float64(pi * 0.5), R_m, Float64(asin(fma(t_0, t_1, t_2)) * Float64(-R_m)));
	elseif (phi1 <= 0.057)
		tmp = Float64(R_m * acos(fma(Float64(cos(phi2) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2)))), fma(Float64(phi1 * phi1), fma(Float64(phi1 * phi1), fma(Float64(phi1 * phi1), -0.001388888888888889, 0.041666666666666664), -0.5), 1.0), t_2)));
	else
		tmp = Float64(R_m * acos(fma(sin(phi2), sin(phi1), Float64(t_1 * t_0))));
	end
	return Float64(R_s * tmp)
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi1, -2.6e-21], N[(N[(Pi * 0.5), $MachinePrecision] * R$95$m + N[(N[ArcSin[N[(t$95$0 * t$95$1 + t$95$2), $MachinePrecision]], $MachinePrecision] * (-R$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.057], N[(R$95$m * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(phi1 * phi1), $MachinePrecision] * N[(N[(phi1 * phi1), $MachinePrecision] * N[(N[(phi1 * phi1), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.60000000000000017e-21

   1. Initial program 86.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-acos.f64 N/A

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

      12. *-commutative N/A

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

   4. Applied egg-rr 86.4%

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

   if -2.60000000000000017e-21 < phi1 < 0.0570000000000000021

   1. Initial program 69.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 69.7

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

   4. Applied egg-rr 69.7%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 89.2

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

   6. Applied egg-rr 89.2%

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

   7. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \color{blue}{1 + {\phi_1}^{2} \cdot \left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}\right)}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \color{blue}{{\phi_1}^{2} \cdot \left({\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}\right) + 1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \color{blue}{\mathsf{fma}\left({\phi_1}^{2}, {\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}, 1\right)}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      3. unpow2 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \mathsf{fma}\left(\color{blue}{\phi_1 \cdot \phi_1}, {\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}, 1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \mathsf{fma}\left(\color{blue}{\phi_1 \cdot \phi_1}, {\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) - \frac{1}{2}, 1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \color{blue}{{\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \mathsf{fma}\left(\phi_1 \cdot \phi_1, {\phi_1}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}\right) + \color{blue}{\frac{-1}{2}}, 1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \color{blue}{\mathsf{fma}\left({\phi_1}^{2}, \frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}, \frac{-1}{2}\right)}, 1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. unpow2 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\color{blue}{\phi_1 \cdot \phi_1}, \frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}, \frac{-1}{2}\right), 1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      9. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\color{blue}{\phi_1 \cdot \phi_1}, \frac{1}{24} + \frac{-1}{720} \cdot {\phi_1}^{2}, \frac{-1}{2}\right), 1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \color{blue}{\frac{-1}{720} \cdot {\phi_1}^{2} + \frac{1}{24}}, \frac{-1}{2}\right), 1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      11. *-commutative N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \color{blue}{{\phi_1}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}, \frac{-1}{2}\right), 1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      12. lower-fma.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \color{blue}{\mathsf{fma}\left({\phi_1}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      13. unpow2 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\color{blue}{\phi_1 \cdot \phi_1}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      14. lower-*.f64 89.2

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\color{blue}{\phi_1 \cdot \phi_1}, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   9. Simplified 89.2%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \color{blue}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   if 0.0570000000000000021 < phi1

   1. Initial program 82.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 82.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 82.3

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

   4. Applied egg-rr 82.3%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R, \sin^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot \left(-R\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.057:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := \sin \phi_1 \cdot \sin \phi_2\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R\_m, \sin^{-1} \left(\mathsf{fma}\left(t\_0, t\_1, t\_2\right)\right) \cdot \left(-R\_m\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.045:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1 \cdot \phi_1, 0.041666666666666664, -0.5\right), 1\right), t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, t\_1 \cdot t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi1) (cos phi2)))
        (t_2 (* (sin phi1) (sin phi2))))
   (*
    R_s
    (if (<= phi1 -2.6e-21)
      (fma (* PI 0.5) R_m (* (asin (fma t_0 t_1 t_2)) (- R_m)))
      (if (<= phi1 0.045)
        (*
         R_m
         (acos
          (fma
           (*
            (cos phi2)
            (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2))))
           (fma
            (* phi1 phi1)
            (fma (* phi1 phi1) 0.041666666666666664 -0.5)
            1.0)
           t_2)))
        (* R_m (acos (fma (sin phi2) (sin phi1) (* t_1 t_0)))))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = cos(phi1) * cos(phi2);
	double t_2 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi1 <= -2.6e-21) {
		tmp = fma((((double) M_PI) * 0.5), R_m, (asin(fma(t_0, t_1, t_2)) * -R_m));
	} else if (phi1 <= 0.045) {
		tmp = R_m * acos(fma((cos(phi2) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2)))), fma((phi1 * phi1), fma((phi1 * phi1), 0.041666666666666664, -0.5), 1.0), t_2));
	} else {
		tmp = R_m * acos(fma(sin(phi2), sin(phi1), (t_1 * t_0)));
	}
	return R_s * tmp;
}

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi1) * cos(phi2))
	t_2 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi1 <= -2.6e-21)
		tmp = fma(Float64(pi * 0.5), R_m, Float64(asin(fma(t_0, t_1, t_2)) * Float64(-R_m)));
	elseif (phi1 <= 0.045)
		tmp = Float64(R_m * acos(fma(Float64(cos(phi2) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2)))), fma(Float64(phi1 * phi1), fma(Float64(phi1 * phi1), 0.041666666666666664, -0.5), 1.0), t_2)));
	else
		tmp = Float64(R_m * acos(fma(sin(phi2), sin(phi1), Float64(t_1 * t_0))));
	end
	return Float64(R_s * tmp)
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi1, -2.6e-21], N[(N[(Pi * 0.5), $MachinePrecision] * R$95$m + N[(N[ArcSin[N[(t$95$0 * t$95$1 + t$95$2), $MachinePrecision]], $MachinePrecision] * (-R$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.045], N[(R$95$m * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(phi1 * phi1), $MachinePrecision] * N[(N[(phi1 * phi1), $MachinePrecision] * 0.041666666666666664 + -0.5), $MachinePrecision] + 1.0), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.60000000000000017e-21

   1. Initial program 86.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-acos.f64 N/A

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

      12. *-commutative N/A

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

   4. Applied egg-rr 86.4%

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

   if -2.60000000000000017e-21 < phi1 < 0.044999999999999998

   1. Initial program 69.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 69.7

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

   4. Applied egg-rr 69.7%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 89.2

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

   6. Applied egg-rr 89.2%

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

   7. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \color{blue}{1 + {\phi_1}^{2} \cdot \left(\frac{1}{24} \cdot {\phi_1}^{2} - \frac{1}{2}\right)}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \color{blue}{{\phi_1}^{2} \cdot \left(\frac{1}{24} \cdot {\phi_1}^{2} - \frac{1}{2}\right) + 1}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \color{blue}{\mathsf{fma}\left({\phi_1}^{2}, \frac{1}{24} \cdot {\phi_1}^{2} - \frac{1}{2}, 1\right)}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      3. unpow2 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \mathsf{fma}\left(\color{blue}{\phi_1 \cdot \phi_1}, \frac{1}{24} \cdot {\phi_1}^{2} - \frac{1}{2}, 1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \mathsf{fma}\left(\color{blue}{\phi_1 \cdot \phi_1}, \frac{1}{24} \cdot {\phi_1}^{2} - \frac{1}{2}, 1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      5. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \color{blue}{\frac{1}{24} \cdot {\phi_1}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      6. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \color{blue}{{\phi_1}^{2} \cdot \frac{1}{24}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      7. metadata-eval N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \mathsf{fma}\left(\phi_1 \cdot \phi_1, {\phi_1}^{2} \cdot \frac{1}{24} + \color{blue}{\frac{-1}{2}}, 1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      8. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \color{blue}{\mathsf{fma}\left({\phi_1}^{2}, \frac{1}{24}, \frac{-1}{2}\right)}, 1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      9. unpow2 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\color{blue}{\phi_1 \cdot \phi_1}, \frac{1}{24}, \frac{-1}{2}\right), 1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      10. lower-*.f64 89.2

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\color{blue}{\phi_1 \cdot \phi_1}, 0.041666666666666664, -0.5\right), 1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   9. Simplified 89.2%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \color{blue}{\mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1 \cdot \phi_1, 0.041666666666666664, -0.5\right), 1\right)}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   if 0.044999999999999998 < phi1

   1. Initial program 82.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 82.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 82.3

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

   4. Applied egg-rr 82.3%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R, \sin^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot \left(-R\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.045:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(\phi_1 \cdot \phi_1, 0.041666666666666664, -0.5\right), 1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R\_m, \sin^{-1} \left(\mathsf{fma}\left(t\_0, t\_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot \left(-R\_m\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.045:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1, \phi_1 \cdot \left(\sin \phi_2 \cdot \mathsf{fma}\left(-0.16666666666666666, \phi_1 \cdot \phi_1, 1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, t\_1 \cdot t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))) (t_1 (* (cos phi1) (cos phi2))))
   (*
    R_s
    (if (<= phi1 -2.6e-21)
      (fma
       (* PI 0.5)
       R_m
       (* (asin (fma t_0 t_1 (* (sin phi1) (sin phi2)))) (- R_m)))
      (if (<= phi1 0.045)
        (*
         R_m
         (acos
          (fma
           (*
            (cos phi2)
            (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2))))
           (cos phi1)
           (*
            phi1
            (* (sin phi2) (fma -0.16666666666666666 (* phi1 phi1) 1.0))))))
        (* R_m (acos (fma (sin phi2) (sin phi1) (* t_1 t_0)))))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = cos(phi1) * cos(phi2);
	double tmp;
	if (phi1 <= -2.6e-21) {
		tmp = fma((((double) M_PI) * 0.5), R_m, (asin(fma(t_0, t_1, (sin(phi1) * sin(phi2)))) * -R_m));
	} else if (phi1 <= 0.045) {
		tmp = R_m * acos(fma((cos(phi2) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2)))), cos(phi1), (phi1 * (sin(phi2) * fma(-0.16666666666666666, (phi1 * phi1), 1.0)))));
	} else {
		tmp = R_m * acos(fma(sin(phi2), sin(phi1), (t_1 * t_0)));
	}
	return R_s * tmp;
}

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if (phi1 <= -2.6e-21)
		tmp = fma(Float64(pi * 0.5), R_m, Float64(asin(fma(t_0, t_1, Float64(sin(phi1) * sin(phi2)))) * Float64(-R_m)));
	elseif (phi1 <= 0.045)
		tmp = Float64(R_m * acos(fma(Float64(cos(phi2) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2)))), cos(phi1), Float64(phi1 * Float64(sin(phi2) * fma(-0.16666666666666666, Float64(phi1 * phi1), 1.0))))));
	else
		tmp = Float64(R_m * acos(fma(sin(phi2), sin(phi1), Float64(t_1 * t_0))));
	end
	return Float64(R_s * tmp)
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi1, -2.6e-21], N[(N[(Pi * 0.5), $MachinePrecision] * R$95$m + N[(N[ArcSin[N[(t$95$0 * t$95$1 + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-R$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.045], N[(R$95$m * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(phi1 * N[(N[Sin[phi2], $MachinePrecision] * N[(-0.16666666666666666 * N[(phi1 * phi1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.60000000000000017e-21

   1. Initial program 86.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-acos.f64 N/A

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

      12. *-commutative N/A

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

   4. Applied egg-rr 86.4%

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

   if -2.60000000000000017e-21 < phi1 < 0.044999999999999998

   1. Initial program 69.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 69.7

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

   4. Applied egg-rr 69.7%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 89.2

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

   6. Applied egg-rr 89.2%

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

   7. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \cos \phi_1, \color{blue}{\phi_1 \cdot \left(\sin \phi_2 + \frac{-1}{6} \cdot \left({\phi_1}^{2} \cdot \sin \phi_2\right)\right)}\right)\right) \cdot R \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \cos \phi_1, \color{blue}{\phi_1 \cdot \left(\sin \phi_2 + \frac{-1}{6} \cdot \left({\phi_1}^{2} \cdot \sin \phi_2\right)\right)}\right)\right) \cdot R \]

      2. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \cos \phi_1, \phi_1 \cdot \left(\sin \phi_2 + \color{blue}{\left(\frac{-1}{6} \cdot {\phi_1}^{2}\right) \cdot \sin \phi_2}\right)\right)\right) \cdot R \]

      3. distribute-rgt1-in N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \cos \phi_1, \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {\phi_1}^{2} + 1\right) \cdot \sin \phi_2\right)}\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \cos \phi_1, \phi_1 \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot {\phi_1}^{2} + 1\right) \cdot \sin \phi_2\right)}\right)\right) \cdot R \]

      5. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \cos \phi_1, \phi_1 \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {\phi_1}^{2}, 1\right)} \cdot \sin \phi_2\right)\right)\right) \cdot R \]

      6. unpow2 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \cos \phi_1, \phi_1 \cdot \left(\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{\phi_1 \cdot \phi_1}, 1\right) \cdot \sin \phi_2\right)\right)\right) \cdot R \]

      7. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \cos \phi_1, \phi_1 \cdot \left(\mathsf{fma}\left(\frac{-1}{6}, \color{blue}{\phi_1 \cdot \phi_1}, 1\right) \cdot \sin \phi_2\right)\right)\right) \cdot R \]

      8. lower-sin.f64 89.2

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \cos \phi_1, \phi_1 \cdot \left(\mathsf{fma}\left(-0.16666666666666666, \phi_1 \cdot \phi_1, 1\right) \cdot \color{blue}{\sin \phi_2}\right)\right)\right) \cdot R \]

   9. Simplified 89.2%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right) \cdot \cos \phi_2, \cos \phi_1, \color{blue}{\phi_1 \cdot \left(\mathsf{fma}\left(-0.16666666666666666, \phi_1 \cdot \phi_1, 1\right) \cdot \sin \phi_2\right)}\right)\right) \cdot R \]

   if 0.044999999999999998 < phi1

   1. Initial program 82.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 82.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 82.3

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

   4. Applied egg-rr 82.3%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R, \sin^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot \left(-R\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.045:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1, \phi_1 \cdot \left(\sin \phi_2 \cdot \mathsf{fma}\left(-0.16666666666666666, \phi_1 \cdot \phi_1, 1\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := \sin \phi_1 \cdot \sin \phi_2\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R\_m, \sin^{-1} \left(\mathsf{fma}\left(t\_0, t\_1, t\_2\right)\right) \cdot \left(-R\_m\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.0155:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right), t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, t\_1 \cdot t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2)))
        (t_1 (* (cos phi1) (cos phi2)))
        (t_2 (* (sin phi1) (sin phi2))))
   (*
    R_s
    (if (<= phi1 -2.6e-21)
      (fma (* PI 0.5) R_m (* (asin (fma t_0 t_1 t_2)) (- R_m)))
      (if (<= phi1 0.0155)
        (*
         R_m
         (acos
          (fma
           (*
            (cos phi2)
            (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2))))
           (fma (* phi1 phi1) -0.5 1.0)
           t_2)))
        (* R_m (acos (fma (sin phi2) (sin phi1) (* t_1 t_0)))))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = cos(phi1) * cos(phi2);
	double t_2 = sin(phi1) * sin(phi2);
	double tmp;
	if (phi1 <= -2.6e-21) {
		tmp = fma((((double) M_PI) * 0.5), R_m, (asin(fma(t_0, t_1, t_2)) * -R_m));
	} else if (phi1 <= 0.0155) {
		tmp = R_m * acos(fma((cos(phi2) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2)))), fma((phi1 * phi1), -0.5, 1.0), t_2));
	} else {
		tmp = R_m * acos(fma(sin(phi2), sin(phi1), (t_1 * t_0)));
	}
	return R_s * tmp;
}

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi1) * cos(phi2))
	t_2 = Float64(sin(phi1) * sin(phi2))
	tmp = 0.0
	if (phi1 <= -2.6e-21)
		tmp = fma(Float64(pi * 0.5), R_m, Float64(asin(fma(t_0, t_1, t_2)) * Float64(-R_m)));
	elseif (phi1 <= 0.0155)
		tmp = Float64(R_m * acos(fma(Float64(cos(phi2) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2)))), fma(Float64(phi1 * phi1), -0.5, 1.0), t_2)));
	else
		tmp = Float64(R_m * acos(fma(sin(phi2), sin(phi1), Float64(t_1 * t_0))));
	end
	return Float64(R_s * tmp)
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi1, -2.6e-21], N[(N[(Pi * 0.5), $MachinePrecision] * R$95$m + N[(N[ArcSin[N[(t$95$0 * t$95$1 + t$95$2), $MachinePrecision]], $MachinePrecision] * (-R$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.0155], N[(R$95$m * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(phi1 * phi1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.60000000000000017e-21

   1. Initial program 86.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-acos.f64 N/A

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

      12. *-commutative N/A

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

   4. Applied egg-rr 86.4%

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

   if -2.60000000000000017e-21 < phi1 < 0.0155

   1. Initial program 69.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 69.7

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

   4. Applied egg-rr 69.7%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 89.2

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

   6. Applied egg-rr 89.2%

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

   7. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 89.2

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

   9. Simplified 89.2%

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

   if 0.0155 < phi1

   1. Initial program 82.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 82.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 82.3

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

   4. Applied egg-rr 82.3%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R, \sin^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot \left(-R\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.0155:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R\_m, \sin^{-1} \left(\mathsf{fma}\left(t\_0, t\_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot \left(-R\_m\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.0155:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1, \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, t\_1 \cdot t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))) (t_1 (* (cos phi1) (cos phi2))))
   (*
    R_s
    (if (<= phi1 -2.6e-21)
      (fma
       (* PI 0.5)
       R_m
       (* (asin (fma t_0 t_1 (* (sin phi1) (sin phi2)))) (- R_m)))
      (if (<= phi1 0.0155)
        (*
         R_m
         (acos
          (fma
           (*
            (cos phi2)
            (fma (sin lambda2) (sin lambda1) (* (cos lambda1) (cos lambda2))))
           (cos phi1)
           (* phi1 (sin phi2)))))
        (* R_m (acos (fma (sin phi2) (sin phi1) (* t_1 t_0)))))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = cos(phi1) * cos(phi2);
	double tmp;
	if (phi1 <= -2.6e-21) {
		tmp = fma((((double) M_PI) * 0.5), R_m, (asin(fma(t_0, t_1, (sin(phi1) * sin(phi2)))) * -R_m));
	} else if (phi1 <= 0.0155) {
		tmp = R_m * acos(fma((cos(phi2) * fma(sin(lambda2), sin(lambda1), (cos(lambda1) * cos(lambda2)))), cos(phi1), (phi1 * sin(phi2))));
	} else {
		tmp = R_m * acos(fma(sin(phi2), sin(phi1), (t_1 * t_0)));
	}
	return R_s * tmp;
}

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if (phi1 <= -2.6e-21)
		tmp = fma(Float64(pi * 0.5), R_m, Float64(asin(fma(t_0, t_1, Float64(sin(phi1) * sin(phi2)))) * Float64(-R_m)));
	elseif (phi1 <= 0.0155)
		tmp = Float64(R_m * acos(fma(Float64(cos(phi2) * fma(sin(lambda2), sin(lambda1), Float64(cos(lambda1) * cos(lambda2)))), cos(phi1), Float64(phi1 * sin(phi2)))));
	else
		tmp = Float64(R_m * acos(fma(sin(phi2), sin(phi1), Float64(t_1 * t_0))));
	end
	return Float64(R_s * tmp)
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi1, -2.6e-21], N[(N[(Pi * 0.5), $MachinePrecision] * R$95$m + N[(N[ArcSin[N[(t$95$0 * t$95$1 + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-R$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.0155], N[(R$95$m * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(phi1 * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.60000000000000017e-21

   1. Initial program 86.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-acos.f64 N/A

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

      12. *-commutative N/A

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

   4. Applied egg-rr 86.4%

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

   if -2.60000000000000017e-21 < phi1 < 0.0155

   1. Initial program 69.7%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 69.7

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

   4. Applied egg-rr 69.7%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-sin.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 89.2

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

   6. Applied egg-rr 89.2%

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

   7. Taylor expanded in phi1 around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 89.2

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

   9. Simplified 89.2%

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

   if 0.0155 < phi1

   1. Initial program 82.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 82.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 82.3

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

   4. Applied egg-rr 82.3%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R, \sin^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot \left(-R\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.0155:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_2, \sin \lambda_1, \cos \lambda_1 \cdot \cos \lambda_2\right), \cos \phi_1, \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R\_m, \sin^{-1} \left(\mathsf{fma}\left(t\_0, t\_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot \left(-R\_m\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.0155:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, t\_1 \cdot t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))) (t_1 (* (cos phi1) (cos phi2))))
   (*
    R_s
    (if (<= phi1 -2.6e-21)
      (fma
       (* PI 0.5)
       R_m
       (* (asin (fma t_0 t_1 (* (sin phi1) (sin phi2)))) (- R_m)))
      (if (<= phi1 0.0155)
        (*
         R_m
         (acos
          (fma
           phi1
           (sin phi2)
           (*
            (cos phi2)
            (fma
             (sin lambda1)
             (sin lambda2)
             (* (cos lambda1) (cos lambda2)))))))
        (* R_m (acos (fma (sin phi2) (sin phi1) (* t_1 t_0)))))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = cos(phi1) * cos(phi2);
	double tmp;
	if (phi1 <= -2.6e-21) {
		tmp = fma((((double) M_PI) * 0.5), R_m, (asin(fma(t_0, t_1, (sin(phi1) * sin(phi2)))) * -R_m));
	} else if (phi1 <= 0.0155) {
		tmp = R_m * acos(fma(phi1, sin(phi2), (cos(phi2) * fma(sin(lambda1), sin(lambda2), (cos(lambda1) * cos(lambda2))))));
	} else {
		tmp = R_m * acos(fma(sin(phi2), sin(phi1), (t_1 * t_0)));
	}
	return R_s * tmp;
}

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if (phi1 <= -2.6e-21)
		tmp = fma(Float64(pi * 0.5), R_m, Float64(asin(fma(t_0, t_1, Float64(sin(phi1) * sin(phi2)))) * Float64(-R_m)));
	elseif (phi1 <= 0.0155)
		tmp = Float64(R_m * acos(fma(phi1, sin(phi2), Float64(cos(phi2) * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda1) * cos(lambda2)))))));
	else
		tmp = Float64(R_m * acos(fma(sin(phi2), sin(phi1), Float64(t_1 * t_0))));
	end
	return Float64(R_s * tmp)
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi1, -2.6e-21], N[(N[(Pi * 0.5), $MachinePrecision] * R$95$m + N[(N[ArcSin[N[(t$95$0 * t$95$1 + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-R$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.0155], N[(R$95$m * N[ArcCos[N[(phi1 * N[Sin[phi2], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.60000000000000017e-21

   1. Initial program 86.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-acos.f64 N/A

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

      12. *-commutative N/A

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

   4. Applied egg-rr 86.4%

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

   if -2.60000000000000017e-21 < phi1 < 0.0155

   1. Initial program 69.7%

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

   4. Applied egg-rr 89.2%

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

   5. Taylor expanded in phi1 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-cos.f64 89.2

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

   7. Simplified 89.2%

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

   if 0.0155 < phi1

   1. Initial program 82.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 82.3

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 82.3

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

   4. Applied egg-rr 82.3%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R, \sin^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot \left(-R\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 0.0155:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\phi_1, \sin \phi_2, \cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R\_m, \sin^{-1} \left(\mathsf{fma}\left(t\_0, t\_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot \left(-R\_m\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 3.7 \cdot 10^{-18}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, t\_1 \cdot t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))) (t_1 (* (cos phi1) (cos phi2))))
   (*
    R_s
    (if (<= phi1 -2.6e-21)
      (fma
       (* PI 0.5)
       R_m
       (* (asin (fma t_0 t_1 (* (sin phi1) (sin phi2)))) (- R_m)))
      (if (<= phi1 3.7e-18)
        (*
         R_m
         (acos
          (*
           (cos phi2)
           (fma (sin lambda1) (sin lambda2) (* (cos lambda1) (cos lambda2))))))
        (* R_m (acos (fma (sin phi2) (sin phi1) (* t_1 t_0)))))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double t_1 = cos(phi1) * cos(phi2);
	double tmp;
	if (phi1 <= -2.6e-21) {
		tmp = fma((((double) M_PI) * 0.5), R_m, (asin(fma(t_0, t_1, (sin(phi1) * sin(phi2)))) * -R_m));
	} else if (phi1 <= 3.7e-18) {
		tmp = R_m * acos((cos(phi2) * fma(sin(lambda1), sin(lambda2), (cos(lambda1) * cos(lambda2)))));
	} else {
		tmp = R_m * acos(fma(sin(phi2), sin(phi1), (t_1 * t_0)));
	}
	return R_s * tmp;
}

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	t_1 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if (phi1 <= -2.6e-21)
		tmp = fma(Float64(pi * 0.5), R_m, Float64(asin(fma(t_0, t_1, Float64(sin(phi1) * sin(phi2)))) * Float64(-R_m)));
	elseif (phi1 <= 3.7e-18)
		tmp = Float64(R_m * acos(Float64(cos(phi2) * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda1) * cos(lambda2))))));
	else
		tmp = Float64(R_m * acos(fma(sin(phi2), sin(phi1), Float64(t_1 * t_0))));
	end
	return Float64(R_s * tmp)
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi1, -2.6e-21], N[(N[(Pi * 0.5), $MachinePrecision] * R$95$m + N[(N[ArcSin[N[(t$95$0 * t$95$1 + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-R$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 3.7e-18], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.60000000000000017e-21

   1. Initial program 86.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-acos.f64 N/A

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

      12. *-commutative N/A

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

   4. Applied egg-rr 86.4%

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

   if -2.60000000000000017e-21 < phi1 < 3.7000000000000003e-18

   1. Initial program 69.0%

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

   4. Applied egg-rr 88.9%

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

   5. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 88.9

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

   7. Simplified 88.9%

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

   if 3.7000000000000003e-18 < phi1

   1. Initial program 83.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 83.0

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 83.0

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

   4. Applied egg-rr 83.0%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{-21}:\\ \;\;\;\;\mathsf{fma}\left(\pi \cdot 0.5, R, \sin^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot \left(-R\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 3.7 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.72 \cdot 10^{-9}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot t\_0, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 3.7 \cdot 10^{-18}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (*
    R_s
    (if (<= phi1 -1.72e-9)
      (*
       R_m
       (acos (fma (* (cos phi2) t_0) (cos phi1) (* (sin phi1) (sin phi2)))))
      (if (<= phi1 3.7e-18)
        (*
         R_m
         (acos
          (*
           (cos phi2)
           (fma (sin lambda1) (sin lambda2) (* (cos lambda1) (cos lambda2))))))
        (*
         R_m
         (acos
          (fma (sin phi2) (sin phi1) (* (* (cos phi1) (cos phi2)) t_0)))))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (phi1 <= -1.72e-9) {
		tmp = R_m * acos(fma((cos(phi2) * t_0), cos(phi1), (sin(phi1) * sin(phi2))));
	} else if (phi1 <= 3.7e-18) {
		tmp = R_m * acos((cos(phi2) * fma(sin(lambda1), sin(lambda2), (cos(lambda1) * cos(lambda2)))));
	} else {
		tmp = R_m * acos(fma(sin(phi2), sin(phi1), ((cos(phi1) * cos(phi2)) * t_0)));
	}
	return R_s * tmp;
}

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (phi1 <= -1.72e-9)
		tmp = Float64(R_m * acos(fma(Float64(cos(phi2) * t_0), cos(phi1), Float64(sin(phi1) * sin(phi2)))));
	elseif (phi1 <= 3.7e-18)
		tmp = Float64(R_m * acos(Float64(cos(phi2) * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda1) * cos(lambda2))))));
	else
		tmp = Float64(R_m * acos(fma(sin(phi2), sin(phi1), Float64(Float64(cos(phi1) * cos(phi2)) * t_0))));
	end
	return Float64(R_s * tmp)
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi1, -1.72e-9], N[(R$95$m * N[ArcCos[N[(N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 3.7e-18], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.72000000000000006e-9

   1. Initial program 86.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 86.2

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

   4. Applied egg-rr 86.2%

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

   if -1.72000000000000006e-9 < phi1 < 3.7000000000000003e-18

   1. Initial program 69.2%

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

   4. Applied egg-rr 89.0%

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

   5. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 89.0

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

   7. Simplified 89.0%

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

   if 3.7000000000000003e-18 < phi1

   1. Initial program 83.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 83.0

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 83.0

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

   4. Applied egg-rr 83.0%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.72 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 3.7 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.72 \cdot 10^{-9}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 3.7 \cdot 10^{-18}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi1 -1.72e-9)
    (*
     R_m
     (acos
      (fma
       (sin phi2)
       (sin phi1)
       (* (cos phi1) (* (cos phi2) (cos (- lambda2 lambda1)))))))
    (if (<= phi1 3.7e-18)
      (*
       R_m
       (acos
        (*
         (cos phi2)
         (fma (sin lambda1) (sin lambda2) (* (cos lambda1) (cos lambda2))))))
      (*
       R_m
       (acos
        (fma
         (sin phi2)
         (sin phi1)
         (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.72e-9) {
		tmp = R_m * acos(fma(sin(phi2), sin(phi1), (cos(phi1) * (cos(phi2) * cos((lambda2 - lambda1))))));
	} else if (phi1 <= 3.7e-18) {
		tmp = R_m * acos((cos(phi2) * fma(sin(lambda1), sin(lambda2), (cos(lambda1) * cos(lambda2)))));
	} else {
		tmp = R_m * acos(fma(sin(phi2), sin(phi1), ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	}
	return R_s * tmp;
}

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.72e-9)
		tmp = Float64(R_m * acos(fma(sin(phi2), sin(phi1), Float64(cos(phi1) * Float64(cos(phi2) * cos(Float64(lambda2 - lambda1)))))));
	elseif (phi1 <= 3.7e-18)
		tmp = Float64(R_m * acos(Float64(cos(phi2) * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda1) * cos(lambda2))))));
	else
		tmp = Float64(R_m * acos(fma(sin(phi2), sin(phi1), Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))));
	end
	return Float64(R_s * tmp)
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi1, -1.72e-9], N[(R$95$m * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 3.7e-18], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.72000000000000006e-9

   1. Initial program 86.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-*l* N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 86.2

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

   4. Applied egg-rr 86.2%

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

      1. lift--.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-cos.f64 N/A

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

      6. lift--.f64 N/A

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

      7. cos-diff N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-rgt-in N/A

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

      12. lift-sin.f64 N/A

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

      13. lift-sin.f64 N/A

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

      14. associate-*r* N/A

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

      15. lift-*.f64 N/A

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

      16. lift-*.f64 N/A

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

   6. Applied egg-rr 99.1%

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

   8. Applied egg-rr 86.2%

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

   if -1.72000000000000006e-9 < phi1 < 3.7000000000000003e-18

   1. Initial program 69.2%

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

   4. Applied egg-rr 89.0%

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

   5. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 89.0

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

   7. Simplified 89.0%

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

   if 3.7000000000000003e-18 < phi1

   1. Initial program 83.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 83.0

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 83.0

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

   4. Applied egg-rr 83.0%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.72 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 3.7 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.72 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq 3.7 \cdot 10^{-18}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          R_m
          (acos
           (fma
            (sin phi2)
            (sin phi1)
            (* (* (cos phi1) (cos phi2)) (cos (- lambda1 lambda2))))))))
   (*
    R_s
    (if (<= phi1 -1.72e-9)
      t_0
      (if (<= phi1 3.7e-18)
        (*
         R_m
         (acos
          (*
           (cos phi2)
           (fma (sin lambda1) (sin lambda2) (* (cos lambda1) (cos lambda2))))))
        t_0)))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R_m * acos(fma(sin(phi2), sin(phi1), ((cos(phi1) * cos(phi2)) * cos((lambda1 - lambda2)))));
	double tmp;
	if (phi1 <= -1.72e-9) {
		tmp = t_0;
	} else if (phi1 <= 3.7e-18) {
		tmp = R_m * acos((cos(phi2) * fma(sin(lambda1), sin(lambda2), (cos(lambda1) * cos(lambda2)))));
	} else {
		tmp = t_0;
	}
	return R_s * tmp;
}

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R_m * acos(fma(sin(phi2), sin(phi1), Float64(Float64(cos(phi1) * cos(phi2)) * cos(Float64(lambda1 - lambda2))))))
	tmp = 0.0
	if (phi1 <= -1.72e-9)
		tmp = t_0;
	elseif (phi1 <= 3.7e-18)
		tmp = Float64(R_m * acos(Float64(cos(phi2) * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda1) * cos(lambda2))))));
	else
		tmp = t_0;
	end
	return Float64(R_s * tmp)
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(R$95$m * N[ArcCos[N[(N[Sin[phi2], $MachinePrecision] * N[Sin[phi1], $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi1, -1.72e-9], t$95$0, If[LessEqual[phi1, 3.7e-18], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.72000000000000006e-9 or 3.7000000000000003e-18 < phi1

   1. Initial program 84.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 84.5

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 84.5

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

   4. Applied egg-rr 84.5%

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

   if -1.72000000000000006e-9 < phi1 < 3.7000000000000003e-18

   1. Initial program 69.2%

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

   4. Applied egg-rr 89.0%

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

   5. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 89.0

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

   7. Simplified 89.0%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.72 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 3.7 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\sin \phi_2, \sin \phi_1, \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \sin \phi_1 \cdot \sin \phi_2\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -6.7 \cdot 10^{-6}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 9 \cdot 10^{-5}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, t\_1, t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, t\_1, t\_0\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin phi1) (sin phi2))) (t_1 (* (cos phi1) (cos phi2))))
   (*
    R_s
    (if (<= lambda2 -6.7e-6)
      (*
       R_m
       (acos
        (*
         (cos phi1)
         (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1))))))
      (if (<= lambda2 9e-5)
        (* R_m (acos (fma (cos lambda1) t_1 t_0)))
        (* R_m (acos (fma (cos lambda2) t_1 t_0))))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, 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 (lambda2 <= -6.7e-6) {
		tmp = R_m * acos((cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1)))));
	} else if (lambda2 <= 9e-5) {
		tmp = R_m * acos(fma(cos(lambda1), t_1, t_0));
	} else {
		tmp = R_m * acos(fma(cos(lambda2), t_1, t_0));
	}
	return R_s * tmp;
}

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(phi1) * sin(phi2))
	t_1 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if (lambda2 <= -6.7e-6)
		tmp = Float64(R_m * acos(Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1))))));
	elseif (lambda2 <= 9e-5)
		tmp = Float64(R_m * acos(fma(cos(lambda1), t_1, t_0)));
	else
		tmp = Float64(R_m * acos(fma(cos(lambda2), t_1, t_0)));
	end
	return Float64(R_s * tmp)
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, 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]}, N[(R$95$s * If[LessEqual[lambda2, -6.7e-6], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 9e-5], N[(R$95$m * N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[lambda2], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -6.7e-6

   1. Initial program 62.7%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 38.8

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

   5. Simplified 38.8%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lower-*.f64 59.1

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

   7. Applied egg-rr 59.1%

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

   if -6.7e-6 < lambda2 < 9.00000000000000057e-5

   1. Initial program 90.3%

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

   3. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 90.4

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

   5. Simplified 90.4%

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

   if 9.00000000000000057e-5 < lambda2

   1. Initial program 58.7%

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

   3. Taylor expanded in lambda1 around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. cos-neg N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-sin.f64 59.0

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

   5. Simplified 59.0%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -6.7 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{elif}\;\lambda_2 \leq 9 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_2, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6.2 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq 3.7 \cdot 10^{-18}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (*
          R_m
          (acos
           (fma
            (cos lambda1)
            (* (cos phi1) (cos phi2))
            (* (sin phi1) (sin phi2)))))))
   (*
    R_s
    (if (<= phi1 -6.2e-6)
      t_0
      (if (<= phi1 3.7e-18)
        (*
         R_m
         (acos
          (*
           (cos phi2)
           (fma (sin lambda1) (sin lambda2) (* (cos lambda1) (cos lambda2))))))
        t_0)))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R_m * acos(fma(cos(lambda1), (cos(phi1) * cos(phi2)), (sin(phi1) * sin(phi2))));
	double tmp;
	if (phi1 <= -6.2e-6) {
		tmp = t_0;
	} else if (phi1 <= 3.7e-18) {
		tmp = R_m * acos((cos(phi2) * fma(sin(lambda1), sin(lambda2), (cos(lambda1) * cos(lambda2)))));
	} else {
		tmp = t_0;
	}
	return R_s * tmp;
}

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R_m * acos(fma(cos(lambda1), Float64(cos(phi1) * cos(phi2)), Float64(sin(phi1) * sin(phi2)))))
	tmp = 0.0
	if (phi1 <= -6.2e-6)
		tmp = t_0;
	elseif (phi1 <= 3.7e-18)
		tmp = Float64(R_m * acos(Float64(cos(phi2) * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda1) * cos(lambda2))))));
	else
		tmp = t_0;
	end
	return Float64(R_s * tmp)
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(R$95$m * N[ArcCos[N[(N[Cos[lambda1], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi1, -6.2e-6], t$95$0, If[LessEqual[phi1, 3.7e-18], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -6.1999999999999999e-6 or 3.7000000000000003e-18 < phi1

   1. Initial program 84.3%

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

   3. Taylor expanded in lambda2 around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-sin.f64 66.4

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

   5. Simplified 66.4%

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

   if -6.1999999999999999e-6 < phi1 < 3.7000000000000003e-18

   1. Initial program 69.7%

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

   4. Applied egg-rr 89.2%

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

   5. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 88.6

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

   7. Simplified 88.6%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6.2 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\phi_1 \leq 3.7 \cdot 10^{-18}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 63.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 6.5 \cdot 10^{-6}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi2 6.5e-6)
    (*
     R_m
     (acos
      (*
       (cos phi1)
       (fma (cos lambda2) (cos lambda1) (* (sin lambda2) (sin lambda1))))))
    (*
     R_m
     (acos
      (*
       (cos phi2)
       (fma (sin lambda1) (sin lambda2) (* (cos lambda1) (cos lambda2)))))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 6.5e-6) {
		tmp = R_m * acos((cos(phi1) * fma(cos(lambda2), cos(lambda1), (sin(lambda2) * sin(lambda1)))));
	} else {
		tmp = R_m * acos((cos(phi2) * fma(sin(lambda1), sin(lambda2), (cos(lambda1) * cos(lambda2)))));
	}
	return R_s * tmp;
}

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 6.5e-6)
		tmp = Float64(R_m * acos(Float64(cos(phi1) * fma(cos(lambda2), cos(lambda1), Float64(sin(lambda2) * sin(lambda1))))));
	else
		tmp = Float64(R_m * acos(Float64(cos(phi2) * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda1) * cos(lambda2))))));
	end
	return Float64(R_s * tmp)
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi2, 6.5e-6], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[lambda2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision] + N[(N[Sin[lambda2], $MachinePrecision] * N[Sin[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi2 < 6.4999999999999996e-6

   1. Initial program 77.5%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 51.2

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

   5. Simplified 51.2%

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

      1. cos-diff N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-sin.f64 N/A

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

      12. lower-*.f64 61.8

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

   7. Applied egg-rr 61.8%

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

   if 6.4999999999999996e-6 < phi2

   1. Initial program 77.0%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 58.9

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

   7. Simplified 58.9%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 6.5 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\cos \lambda_2, \cos \lambda_1, \sin \lambda_2 \cdot \sin \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 60.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.00011:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi1 -0.00011)
    (* R_m (acos (* (cos phi1) (cos (- lambda2 lambda1)))))
    (*
     R_m
     (acos
      (*
       (cos phi2)
       (fma (sin lambda1) (sin lambda2) (* (cos lambda1) (cos lambda2)))))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.00011) {
		tmp = R_m * acos((cos(phi1) * cos((lambda2 - lambda1))));
	} else {
		tmp = R_m * acos((cos(phi2) * fma(sin(lambda1), sin(lambda2), (cos(lambda1) * cos(lambda2)))));
	}
	return R_s * tmp;
}

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -0.00011)
		tmp = Float64(R_m * acos(Float64(cos(phi1) * cos(Float64(lambda2 - lambda1)))));
	else
		tmp = Float64(R_m * acos(Float64(cos(phi2) * fma(sin(lambda1), sin(lambda2), Float64(cos(lambda1) * cos(lambda2))))));
	end
	return Float64(R_s * tmp)
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi1, -0.00011], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision] + N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.10000000000000004e-4

   1. Initial program 85.7%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. remove-double-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-neg-in N/A

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

      7. +-commutative N/A

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

      8. cos-neg N/A

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

      9. lower-cos.f64 N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. lower--.f64 N/A

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

      13. lower-cos.f64 47.5

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

   5. Simplified 47.5%

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

   if -1.10000000000000004e-4 < phi1

   1. Initial program 74.7%

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

   4. Applied egg-rr 92.9%

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

   5. Taylor expanded in phi1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-cos.f64 N/A

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

      12. lower-cos.f64 62.8

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

   7. Simplified 62.8%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.00011:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \mathsf{fma}\left(\sin \lambda_1, \sin \lambda_2, \cos \lambda_1 \cdot \cos \lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 56.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.25 \cdot 10^{-5}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\lambda_2 \cdot \mathsf{fma}\left(\cos \phi_1, \frac{\cos \phi_2}{\lambda_2}, \sin \phi_1 \cdot \frac{\sin \phi_2}{\lambda_2}\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 0.00065:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(t\_0, \cos \phi_1, \sin \phi_1 \cdot \phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.5 \cdot 10^{+67}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (*
    R_s
    (if (<= phi2 -1.25e-5)
      (*
       R_m
       (acos
        (*
         lambda2
         (fma
          (cos phi1)
          (/ (cos phi2) lambda2)
          (* (sin phi1) (/ (sin phi2) lambda2))))))
      (if (<= phi2 0.00065)
        (* R_m (acos (fma t_0 (cos phi1) (* (sin phi1) phi2))))
        (if (<= phi2 1.5e+67)
          (* R_m (acos (fma (cos phi1) (cos phi2) (* (sin phi1) (sin phi2)))))
          (* R_m (acos (* (cos phi2) t_0)))))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= -1.25e-5) {
		tmp = R_m * acos((lambda2 * fma(cos(phi1), (cos(phi2) / lambda2), (sin(phi1) * (sin(phi2) / lambda2)))));
	} else if (phi2 <= 0.00065) {
		tmp = R_m * acos(fma(t_0, cos(phi1), (sin(phi1) * phi2)));
	} else if (phi2 <= 1.5e+67) {
		tmp = R_m * acos(fma(cos(phi1), cos(phi2), (sin(phi1) * sin(phi2))));
	} else {
		tmp = R_m * acos((cos(phi2) * t_0));
	}
	return R_s * tmp;
}

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi2 <= -1.25e-5)
		tmp = Float64(R_m * acos(Float64(lambda2 * fma(cos(phi1), Float64(cos(phi2) / lambda2), Float64(sin(phi1) * Float64(sin(phi2) / lambda2))))));
	elseif (phi2 <= 0.00065)
		tmp = Float64(R_m * acos(fma(t_0, cos(phi1), Float64(sin(phi1) * phi2))));
	elseif (phi2 <= 1.5e+67)
		tmp = Float64(R_m * acos(fma(cos(phi1), cos(phi2), Float64(sin(phi1) * sin(phi2)))));
	else
		tmp = Float64(R_m * acos(Float64(cos(phi2) * t_0)));
	end
	return Float64(R_s * tmp)
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi2, -1.25e-5], N[(R$95$m * N[ArcCos[N[(lambda2 * N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] / lambda2), $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[(N[Sin[phi2], $MachinePrecision] / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 0.00065], N[(R$95$m * N[ArcCos[N[(t$95$0 * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.5e+67], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if phi2 < -1.25000000000000006e-5

   1. Initial program 86.6%

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

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\lambda_2 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right) + \left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\left(\lambda_2 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right) + \cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]

      2. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \lambda_2 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right)\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \lambda_2 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      4. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1 + \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right) \cdot \lambda_2}\right) + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      5. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1 + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \lambda_1\right)} \cdot \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      6. associate-*l* N/A

         \[\leadsto \cos^{-1} \left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1 + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \lambda_2\right)}\right) + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      7. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\lambda_2 \cdot \sin \lambda_1\right)}\right) + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      8. distribute-lft-out N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 + \lambda_2 \cdot \sin \lambda_1\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      9. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1 + \lambda_2 \cdot \sin \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

   5. Simplified 60.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \mathsf{fma}\left(\lambda_2, \sin \lambda_1, \cos \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

   6. Taylor expanded in lambda2 around inf

      \[\leadsto \cos^{-1} \color{blue}{\left(\lambda_2 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right) + \left(\frac{\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}{\lambda_2} + \frac{\sin \phi_1 \cdot \sin \phi_2}{\lambda_2}\right)\right)\right)} \cdot R \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\lambda_2 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right) + \left(\frac{\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}{\lambda_2} + \frac{\sin \phi_1 \cdot \sin \phi_2}{\lambda_2}\right)\right)\right)} \cdot R \]

      2. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\lambda_2 \cdot \color{blue}{\left(\left(\frac{\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}{\lambda_2} + \frac{\sin \phi_1 \cdot \sin \phi_2}{\lambda_2}\right) + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right)}\right) \cdot R \]

      3. associate-+l+ N/A

         \[\leadsto \cos^{-1} \left(\lambda_2 \cdot \color{blue}{\left(\frac{\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)}{\lambda_2} + \left(\frac{\sin \phi_1 \cdot \sin \phi_2}{\lambda_2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right)\right)}\right) \cdot R \]

      4. associate-/l* N/A

         \[\leadsto \cos^{-1} \left(\lambda_2 \cdot \left(\color{blue}{\cos \lambda_1 \cdot \frac{\cos \phi_1 \cdot \cos \phi_2}{\lambda_2}} + \left(\frac{\sin \phi_1 \cdot \sin \phi_2}{\lambda_2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right)\right)\right) \cdot R \]

      5. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\lambda_2 \cdot \color{blue}{\mathsf{fma}\left(\cos \lambda_1, \frac{\cos \phi_1 \cdot \cos \phi_2}{\lambda_2}, \frac{\sin \phi_1 \cdot \sin \phi_2}{\lambda_2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right)}\right) \cdot R \]

      6. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\lambda_2 \cdot \mathsf{fma}\left(\color{blue}{\cos \lambda_1}, \frac{\cos \phi_1 \cdot \cos \phi_2}{\lambda_2}, \frac{\sin \phi_1 \cdot \sin \phi_2}{\lambda_2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right)\right) \cdot R \]

      7. associate-/l* N/A

         \[\leadsto \cos^{-1} \left(\lambda_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1 \cdot \frac{\cos \phi_2}{\lambda_2}}, \frac{\sin \phi_1 \cdot \sin \phi_2}{\lambda_2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right)\right) \cdot R \]

      8. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\lambda_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1 \cdot \frac{\cos \phi_2}{\lambda_2}}, \frac{\sin \phi_1 \cdot \sin \phi_2}{\lambda_2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right)\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\lambda_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \color{blue}{\cos \phi_1} \cdot \frac{\cos \phi_2}{\lambda_2}, \frac{\sin \phi_1 \cdot \sin \phi_2}{\lambda_2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right)\right) \cdot R \]

      10. lower-/.f64 N/A

          \[\leadsto \cos^{-1} \left(\lambda_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \color{blue}{\frac{\cos \phi_2}{\lambda_2}}, \frac{\sin \phi_1 \cdot \sin \phi_2}{\lambda_2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right)\right) \cdot R \]

      11. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\lambda_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \frac{\color{blue}{\cos \phi_2}}{\lambda_2}, \frac{\sin \phi_1 \cdot \sin \phi_2}{\lambda_2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right)\right) \cdot R \]

   8. Simplified 60.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\lambda_2 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \phi_1 \cdot \frac{\cos \phi_2}{\lambda_2}, \mathsf{fma}\left(\sin \phi_1, \frac{\sin \phi_2}{\lambda_2}, \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right)\right)\right)} \cdot R \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\lambda_2 \cdot \left(\frac{\cos \phi_1 \cdot \cos \phi_2}{\lambda_2} + \frac{\sin \phi_1 \cdot \sin \phi_2}{\lambda_2}\right)\right)} \cdot R \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \color{blue}{\left(\lambda_2 \cdot \left(\frac{\cos \phi_1 \cdot \cos \phi_2}{\lambda_2} + \frac{\sin \phi_1 \cdot \sin \phi_2}{\lambda_2}\right)\right)} \cdot R \]

       2. associate-/l* N/A

          \[\leadsto \cos^{-1} \left(\lambda_2 \cdot \left(\color{blue}{\cos \phi_1 \cdot \frac{\cos \phi_2}{\lambda_2}} + \frac{\sin \phi_1 \cdot \sin \phi_2}{\lambda_2}\right)\right) \cdot R \]

       3. lower-fma.f64 N/A

          \[\leadsto \cos^{-1} \left(\lambda_2 \cdot \color{blue}{\mathsf{fma}\left(\cos \phi_1, \frac{\cos \phi_2}{\lambda_2}, \frac{\sin \phi_1 \cdot \sin \phi_2}{\lambda_2}\right)}\right) \cdot R \]

       4. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\lambda_2 \cdot \mathsf{fma}\left(\color{blue}{\cos \phi_1}, \frac{\cos \phi_2}{\lambda_2}, \frac{\sin \phi_1 \cdot \sin \phi_2}{\lambda_2}\right)\right) \cdot R \]

       5. lower-/.f64 N/A

          \[\leadsto \cos^{-1} \left(\lambda_2 \cdot \mathsf{fma}\left(\cos \phi_1, \color{blue}{\frac{\cos \phi_2}{\lambda_2}}, \frac{\sin \phi_1 \cdot \sin \phi_2}{\lambda_2}\right)\right) \cdot R \]

       6. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\lambda_2 \cdot \mathsf{fma}\left(\cos \phi_1, \frac{\color{blue}{\cos \phi_2}}{\lambda_2}, \frac{\sin \phi_1 \cdot \sin \phi_2}{\lambda_2}\right)\right) \cdot R \]

       7. associate-/l* N/A

          \[\leadsto \cos^{-1} \left(\lambda_2 \cdot \mathsf{fma}\left(\cos \phi_1, \frac{\cos \phi_2}{\lambda_2}, \color{blue}{\sin \phi_1 \cdot \frac{\sin \phi_2}{\lambda_2}}\right)\right) \cdot R \]

       8. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\lambda_2 \cdot \mathsf{fma}\left(\cos \phi_1, \frac{\cos \phi_2}{\lambda_2}, \color{blue}{\sin \phi_1 \cdot \frac{\sin \phi_2}{\lambda_2}}\right)\right) \cdot R \]

       9. lower-sin.f64 N/A

          \[\leadsto \cos^{-1} \left(\lambda_2 \cdot \mathsf{fma}\left(\cos \phi_1, \frac{\cos \phi_2}{\lambda_2}, \color{blue}{\sin \phi_1} \cdot \frac{\sin \phi_2}{\lambda_2}\right)\right) \cdot R \]

       10. lower-/.f64 N/A

           \[\leadsto \cos^{-1} \left(\lambda_2 \cdot \mathsf{fma}\left(\cos \phi_1, \frac{\cos \phi_2}{\lambda_2}, \sin \phi_1 \cdot \color{blue}{\frac{\sin \phi_2}{\lambda_2}}\right)\right) \cdot R \]

       11. lower-sin.f64 47.4

           \[\leadsto \cos^{-1} \left(\lambda_2 \cdot \mathsf{fma}\left(\cos \phi_1, \frac{\cos \phi_2}{\lambda_2}, \sin \phi_1 \cdot \frac{\color{blue}{\sin \phi_2}}{\lambda_2}\right)\right) \cdot R \]

   11. Simplified 47.4%

       \[\leadsto \cos^{-1} \color{blue}{\left(\lambda_2 \cdot \mathsf{fma}\left(\cos \phi_1, \frac{\cos \phi_2}{\lambda_2}, \sin \phi_1 \cdot \frac{\sin \phi_2}{\lambda_2}\right)\right)} \cdot R \]

   if -1.25000000000000006e-5 < phi2 < 6.4999999999999997e-4

   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

      \[\leadsto \cos^{-1} \color{blue}{\left(\phi_2 \cdot \sin \phi_1 + \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_2 \cdot \sin \phi_1\right)} \cdot R \]

      2. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1} + \phi_2 \cdot \sin \phi_1\right) \cdot R \]

      3. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

      4. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}, \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      5. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right), \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      6. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right), \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      7. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}, \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      8. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right), \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      9. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}, \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      10. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}, \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right), \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      12. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}, \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      13. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}, \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      14. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \color{blue}{\cos \phi_1}, \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      15. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_1, \color{blue}{\phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      16. lower-sin.f64 71.9

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_1, \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Simplified 71.9%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   if 6.4999999999999997e-4 < phi2 < 1.50000000000000005e67

   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 lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\lambda_2 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right) + \left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\left(\lambda_2 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right) + \cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]

      2. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \lambda_2 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right)\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \lambda_2 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      4. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1 + \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right) \cdot \lambda_2}\right) + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      5. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1 + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \lambda_1\right)} \cdot \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      6. associate-*l* N/A

         \[\leadsto \cos^{-1} \left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1 + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \lambda_2\right)}\right) + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      7. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\lambda_2 \cdot \sin \lambda_1\right)}\right) + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      8. distribute-lft-out N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 + \lambda_2 \cdot \sin \lambda_1\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      9. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1 + \lambda_2 \cdot \sin \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

   5. Simplified 47.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \mathsf{fma}\left(\lambda_2, \sin \lambda_1, \cos \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_1}, \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right)\right) \cdot R \]

      5. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \color{blue}{\sin \phi_1} \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-sin.f64 48.8

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \sin \phi_1 \cdot \color{blue}{\sin \phi_2}\right)\right) \cdot R \]

   8. Simplified 48.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

   if 1.50000000000000005e67 < phi2

   1. Initial program 76.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_2\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_2\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_2\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_2\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_2\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_2\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_2\right) \cdot R \]

      13. lower-cos.f64 50.4

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]

   5. Simplified 50.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\right)} \cdot R \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.25 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\lambda_2 \cdot \mathsf{fma}\left(\cos \phi_1, \frac{\cos \phi_2}{\lambda_2}, \sin \phi_1 \cdot \frac{\sin \phi_2}{\lambda_2}\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 0.00065:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_1, \sin \phi_1 \cdot \phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.5 \cdot 10^{+67}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 56.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ t_1 := \cos \left(\lambda_2 - \lambda_1\right)\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.25 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 0.00065:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\mathsf{fma}\left(t\_1, \cos \phi_1, \sin \phi_1 \cdot \phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.5 \cdot 10^{+67}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (* R_m (acos (fma (cos phi1) (cos phi2) (* (sin phi1) (sin phi2))))))
        (t_1 (cos (- lambda2 lambda1))))
   (*
    R_s
    (if (<= phi2 -1.25e-5)
      t_0
      (if (<= phi2 0.00065)
        (* R_m (acos (fma t_1 (cos phi1) (* (sin phi1) phi2))))
        (if (<= phi2 1.5e+67) t_0 (* R_m (acos (* (cos phi2) t_1)))))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R_m * acos(fma(cos(phi1), cos(phi2), (sin(phi1) * sin(phi2))));
	double t_1 = cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= -1.25e-5) {
		tmp = t_0;
	} else if (phi2 <= 0.00065) {
		tmp = R_m * acos(fma(t_1, cos(phi1), (sin(phi1) * phi2)));
	} else if (phi2 <= 1.5e+67) {
		tmp = t_0;
	} else {
		tmp = R_m * acos((cos(phi2) * t_1));
	}
	return R_s * tmp;
}

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R_m * acos(fma(cos(phi1), cos(phi2), Float64(sin(phi1) * sin(phi2)))))
	t_1 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi2 <= -1.25e-5)
		tmp = t_0;
	elseif (phi2 <= 0.00065)
		tmp = Float64(R_m * acos(fma(t_1, cos(phi1), Float64(sin(phi1) * phi2))));
	elseif (phi2 <= 1.5e+67)
		tmp = t_0;
	else
		tmp = Float64(R_m * acos(Float64(cos(phi2) * t_1)));
	end
	return Float64(R_s * tmp)
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * N[Sin[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi2, -1.25e-5], t$95$0, If[LessEqual[phi2, 0.00065], N[(R$95$m * N[ArcCos[N[(t$95$1 * N[Cos[phi1], $MachinePrecision] + N[(N[Sin[phi1], $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.5e+67], t$95$0, N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.25000000000000006e-5 or 6.4999999999999997e-4 < phi2 < 1.50000000000000005e67

   1. Initial program 85.3%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\lambda_2 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right) + \left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\left(\lambda_2 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right) + \cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]

      2. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \lambda_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right) + \lambda_2 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right)\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      3. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1} + \lambda_2 \cdot \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right)\right) + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      4. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1 + \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \sin \lambda_1\right)\right) \cdot \lambda_2}\right) + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      5. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1 + \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \lambda_1\right)} \cdot \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      6. associate-*l* N/A

         \[\leadsto \cos^{-1} \left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1 + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \lambda_1 \cdot \lambda_2\right)}\right) + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      7. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \lambda_1 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\lambda_2 \cdot \sin \lambda_1\right)}\right) + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      8. distribute-lft-out N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \lambda_1 + \lambda_2 \cdot \sin \lambda_1\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      9. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \cos \lambda_1 + \lambda_2 \cdot \sin \lambda_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

   5. Simplified 58.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \mathsf{fma}\left(\lambda_2, \sin \lambda_1, \cos \lambda_1\right), \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2 + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      2. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \phi_1}, \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2}, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      4. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \color{blue}{\sin \phi_1 \cdot \sin \phi_2}\right)\right) \cdot R \]

      5. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \color{blue}{\sin \phi_1} \cdot \sin \phi_2\right)\right) \cdot R \]

      6. lower-sin.f64 47.7

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \sin \phi_1 \cdot \color{blue}{\sin \phi_2}\right)\right) \cdot R \]

   8. Simplified 47.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

   if -1.25000000000000006e-5 < phi2 < 6.4999999999999997e-4

   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

      \[\leadsto \cos^{-1} \color{blue}{\left(\phi_2 \cdot \sin \phi_1 + \cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right) + \phi_2 \cdot \sin \phi_1\right)} \cdot R \]

      2. *-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1} + \phi_2 \cdot \sin \phi_1\right) \cdot R \]

      3. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

      4. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}, \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      5. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right), \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      6. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right), \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      7. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)}, \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      8. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right), \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      9. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}, \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      10. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)}, \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      11. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right), \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      12. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}, \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      13. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)}, \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      14. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \color{blue}{\cos \phi_1}, \phi_2 \cdot \sin \phi_1\right)\right) \cdot R \]

      15. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_1, \color{blue}{\phi_2 \cdot \sin \phi_1}\right)\right) \cdot R \]

      16. lower-sin.f64 71.9

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_1, \phi_2 \cdot \color{blue}{\sin \phi_1}\right)\right) \cdot R \]

   5. Simplified 71.9%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_1, \phi_2 \cdot \sin \phi_1\right)\right)} \cdot R \]

   if 1.50000000000000005e67 < phi2

   1. Initial program 76.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_2\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_2\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_2\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_2\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_2\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_2\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_2\right) \cdot R \]

      13. lower-cos.f64 50.4

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]

   5. Simplified 50.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\right)} \cdot R \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.25 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 0.00065:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \cos \phi_1, \sin \phi_1 \cdot \phi_2\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.5 \cdot 10^{+67}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2, \sin \phi_1 \cdot \sin \phi_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 33.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_1 - \lambda_2\right)\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0.999999:\\ \;\;\;\;R\_m \cdot \cos^{-1} t\_0\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\lambda_2, \mathsf{fma}\left(\lambda_2, -0.5, \lambda_1\right), 1\right)\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda1 lambda2))))
   (*
    R_s
    (if (<= t_0 0.999999)
      (* R_m (acos t_0))
      (*
       R_m
       (acos (* (cos phi1) (fma lambda2 (fma lambda2 -0.5 lambda1) 1.0))))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda1 - lambda2));
	double tmp;
	if (t_0 <= 0.999999) {
		tmp = R_m * acos(t_0);
	} else {
		tmp = R_m * acos((cos(phi1) * fma(lambda2, fma(lambda2, -0.5, lambda1), 1.0)));
	}
	return R_s * tmp;
}

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda1 - lambda2))
	tmp = 0.0
	if (t_0 <= 0.999999)
		tmp = Float64(R_m * acos(t_0));
	else
		tmp = Float64(R_m * acos(Float64(cos(phi1) * fma(lambda2, fma(lambda2, -0.5, lambda1), 1.0))));
	end
	return Float64(R_s * tmp)
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]}, N[(R$95$s * If[LessEqual[t$95$0, 0.999999], N[(R$95$m * N[ArcCos[t$95$0], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[(lambda2 * N[(lambda2 * -0.5 + lambda1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (-.f64 lambda1 lambda2)) < 0.999998999999999971

   1. Initial program 74.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. lower-cos.f64 46.0

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Simplified 46.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

      2. remove-double-neg N/A

         \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

      3. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

      4. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

      5. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      8. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

      9. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      10. lower--.f64 32.8

          \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   8. Simplified 32.8%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

   if 0.999998999999999971 < (cos.f64 (-.f64 lambda1 lambda2))

   1. Initial program 83.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. lower-cos.f64 35.6

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Simplified 35.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \lambda_2 + \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\mathsf{fma}\left(\lambda_1, \sin \lambda_2, \cos \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      3. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\lambda_1, \color{blue}{\sin \lambda_2}, \cos \lambda_2\right) \cdot \cos \phi_1\right) \cdot R \]

      4. lower-cos.f64 35.5

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\lambda_1, \sin \lambda_2, \color{blue}{\cos \lambda_2}\right) \cdot \cos \phi_1\right) \cdot R \]

   8. Simplified 35.5%

      \[\leadsto \cos^{-1} \left(\color{blue}{\mathsf{fma}\left(\lambda_1, \sin \lambda_2, \cos \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\color{blue}{\left(1 + \lambda_2 \cdot \left(\lambda_1 + \frac{-1}{2} \cdot \lambda_2\right)\right)} \cdot \cos \phi_1\right) \cdot R \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\left(\lambda_2 \cdot \left(\lambda_1 + \frac{-1}{2} \cdot \lambda_2\right) + 1\right)} \cdot \cos \phi_1\right) \cdot R \]

       2. lower-fma.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\mathsf{fma}\left(\lambda_2, \lambda_1 + \frac{-1}{2} \cdot \lambda_2, 1\right)} \cdot \cos \phi_1\right) \cdot R \]

       3. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\lambda_2, \color{blue}{\frac{-1}{2} \cdot \lambda_2 + \lambda_1}, 1\right) \cdot \cos \phi_1\right) \cdot R \]

       4. *-commutative N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\lambda_2, \color{blue}{\lambda_2 \cdot \frac{-1}{2}} + \lambda_1, 1\right) \cdot \cos \phi_1\right) \cdot R \]

       5. lower-fma.f64 35.5

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\lambda_2, \color{blue}{\mathsf{fma}\left(\lambda_2, -0.5, \lambda_1\right)}, 1\right) \cdot \cos \phi_1\right) \cdot R \]

   11. Simplified 35.5%

       \[\leadsto \cos^{-1} \left(\color{blue}{\mathsf{fma}\left(\lambda_2, \mathsf{fma}\left(\lambda_2, -0.5, \lambda_1\right), 1\right)} \cdot \cos \phi_1\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\lambda_1 - \lambda_2\right) \leq 0.999999:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \mathsf{fma}\left(\lambda_2, \mathsf{fma}\left(\lambda_2, -0.5, \lambda_1\right), 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 36.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.002:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\phi_1 \leq -5.2 \cdot 10^{-109}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi1 -0.002)
    (* R_m (acos (* (cos phi1) (cos lambda1))))
    (if (<= phi1 -5.2e-109)
      (* R_m (acos (* (cos (- lambda1 lambda2)) (fma (* phi1 phi1) -0.5 1.0))))
      (* R_m (acos (* (cos phi2) (cos lambda1))))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.002) {
		tmp = R_m * acos((cos(phi1) * cos(lambda1)));
	} else if (phi1 <= -5.2e-109) {
		tmp = R_m * acos((cos((lambda1 - lambda2)) * fma((phi1 * phi1), -0.5, 1.0)));
	} else {
		tmp = R_m * acos((cos(phi2) * cos(lambda1)));
	}
	return R_s * tmp;
}

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -0.002)
		tmp = Float64(R_m * acos(Float64(cos(phi1) * cos(lambda1))));
	elseif (phi1 <= -5.2e-109)
		tmp = Float64(R_m * acos(Float64(cos(Float64(lambda1 - lambda2)) * fma(Float64(phi1 * phi1), -0.5, 1.0))));
	else
		tmp = Float64(R_m * acos(Float64(cos(phi2) * cos(lambda1))));
	end
	return Float64(R_s * tmp)
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi1, -0.002], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, -5.2e-109], N[(R$95$m * N[ArcCos[N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * N[(N[(phi1 * phi1), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2e-3

   1. Initial program 85.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. lower-cos.f64 47.5

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Simplified 47.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\lambda_1\right)\right)} \cdot \cos \phi_1\right) \cdot R \]
   7. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_1} \cdot \cos \phi_1\right) \cdot R \]

      2. lower-cos.f64 43.6

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_1} \cdot \cos \phi_1\right) \cdot R \]

   8. Simplified 43.6%

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_1} \cdot \cos \phi_1\right) \cdot R \]

   if -2e-3 < phi1 < -5.1999999999999997e-109

   1. Initial program 84.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. lower-cos.f64 52.4

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Simplified 52.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) + \frac{-1}{2} \cdot \left({\phi_1}^{2} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\right)} \cdot R \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) + \color{blue}{\left(\frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)}\right) \cdot R \]

      2. distribute-rgt1-in N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\left(\frac{-1}{2} \cdot {\phi_1}^{2} + 1\right) \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)} \cdot R \]

      3. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right)} \cdot \cos \left(\lambda_2 - \lambda_1\right)\right) \cdot R \]

      4. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)}\right) \cdot R \]

      5. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_2 + \color{blue}{-1 \cdot \lambda_1}\right)\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\left(\lambda_2 + -1 \cdot \lambda_1\right)\right)\right)}\right) \cdot R \]

      7. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\mathsf{neg}\left(\left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right)\right)\right)\right) \cdot R \]

      8. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\left(\left(\mathsf{neg}\left(\lambda_1\right)\right) + \lambda_2\right)}\right)\right)\right) \cdot R \]

      9. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \color{blue}{\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

      10. remove-double-neg N/A

          \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\color{blue}{\lambda_1} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right) \cdot R \]

      11. sub-neg N/A

          \[\leadsto \cos^{-1} \left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      12. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \color{blue}{\left(\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

      13. +-commutative N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\left(\frac{-1}{2} \cdot {\phi_1}^{2} + 1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      14. *-commutative N/A

          \[\leadsto \cos^{-1} \left(\left(\color{blue}{{\phi_1}^{2} \cdot \frac{-1}{2}} + 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      15. lower-fma.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\mathsf{fma}\left({\phi_1}^{2}, \frac{-1}{2}, 1\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      16. unpow2 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\phi_1 \cdot \phi_1}, \frac{-1}{2}, 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      17. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\phi_1 \cdot \phi_1}, \frac{-1}{2}, 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      18. sub-neg N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1 \cdot \phi_1, \frac{-1}{2}, 1\right) \cdot \cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right) \cdot R \]

      19. neg-mul-1 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1 \cdot \phi_1, \frac{-1}{2}, 1\right) \cdot \cos \left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right) \cdot R \]

      20. lower-cos.f64 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1 \cdot \phi_1, \frac{-1}{2}, 1\right) \cdot \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)}\right) \cdot R \]

      21. neg-mul-1 N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\phi_1 \cdot \phi_1, \frac{-1}{2}, 1\right) \cdot \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right)\right) \cdot R \]

   8. Simplified 52.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]

   if -5.1999999999999997e-109 < phi1

   1. Initial program 73.6%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\sin \phi_1} \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \color{blue}{\sin \phi_2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      5. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      6. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      7. lift--.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      8. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      9. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]

      11. lift-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      12. lift-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      13. associate-*l* N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      14. *-commutative N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      15. lower-fma.f64 N/A

          \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      16. *-commutative N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      17. lower-*.f64 73.6

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   4. Applied egg-rr 73.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

   5. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \lambda_1} \cdot \cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-cos.f64 55.8

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \lambda_1} \cdot \cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   7. Simplified 55.8%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \lambda_1} \cdot \cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_2\right)} \cdot R \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \lambda_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \lambda_1\right)} \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \lambda_1\right) \cdot R \]

      4. lower-cos.f64 36.2

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \lambda_1}\right) \cdot R \]

   10. Simplified 36.2%

       \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \lambda_1\right)} \cdot R \]
3. Recombined 3 regimes into one program.

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.002:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{elif}\;\phi_1 \leq -5.2 \cdot 10^{-109}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\phi_1 \cdot \phi_1, -0.5, 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 50.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 10^{-5}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot t\_0\right)\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1))))
   (*
    R_s
    (if (<= phi2 1e-5)
      (* R_m (acos (* (cos phi1) t_0)))
      (* R_m (acos (* (cos phi2) t_0)))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 1e-5) {
		tmp = R_m * acos((cos(phi1) * t_0));
	} else {
		tmp = R_m * acos((cos(phi2) * t_0));
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    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 <= 1d-5) then
        tmp = r_m * acos((cos(phi1) * t_0))
    else
        tmp = r_m * acos((cos(phi2) * t_0))
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((lambda2 - lambda1));
	double tmp;
	if (phi2 <= 1e-5) {
		tmp = R_m * Math.acos((Math.cos(phi1) * t_0));
	} else {
		tmp = R_m * Math.acos((Math.cos(phi2) * t_0));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((lambda2 - lambda1))
	tmp = 0
	if phi2 <= 1e-5:
		tmp = R_m * math.acos((math.cos(phi1) * t_0))
	else:
		tmp = R_m * math.acos((math.cos(phi2) * t_0))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	tmp = 0.0
	if (phi2 <= 1e-5)
		tmp = Float64(R_m * acos(Float64(cos(phi1) * t_0)));
	else
		tmp = Float64(R_m * acos(Float64(cos(phi2) * t_0)));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = cos((lambda2 - lambda1));
	tmp = 0.0;
	if (phi2 <= 1e-5)
		tmp = R_m * acos((cos(phi1) * t_0));
	else
		tmp = R_m * acos((cos(phi2) * t_0));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi2, 1e-5], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.00000000000000008e-5

   1. Initial program 77.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. lower-cos.f64 51.2

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Simplified 51.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   if 1.00000000000000008e-5 < phi2

   1. Initial program 77.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_2\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_2\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_2\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_2\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_2\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_2\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_2\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_2\right) \cdot R \]

      13. lower-cos.f64 46.1

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_2}\right) \cdot R \]

   5. Simplified 46.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_2\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 48.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi2 1.3e-5)
    (* R_m (acos (* (cos phi1) (cos (- lambda2 lambda1)))))
    (* R_m (acos (* (cos phi2) (cos lambda1)))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.3e-5) {
		tmp = R_m * acos((cos(phi1) * cos((lambda2 - lambda1))));
	} else {
		tmp = R_m * acos((cos(phi2) * cos(lambda1)));
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 1.3d-5) then
        tmp = r_m * acos((cos(phi1) * cos((lambda2 - lambda1))))
    else
        tmp = r_m * acos((cos(phi2) * cos(lambda1)))
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.3e-5) {
		tmp = R_m * Math.acos((Math.cos(phi1) * Math.cos((lambda2 - lambda1))));
	} else {
		tmp = R_m * Math.acos((Math.cos(phi2) * Math.cos(lambda1)));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.3e-5:
		tmp = R_m * math.acos((math.cos(phi1) * math.cos((lambda2 - lambda1))))
	else:
		tmp = R_m * math.acos((math.cos(phi2) * math.cos(lambda1)))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.3e-5)
		tmp = Float64(R_m * acos(Float64(cos(phi1) * cos(Float64(lambda2 - lambda1)))));
	else
		tmp = Float64(R_m * acos(Float64(cos(phi2) * cos(lambda1))));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.3e-5)
		tmp = R_m * acos((cos(phi1) * cos((lambda2 - lambda1))));
	else
		tmp = R_m * acos((cos(phi2) * cos(lambda1)));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi2, 1.3e-5], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[phi2], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.29999999999999992e-5

   1. Initial program 77.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. lower-cos.f64 51.2

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Simplified 51.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   if 1.29999999999999992e-5 < phi2

   1. Initial program 77.0%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\sin \phi_1} \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      2. lift-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \color{blue}{\sin \phi_2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      3. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\sin \phi_1 \cdot \sin \phi_2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      4. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\color{blue}{\cos \phi_1} \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      5. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \color{blue}{\cos \phi_2}\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      6. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]

      7. lift--.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      8. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      9. lift-*.f64 N/A

         \[\leadsto \cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right) \cdot R \]

      10. +-commutative N/A

          \[\leadsto \cos^{-1} \color{blue}{\left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right)} \cdot R \]

      11. lift-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      12. lift-*.f64 N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right)} \cdot \cos \left(\lambda_1 - \lambda_2\right) + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      13. associate-*l* N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      14. *-commutative N/A

          \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot \cos \phi_1} + \sin \phi_1 \cdot \sin \phi_2\right) \cdot R \]

      15. lower-fma.f64 N/A

          \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \phi_2 \cdot \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

      16. *-commutative N/A

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

      17. lower-*.f64 77.1

          \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2}, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   4. Applied egg-rr 77.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\mathsf{fma}\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right)} \cdot R \]

   5. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \lambda_1} \cdot \cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]
   6. Step-by-step derivation

      1. lower-cos.f64 58.8

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \lambda_1} \cdot \cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   7. Simplified 58.8%

      \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\color{blue}{\cos \lambda_1} \cdot \cos \phi_2, \cos \phi_1, \sin \phi_1 \cdot \sin \phi_2\right)\right) \cdot R \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \cos \phi_2\right)} \cdot R \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \lambda_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \lambda_1\right)} \cdot R \]

      3. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \phi_2} \cdot \cos \lambda_1\right) \cdot R \]

      4. lower-cos.f64 39.5

         \[\leadsto \cos^{-1} \left(\cos \phi_2 \cdot \color{blue}{\cos \lambda_1}\right) \cdot R \]

   10. Simplified 39.5%

       \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_2 \cdot \cos \lambda_1\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.3 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \left(\lambda_2 - \lambda_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_2 \cdot \cos \lambda_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 42.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.05 \cdot 10^{-42}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda1 -1.05e-42)
    (* R_m (acos (* (cos phi1) (cos lambda1))))
    (* R_m (acos (* (cos phi1) (cos lambda2)))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.05e-42) {
		tmp = R_m * acos((cos(phi1) * cos(lambda1)));
	} else {
		tmp = R_m * acos((cos(phi1) * cos(lambda2)));
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    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 <= (-1.05d-42)) then
        tmp = r_m * acos((cos(phi1) * cos(lambda1)))
    else
        tmp = r_m * acos((cos(phi1) * cos(lambda2)))
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.05e-42) {
		tmp = R_m * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R_m * Math.acos((Math.cos(phi1) * Math.cos(lambda2)));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -1.05e-42:
		tmp = R_m * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R_m * math.acos((math.cos(phi1) * math.cos(lambda2)))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.05e-42)
		tmp = Float64(R_m * acos(Float64(cos(phi1) * cos(lambda1))));
	else
		tmp = Float64(R_m * acos(Float64(cos(phi1) * cos(lambda2))));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -1.05e-42)
		tmp = R_m * acos((cos(phi1) * cos(lambda1)));
	else
		tmp = R_m * acos((cos(phi1) * cos(lambda2)));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda1, -1.05e-42], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.05000000000000003e-42

   1. Initial program 66.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. lower-cos.f64 36.6

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Simplified 36.6%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\lambda_1\right)\right)} \cdot \cos \phi_1\right) \cdot R \]
   7. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_1} \cdot \cos \phi_1\right) \cdot R \]

      2. lower-cos.f64 36.9

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_1} \cdot \cos \phi_1\right) \cdot R \]

   8. Simplified 36.9%

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_1} \cdot \cos \phi_1\right) \cdot R \]

   if -1.05000000000000003e-42 < lambda1

   1. Initial program 81.8%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. lower-cos.f64 45.5

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Simplified 45.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_2} \cdot \cos \phi_1\right) \cdot R \]
   7. Step-by-step derivation

      1. lower-cos.f64 35.7

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_2} \cdot \cos \phi_1\right) \cdot R \]

   8. Simplified 35.7%

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_2} \cdot \cos \phi_1\right) \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.05 \cdot 10^{-42}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 32.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.8 \cdot 10^{-8}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi1 -3.8e-8)
    (* R_m (acos (* (cos phi1) (cos lambda1))))
    (* R_m (acos (cos (- lambda1 lambda2)))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.8e-8) {
		tmp = R_m * acos((cos(phi1) * cos(lambda1)));
	} else {
		tmp = R_m * acos(cos((lambda1 - lambda2)));
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-3.8d-8)) then
        tmp = r_m * acos((cos(phi1) * cos(lambda1)))
    else
        tmp = r_m * acos(cos((lambda1 - lambda2)))
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.8e-8) {
		tmp = R_m * Math.acos((Math.cos(phi1) * Math.cos(lambda1)));
	} else {
		tmp = R_m * Math.acos(Math.cos((lambda1 - lambda2)));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -3.8e-8:
		tmp = R_m * math.acos((math.cos(phi1) * math.cos(lambda1)))
	else:
		tmp = R_m * math.acos(math.cos((lambda1 - lambda2)))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -3.8e-8)
		tmp = Float64(R_m * acos(Float64(cos(phi1) * cos(lambda1))));
	else
		tmp = Float64(R_m * acos(cos(Float64(lambda1 - lambda2))));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -3.8e-8)
		tmp = R_m * acos((cos(phi1) * cos(lambda1)));
	else
		tmp = R_m * acos(cos((lambda1 - lambda2)));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi1, -3.8e-8], N[(R$95$m * N[ArcCos[N[(N[Cos[phi1], $MachinePrecision] * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.80000000000000028e-8

   1. Initial program 86.2%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. lower-cos.f64 47.9

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Simplified 47.9%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\mathsf{neg}\left(\lambda_1\right)\right)} \cdot \cos \phi_1\right) \cdot R \]
   7. Step-by-step derivation

      1. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_1} \cdot \cos \phi_1\right) \cdot R \]

      2. lower-cos.f64 44.1

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_1} \cdot \cos \phi_1\right) \cdot R \]

   8. Simplified 44.1%

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_1} \cdot \cos \phi_1\right) \cdot R \]

   if -3.80000000000000028e-8 < phi1

   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

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. lower-cos.f64 41.3

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Simplified 41.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

      2. remove-double-neg N/A

         \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

      3. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

      4. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

      5. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      8. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

      9. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      10. lower--.f64 27.6

          \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   8. Simplified 27.6%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.8 \cdot 10^{-8}:\\ \;\;\;\;R \cdot \cos^{-1} \left(\cos \phi_1 \cdot \cos \lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 28.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -1.9 \cdot 10^{-299}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{elif}\;\lambda_2 \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \cos \phi_1\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda2 -1.9e-299)
    (* R_m (acos (cos lambda1)))
    (if (<= lambda2 4.5e-7)
      (* R_m (acos (cos phi1)))
      (* R_m (acos (cos lambda2)))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -1.9e-299) {
		tmp = R_m * acos(cos(lambda1));
	} else if (lambda2 <= 4.5e-7) {
		tmp = R_m * acos(cos(phi1));
	} else {
		tmp = R_m * acos(cos(lambda2));
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    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 <= (-1.9d-299)) then
        tmp = r_m * acos(cos(lambda1))
    else if (lambda2 <= 4.5d-7) then
        tmp = r_m * acos(cos(phi1))
    else
        tmp = r_m * acos(cos(lambda2))
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -1.9e-299) {
		tmp = R_m * Math.acos(Math.cos(lambda1));
	} else if (lambda2 <= 4.5e-7) {
		tmp = R_m * Math.acos(Math.cos(phi1));
	} else {
		tmp = R_m * Math.acos(Math.cos(lambda2));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= -1.9e-299:
		tmp = R_m * math.acos(math.cos(lambda1))
	elif lambda2 <= 4.5e-7:
		tmp = R_m * math.acos(math.cos(phi1))
	else:
		tmp = R_m * math.acos(math.cos(lambda2))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= -1.9e-299)
		tmp = Float64(R_m * acos(cos(lambda1)));
	elseif (lambda2 <= 4.5e-7)
		tmp = Float64(R_m * acos(cos(phi1)));
	else
		tmp = Float64(R_m * acos(cos(lambda2)));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= -1.9e-299)
		tmp = R_m * acos(cos(lambda1));
	elseif (lambda2 <= 4.5e-7)
		tmp = R_m * acos(cos(phi1));
	else
		tmp = R_m * acos(cos(lambda2));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda2, -1.9e-299], N[(R$95$m * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 4.5e-7], N[(R$95$m * N[ArcCos[N[Cos[phi1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[Cos[lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -1.9000000000000001e-299

   1. Initial program 76.1%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. lower-cos.f64 43.2

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Simplified 43.2%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

      2. remove-double-neg N/A

         \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

      3. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

      4. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

      5. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      8. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

      9. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      10. lower--.f64 27.4

          \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   8. Simplified 27.4%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]
   10. Step-by-step derivation

       1. lower-cos.f64 18.1

          \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]

   11. Simplified 18.1%

       \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]

   if -1.9000000000000001e-299 < lambda2 < 4.4999999999999998e-7

   1. Initial program 91.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. lower-cos.f64 48.3

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Simplified 48.3%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \lambda_2 + \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\mathsf{fma}\left(\lambda_1, \sin \lambda_2, \cos \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      3. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\lambda_1, \color{blue}{\sin \lambda_2}, \cos \lambda_2\right) \cdot \cos \phi_1\right) \cdot R \]

      4. lower-cos.f64 22.0

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\lambda_1, \sin \lambda_2, \color{blue}{\cos \lambda_2}\right) \cdot \cos \phi_1\right) \cdot R \]

   8. Simplified 22.0%

      \[\leadsto \cos^{-1} \left(\color{blue}{\mathsf{fma}\left(\lambda_1, \sin \lambda_2, \cos \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \phi_1} \cdot R \]
   10. Step-by-step derivation

       1. lower-cos.f64 25.6

          \[\leadsto \cos^{-1} \color{blue}{\cos \phi_1} \cdot R \]

   11. Simplified 25.6%

       \[\leadsto \cos^{-1} \color{blue}{\cos \phi_1} \cdot R \]

   if 4.4999999999999998e-7 < lambda2

   1. Initial program 58.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. lower-cos.f64 33.8

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Simplified 33.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

      2. remove-double-neg N/A

         \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

      3. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

      4. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

      5. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      8. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

      9. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      10. lower--.f64 24.9

          \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   8. Simplified 24.9%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot R \]
   10. Step-by-step derivation

       1. cos-neg N/A

          \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_2} \cdot R \]

       2. lower-cos.f64 24.7

          \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_2} \cdot R \]

   11. Simplified 24.7%

       \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_2} \cdot R \]
3. Recombined 3 regimes into one program.

4. Final simplification 21.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -1.9 \cdot 10^{-299}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{elif}\;\lambda_2 \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 30.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.00045:\\ \;\;\;\;R\_m \cdot \cos^{-1} \cos \phi_1\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi1 -0.00045)
    (* R_m (acos (cos phi1)))
    (* R_m (acos (cos (- lambda1 lambda2)))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.00045) {
		tmp = R_m * acos(cos(phi1));
	} else {
		tmp = R_m * acos(cos((lambda1 - lambda2)));
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-0.00045d0)) then
        tmp = r_m * acos(cos(phi1))
    else
        tmp = r_m * acos(cos((lambda1 - lambda2)))
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.00045) {
		tmp = R_m * Math.acos(Math.cos(phi1));
	} else {
		tmp = R_m * Math.acos(Math.cos((lambda1 - lambda2)));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -0.00045:
		tmp = R_m * math.acos(math.cos(phi1))
	else:
		tmp = R_m * math.acos(math.cos((lambda1 - lambda2)))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -0.00045)
		tmp = Float64(R_m * acos(cos(phi1)));
	else
		tmp = Float64(R_m * acos(cos(Float64(lambda1 - lambda2))));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -0.00045)
		tmp = R_m * acos(cos(phi1));
	else
		tmp = R_m * acos(cos((lambda1 - lambda2)));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi1, -0.00045], N[(R$95$m * N[ArcCos[N[Cos[phi1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.4999999999999999e-4

   1. Initial program 85.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. lower-cos.f64 47.5

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Simplified 47.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \left(\color{blue}{\left(\cos \lambda_2 + \lambda_1 \cdot \sin \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\left(\lambda_1 \cdot \sin \lambda_2 + \cos \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      2. lower-fma.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\mathsf{fma}\left(\lambda_1, \sin \lambda_2, \cos \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

      3. lower-sin.f64 N/A

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\lambda_1, \color{blue}{\sin \lambda_2}, \cos \lambda_2\right) \cdot \cos \phi_1\right) \cdot R \]

      4. lower-cos.f64 26.9

         \[\leadsto \cos^{-1} \left(\mathsf{fma}\left(\lambda_1, \sin \lambda_2, \color{blue}{\cos \lambda_2}\right) \cdot \cos \phi_1\right) \cdot R \]

   8. Simplified 26.9%

      \[\leadsto \cos^{-1} \left(\color{blue}{\mathsf{fma}\left(\lambda_1, \sin \lambda_2, \cos \lambda_2\right)} \cdot \cos \phi_1\right) \cdot R \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \phi_1} \cdot R \]
   10. Step-by-step derivation

       1. lower-cos.f64 30.1

          \[\leadsto \cos^{-1} \color{blue}{\cos \phi_1} \cdot R \]

   11. Simplified 30.1%

       \[\leadsto \cos^{-1} \color{blue}{\cos \phi_1} \cdot R \]

   if -4.4999999999999999e-4 < phi1

   1. Initial program 74.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. lower-cos.f64 41.5

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Simplified 41.5%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

      2. remove-double-neg N/A

         \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

      3. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

      4. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

      5. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      8. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

      9. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      10. lower--.f64 27.9

          \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   8. Simplified 27.9%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.00045:\\ \;\;\;\;R \cdot \cos^{-1} \cos \phi_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 26.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 9 \cdot 10^{-5}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda2 9e-5)
    (* R_m (acos (cos lambda1)))
    (* R_m (acos (cos lambda2))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 9e-5) {
		tmp = R_m * acos(cos(lambda1));
	} else {
		tmp = R_m * acos(cos(lambda2));
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    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 <= 9d-5) then
        tmp = r_m * acos(cos(lambda1))
    else
        tmp = r_m * acos(cos(lambda2))
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 9e-5) {
		tmp = R_m * Math.acos(Math.cos(lambda1));
	} else {
		tmp = R_m * Math.acos(Math.cos(lambda2));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 9e-5:
		tmp = R_m * math.acos(math.cos(lambda1))
	else:
		tmp = R_m * math.acos(math.cos(lambda2))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 9e-5)
		tmp = Float64(R_m * acos(cos(lambda1)));
	else
		tmp = Float64(R_m * acos(cos(lambda2)));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 9e-5)
		tmp = R_m * acos(cos(lambda1));
	else
		tmp = R_m * acos(cos(lambda2));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda2, 9e-5], N[(R$95$m * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[ArcCos[N[Cos[lambda2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 9.00000000000000057e-5

   1. Initial program 81.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. lower-cos.f64 45.0

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Simplified 45.0%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

      2. remove-double-neg N/A

         \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

      3. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

      4. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

      5. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      8. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

      9. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      10. lower--.f64 24.8

          \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   8. Simplified 24.8%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]
   10. Step-by-step derivation

       1. lower-cos.f64 18.8

          \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]

   11. Simplified 18.8%

       \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]

   if 9.00000000000000057e-5 < lambda2

   1. Initial program 58.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. lower-cos.f64 33.8

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Simplified 33.8%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

      2. remove-double-neg N/A

         \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

      3. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

      4. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

      5. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      8. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

      9. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      10. lower--.f64 24.9

          \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   8. Simplified 24.9%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

   9. Taylor expanded in lambda1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\mathsf{neg}\left(\lambda_2\right)\right)} \cdot R \]
   10. Step-by-step derivation

       1. cos-neg N/A

          \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_2} \cdot R \]

       2. lower-cos.f64 24.7

          \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_2} \cdot R \]

   11. Simplified 24.7%

       \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_2} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 9 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 24.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -6200000000:\\ \;\;\;\;R\_m \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \left|\left(\left(\lambda_2 - \lambda_1\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda1 -6200000000.0)
    (* R_m (acos (cos lambda1)))
    (* R_m (fabs (remainder (- lambda2 lambda1) (* PI 2.0)))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -6200000000.0) {
		tmp = R_m * acos(cos(lambda1));
	} else {
		tmp = R_m * fabs(remainder((lambda2 - lambda1), (((double) M_PI) * 2.0)));
	}
	return R_s * tmp;
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -6200000000.0) {
		tmp = R_m * Math.acos(Math.cos(lambda1));
	} else {
		tmp = R_m * Math.abs(Math.IEEEremainder((lambda2 - lambda1), (Math.PI * 2.0)));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -6200000000.0:
		tmp = R_m * math.acos(math.cos(lambda1))
	else:
		tmp = R_m * math.fabs(math.remainder((lambda2 - lambda1), (math.pi * 2.0)))
	return R_s * tmp

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda1, -6200000000.0], N[(R$95$m * N[ArcCos[N[Cos[lambda1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[Abs[N[With[{TMP1 = N[(lambda2 - lambda1), $MachinePrecision], TMP2 = N[(Pi * 2.0), $MachinePrecision]}, TMP1 - Round[TMP1 / TMP2] * TMP2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -6.2e9

   1. Initial program 64.7%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. lower-cos.f64 38.4

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Simplified 38.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

      2. remove-double-neg N/A

         \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

      3. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

      4. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

      5. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      8. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

      9. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      10. lower--.f64 30.6

          \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   8. Simplified 30.6%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

   9. Taylor expanded in lambda2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]
   10. Step-by-step derivation

       1. lower-cos.f64 30.8

          \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]

   11. Simplified 30.8%

       \[\leadsto \cos^{-1} \color{blue}{\cos \lambda_1} \cdot R \]

   if -6.2e9 < lambda1

   1. Initial program 81.5%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. lower-cos.f64 44.4

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Simplified 44.4%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

      2. remove-double-neg N/A

         \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

      3. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

      4. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

      5. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      8. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

      9. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      10. lower--.f64 22.9

          \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   8. Simplified 22.9%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      2. acos-cos-s N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

      3. lift-cos.f64 N/A

         \[\leadsto \cos^{-1} \cos \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

      4. lift-acos.f64 N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

      5. acos-cos N/A

         \[\leadsto \color{blue}{\left|\left(\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right|} \cdot R \]

      6. lower-fabs.f64 N/A

         \[\leadsto \color{blue}{\left|\left(\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right|} \cdot R \]

      7. lower-remainder.f64 N/A

         \[\leadsto \left|\color{blue}{\left(\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)}\right| \cdot R \]

   10. Applied egg-rr 18.6%

       \[\leadsto \color{blue}{\left|\left(\left(\lambda_2 - \lambda_1\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right|} \cdot R \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -6200000000:\\ \;\;\;\;R \cdot \cos^{-1} \cos \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left|\left(\left(\lambda_2 - \lambda_1\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 28: 19.7% accurate, 5.4× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \left(R\_m \cdot \left|\left(\left(\lambda_2 - \lambda_1\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right|\right) \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R_s (* R_m (fabs (remainder (- lambda2 lambda1) (* PI 2.0))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (R_m * fabs(remainder((lambda2 - lambda1), (((double) M_PI) * 2.0))));
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (R_m * Math.abs(Math.IEEEremainder((lambda2 - lambda1), (Math.PI * 2.0))));
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	return R_s * (R_m * math.fabs(math.remainder((lambda2 - lambda1), (math.pi * 2.0))))

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * N[(R$95$m * N[Abs[N[With[{TMP1 = N[(lambda2 - lambda1), $MachinePrecision], TMP2 = N[(Pi * 2.0), $MachinePrecision]}, TMP1 - Round[TMP1 / TMP2] * TMP2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Taylor expanded in phi2 around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   2. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   3. sub-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

   4. remove-double-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   5. mul-1-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   6. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

   7. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   8. cos-neg N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

   9. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

   10. mul-1-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

   11. unsub-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

   12. lower--.f64 N/A

       \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

   13. lower-cos.f64 42.9

       \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

5. Simplified 42.9%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

6. Taylor expanded in phi1 around 0

   \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

   2. remove-double-neg N/A

      \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

   3. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

   4. +-commutative N/A

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

   5. neg-mul-1 N/A

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

   6. cos-neg N/A

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

   7. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

   8. neg-mul-1 N/A

      \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

   9. sub-neg N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   10. lower--.f64 24.8

       \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

8. Simplified 24.8%

   \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
9. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   2. acos-cos-s N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

   3. lift-cos.f64 N/A

      \[\leadsto \cos^{-1} \cos \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

   4. lift-acos.f64 N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

   5. acos-cos N/A

      \[\leadsto \color{blue}{\left|\left(\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right|} \cdot R \]

   6. lower-fabs.f64 N/A

      \[\leadsto \color{blue}{\left|\left(\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)\right|} \cdot R \]

   7. lower-remainder.f64 N/A

      \[\leadsto \left|\color{blue}{\left(\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right) \mathsf{rem} \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)}\right| \cdot R \]

10. Applied egg-rr 18.9%

    \[\leadsto \color{blue}{\left|\left(\left(\lambda_2 - \lambda_1\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right|} \cdot R \]

11. Final simplification 18.9%

    \[\leadsto R \cdot \left|\left(\left(\lambda_2 - \lambda_1\right) \mathsf{rem} \left(\pi \cdot 2\right)\right)\right| \]
12. Add Preprocessing

Alternative 29: 8.8% accurate, 44.7× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3 \cdot 10^{-172}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\_m\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R_s (if (<= lambda1 -3e-172) (* lambda1 (- R_m)) (* lambda2 R_m))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -3e-172) {
		tmp = lambda1 * -R_m;
	} else {
		tmp = lambda2 * R_m;
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    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 <= (-3d-172)) then
        tmp = lambda1 * -r_m
    else
        tmp = lambda2 * r_m
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -3e-172) {
		tmp = lambda1 * -R_m;
	} else {
		tmp = lambda2 * R_m;
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -3e-172:
		tmp = lambda1 * -R_m
	else:
		tmp = lambda2 * R_m
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -3e-172)
		tmp = Float64(lambda1 * Float64(-R_m));
	else
		tmp = Float64(lambda2 * R_m);
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -3e-172)
		tmp = lambda1 * -R_m;
	else
		tmp = lambda2 * R_m;
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda1, -3e-172], N[(lambda1 * (-R$95$m)), $MachinePrecision], N[(lambda2 * R$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2.99999999999999984e-172

   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

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. lower-cos.f64 42.7

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Simplified 42.7%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

      2. remove-double-neg N/A

         \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

      3. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

      4. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

      5. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      8. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

      9. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      10. lower--.f64 27.2

          \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   8. Simplified 27.2%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
   9. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\frac{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}{\lambda_1 + \lambda_2}\right)} \cdot R \]

      2. clear-num N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\frac{1}{\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}\right)} \cdot R \]

      3. lower-/.f64 N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\frac{1}{\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}\right)} \cdot R \]

      4. lower-/.f64 N/A

         \[\leadsto \cos^{-1} \cos \left(\frac{1}{\color{blue}{\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}}\right) \cdot R \]

      5. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\color{blue}{\lambda_2 + \lambda_1}}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}\right) \cdot R \]

      6. lower-+.f64 N/A

         \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\color{blue}{\lambda_2 + \lambda_1}}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}\right) \cdot R \]

      7. difference-of-squares N/A

         \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\color{blue}{\left(\lambda_1 + \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

      8. lift--.f64 N/A

         \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_1 + \lambda_2\right) \cdot \color{blue}{\left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

      9. acos-cos-s N/A

         \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_1 + \lambda_2\right) \cdot \color{blue}{\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

      10. lift-cos.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_1 + \lambda_2\right) \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

      11. lift-acos.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_1 + \lambda_2\right) \cdot \color{blue}{\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

      12. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\color{blue}{\left(\lambda_1 + \lambda_2\right) \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

      13. +-commutative N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\color{blue}{\left(\lambda_2 + \lambda_1\right)} \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)}}\right) \cdot R \]

      14. lower-+.f64 27.1

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\color{blue}{\left(\lambda_2 + \lambda_1\right)} \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)}}\right) \cdot R \]

      15. lift-acos.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \color{blue}{\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

      16. lift-cos.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

      17. lift--.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

      18. cos-diff N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}}}\right) \cdot R \]

      19. lift-cos.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\color{blue}{\cos \lambda_1} \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}}\right) \cdot R \]

      20. lift-cos.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \lambda_2} + \sin \lambda_1 \cdot \sin \lambda_2\right)}}\right) \cdot R \]

      21. *-commutative N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\color{blue}{\cos \lambda_2 \cdot \cos \lambda_1} + \sin \lambda_1 \cdot \sin \lambda_2\right)}}\right) \cdot R \]

      22. lift-cos.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\color{blue}{\cos \lambda_2} \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}}\right) \cdot R \]

      23. lift-cos.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\cos \lambda_2 \cdot \color{blue}{\cos \lambda_1} + \sin \lambda_1 \cdot \sin \lambda_2\right)}}\right) \cdot R \]

      24. lift-sin.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_1} \cdot \sin \lambda_2\right)}}\right) \cdot R \]

      25. lift-sin.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \color{blue}{\sin \lambda_2}\right)}}\right) \cdot R \]

      26. *-commutative N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)}}\right) \cdot R \]

      27. lift-sin.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right)}}\right) \cdot R \]

      28. lift-sin.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)}}\right) \cdot R \]

   10. Applied egg-rr 9.6%

       \[\leadsto \cos^{-1} \cos \color{blue}{\left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \left(\lambda_2 - \lambda_1\right)}}\right)} \cdot R \]

   11. Taylor expanded in lambda2 around 0

       \[\leadsto \color{blue}{\left(-1 \cdot \lambda_1\right)} \cdot R \]
   12. Step-by-step derivation

       1. neg-mul-1 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)} \cdot R \]

       2. lower-neg.f64 8.9

          \[\leadsto \color{blue}{\left(-\lambda_1\right)} \cdot R \]

   13. Simplified 8.9%

       \[\leadsto \color{blue}{\left(-\lambda_1\right)} \cdot R \]

   if -2.99999999999999984e-172 < lambda1

   1. Initial program 79.9%

      \[\cos^{-1} \left(\sin \phi_1 \cdot \sin \phi_2 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) \cdot R \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      2. lower-*.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

      3. sub-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      4. remove-double-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      5. mul-1-neg N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      6. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

      7. +-commutative N/A

         \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

      8. cos-neg N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      9. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      10. mul-1-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

      11. unsub-neg N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      12. lower--.f64 N/A

          \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

      13. lower-cos.f64 43.1

          \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

   5. Simplified 43.1%

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
   7. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

      2. remove-double-neg N/A

         \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

      3. distribute-neg-in N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

      4. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

      5. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

      6. cos-neg N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      7. lower-cos.f64 N/A

         \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

      8. neg-mul-1 N/A

         \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

      9. sub-neg N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

      10. lower--.f64 23.2

          \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   8. Simplified 23.2%

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
   9. Step-by-step derivation

      1. flip-- N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\frac{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}{\lambda_1 + \lambda_2}\right)} \cdot R \]

      2. clear-num N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\frac{1}{\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}\right)} \cdot R \]

      3. lower-/.f64 N/A

         \[\leadsto \cos^{-1} \cos \color{blue}{\left(\frac{1}{\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}\right)} \cdot R \]

      4. lower-/.f64 N/A

         \[\leadsto \cos^{-1} \cos \left(\frac{1}{\color{blue}{\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}}\right) \cdot R \]

      5. +-commutative N/A

         \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\color{blue}{\lambda_2 + \lambda_1}}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}\right) \cdot R \]

      6. lower-+.f64 N/A

         \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\color{blue}{\lambda_2 + \lambda_1}}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}\right) \cdot R \]

      7. difference-of-squares N/A

         \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\color{blue}{\left(\lambda_1 + \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

      8. lift--.f64 N/A

         \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_1 + \lambda_2\right) \cdot \color{blue}{\left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

      9. acos-cos-s N/A

         \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_1 + \lambda_2\right) \cdot \color{blue}{\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

      10. lift-cos.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_1 + \lambda_2\right) \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

      11. lift-acos.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_1 + \lambda_2\right) \cdot \color{blue}{\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

      12. lower-*.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\color{blue}{\left(\lambda_1 + \lambda_2\right) \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

      13. +-commutative N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\color{blue}{\left(\lambda_2 + \lambda_1\right)} \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)}}\right) \cdot R \]

      14. lower-+.f64 23.2

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\color{blue}{\left(\lambda_2 + \lambda_1\right)} \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)}}\right) \cdot R \]

      15. lift-acos.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \color{blue}{\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

      16. lift-cos.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

      17. lift--.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

      18. cos-diff N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}}}\right) \cdot R \]

      19. lift-cos.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\color{blue}{\cos \lambda_1} \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}}\right) \cdot R \]

      20. lift-cos.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \lambda_2} + \sin \lambda_1 \cdot \sin \lambda_2\right)}}\right) \cdot R \]

      21. *-commutative N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\color{blue}{\cos \lambda_2 \cdot \cos \lambda_1} + \sin \lambda_1 \cdot \sin \lambda_2\right)}}\right) \cdot R \]

      22. lift-cos.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\color{blue}{\cos \lambda_2} \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}}\right) \cdot R \]

      23. lift-cos.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\cos \lambda_2 \cdot \color{blue}{\cos \lambda_1} + \sin \lambda_1 \cdot \sin \lambda_2\right)}}\right) \cdot R \]

      24. lift-sin.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_1} \cdot \sin \lambda_2\right)}}\right) \cdot R \]

      25. lift-sin.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \color{blue}{\sin \lambda_2}\right)}}\right) \cdot R \]

      26. *-commutative N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)}}\right) \cdot R \]

      27. lift-sin.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right)}}\right) \cdot R \]

      28. lift-sin.f64 N/A

          \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)}}\right) \cdot R \]

   10. Applied egg-rr 12.0%

       \[\leadsto \cos^{-1} \cos \color{blue}{\left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \left(\lambda_2 - \lambda_1\right)}}\right)} \cdot R \]

   11. Taylor expanded in lambda2 around inf

       \[\leadsto \color{blue}{R \cdot \lambda_2} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

       2. lower-*.f64 5.6

          \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

   13. Simplified 5.6%

       \[\leadsto \color{blue}{\lambda_2 \cdot R} \]
3. Recombined 2 regimes into one program.

4. Final simplification 6.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3 \cdot 10^{-172}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 30: 9.0% accurate, 69.7× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \left(R\_m \cdot \left(\lambda_2 - \lambda_1\right)\right) \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R_s (* R_m (- lambda2 lambda1))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (R_m * (lambda2 - lambda1));
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r_s * (r_m * (lambda2 - lambda1))
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (R_m * (lambda2 - lambda1));
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	return R_s * (R_m * (lambda2 - lambda1))

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	return Float64(R_s * Float64(R_m * Float64(lambda2 - lambda1)))
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = R_s * (R_m * (lambda2 - lambda1));
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * N[(R$95$m * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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. Taylor expanded in phi2 around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   2. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   3. sub-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

   4. remove-double-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   5. mul-1-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   6. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

   7. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   8. cos-neg N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

   9. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

   10. mul-1-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

   11. unsub-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

   12. lower--.f64 N/A

       \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

   13. lower-cos.f64 42.9

       \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

5. Simplified 42.9%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

6. Taylor expanded in phi1 around 0

   \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

   2. remove-double-neg N/A

      \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

   3. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

   4. +-commutative N/A

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

   5. neg-mul-1 N/A

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

   6. cos-neg N/A

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

   7. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

   8. neg-mul-1 N/A

      \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

   9. sub-neg N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   10. lower--.f64 24.8

       \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

8. Simplified 24.8%

   \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
9. Step-by-step derivation

   1. cos-diff N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)} \cdot R \]

   2. lift-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_1} \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot R \]

   3. lift-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \lambda_2} + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot R \]

   4. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_2 \cdot \cos \lambda_1} + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot R \]

   5. lift-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \lambda_2} \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot R \]

   6. lift-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \color{blue}{\cos \lambda_1} + \sin \lambda_1 \cdot \sin \lambda_2\right) \cdot R \]

   7. lift-sin.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_1} \cdot \sin \lambda_2\right) \cdot R \]

   8. lift-sin.f64 N/A

      \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \color{blue}{\sin \lambda_2}\right) \cdot R \]

   9. *-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right) \cdot R \]

   10. lift-sin.f64 N/A

       \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right) \cdot R \]

   11. lift-sin.f64 N/A

       \[\leadsto \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right) \cdot R \]

   12. cos-diff N/A

       \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]

   13. acos-cos-s N/A

       \[\leadsto \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot R \]

   14. lower--.f64 5.9

       \[\leadsto \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot R \]

10. Applied egg-rr 5.9%

    \[\leadsto \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot R \]

11. Final simplification 5.9%

    \[\leadsto R \cdot \left(\lambda_2 - \lambda_1\right) \]
12. Add Preprocessing

Alternative 31: 6.8% accurate, 104.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \left(\lambda_2 \cdot R\_m\right) \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R_s (* lambda2 R_m)))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (lambda2 * R_m);
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r_s * (lambda2 * r_m)
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (lambda2 * R_m);
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	return R_s * (lambda2 * R_m)

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	return Float64(R_s * Float64(lambda2 * R_m))
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = R_s * (lambda2 * R_m);
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * N[(lambda2 * R$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

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. Taylor expanded in phi2 around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   2. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   3. sub-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

   4. remove-double-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   5. mul-1-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   6. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

   7. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   8. cos-neg N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

   9. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

   10. mul-1-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

   11. unsub-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

   12. lower--.f64 N/A

       \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

   13. lower-cos.f64 42.9

       \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

5. Simplified 42.9%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

6. Taylor expanded in phi1 around 0

   \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

   2. remove-double-neg N/A

      \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

   3. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

   4. +-commutative N/A

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

   5. neg-mul-1 N/A

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

   6. cos-neg N/A

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

   7. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

   8. neg-mul-1 N/A

      \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

   9. sub-neg N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   10. lower--.f64 24.8

       \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

8. Simplified 24.8%

   \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]
9. Step-by-step derivation

   1. flip-- N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\frac{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}{\lambda_1 + \lambda_2}\right)} \cdot R \]

   2. clear-num N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\frac{1}{\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}\right)} \cdot R \]

   3. lower-/.f64 N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\frac{1}{\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}\right)} \cdot R \]

   4. lower-/.f64 N/A

      \[\leadsto \cos^{-1} \cos \left(\frac{1}{\color{blue}{\frac{\lambda_1 + \lambda_2}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}}\right) \cdot R \]

   5. +-commutative N/A

      \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\color{blue}{\lambda_2 + \lambda_1}}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}\right) \cdot R \]

   6. lower-+.f64 N/A

      \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\color{blue}{\lambda_2 + \lambda_1}}{\lambda_1 \cdot \lambda_1 - \lambda_2 \cdot \lambda_2}}\right) \cdot R \]

   7. difference-of-squares N/A

      \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\color{blue}{\left(\lambda_1 + \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

   8. lift--.f64 N/A

      \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_1 + \lambda_2\right) \cdot \color{blue}{\left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

   9. acos-cos-s N/A

      \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_1 + \lambda_2\right) \cdot \color{blue}{\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

   10. lift-cos.f64 N/A

       \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_1 + \lambda_2\right) \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

   11. lift-acos.f64 N/A

       \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_1 + \lambda_2\right) \cdot \color{blue}{\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

   12. lower-*.f64 N/A

       \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\color{blue}{\left(\lambda_1 + \lambda_2\right) \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

   13. +-commutative N/A

       \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\color{blue}{\left(\lambda_2 + \lambda_1\right)} \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)}}\right) \cdot R \]

   14. lower-+.f64 24.8

       \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\color{blue}{\left(\lambda_2 + \lambda_1\right)} \cdot \cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)}}\right) \cdot R \]

   15. lift-acos.f64 N/A

       \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \color{blue}{\cos^{-1} \cos \left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

   16. lift-cos.f64 N/A

       \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

   17. lift--.f64 N/A

       \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)}}}\right) \cdot R \]

   18. cos-diff N/A

       \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \color{blue}{\left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}}}\right) \cdot R \]

   19. lift-cos.f64 N/A

       \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\color{blue}{\cos \lambda_1} \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)}}\right) \cdot R \]

   20. lift-cos.f64 N/A

       \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\cos \lambda_1 \cdot \color{blue}{\cos \lambda_2} + \sin \lambda_1 \cdot \sin \lambda_2\right)}}\right) \cdot R \]

   21. *-commutative N/A

       \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\color{blue}{\cos \lambda_2 \cdot \cos \lambda_1} + \sin \lambda_1 \cdot \sin \lambda_2\right)}}\right) \cdot R \]

   22. lift-cos.f64 N/A

       \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\color{blue}{\cos \lambda_2} \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \sin \lambda_2\right)}}\right) \cdot R \]

   23. lift-cos.f64 N/A

       \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\cos \lambda_2 \cdot \color{blue}{\cos \lambda_1} + \sin \lambda_1 \cdot \sin \lambda_2\right)}}\right) \cdot R \]

   24. lift-sin.f64 N/A

       \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_1} \cdot \sin \lambda_2\right)}}\right) \cdot R \]

   25. lift-sin.f64 N/A

       \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_1 \cdot \color{blue}{\sin \lambda_2}\right)}}\right) \cdot R \]

   26. *-commutative N/A

       \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_2 \cdot \sin \lambda_1}\right)}}\right) \cdot R \]

   27. lift-sin.f64 N/A

       \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \color{blue}{\sin \lambda_2} \cdot \sin \lambda_1\right)}}\right) \cdot R \]

   28. lift-sin.f64 N/A

       \[\leadsto \cos^{-1} \cos \left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \cos^{-1} \left(\cos \lambda_2 \cdot \cos \lambda_1 + \sin \lambda_2 \cdot \color{blue}{\sin \lambda_1}\right)}}\right) \cdot R \]

10. Applied egg-rr 11.0%

    \[\leadsto \cos^{-1} \cos \color{blue}{\left(\frac{1}{\frac{\lambda_2 + \lambda_1}{\left(\lambda_2 + \lambda_1\right) \cdot \left(\lambda_2 - \lambda_1\right)}}\right)} \cdot R \]

11. Taylor expanded in lambda2 around inf

    \[\leadsto \color{blue}{R \cdot \lambda_2} \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

    2. lower-*.f64 4.9

       \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

13. Simplified 4.9%

    \[\leadsto \color{blue}{\lambda_2 \cdot R} \]
14. Add Preprocessing

Alternative 32: 3.6% accurate, 104.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \left(\lambda_1 \cdot R\_m\right) \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R_s (* lambda1 R_m)))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (lambda1 * R_m);
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r_s * (lambda1 * r_m)
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (lambda1 * R_m);
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	return R_s * (lambda1 * R_m)

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	return Float64(R_s * Float64(lambda1 * R_m))
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = R_s * (lambda1 * R_m);
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * N[(lambda1 * R$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

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. Taylor expanded in phi2 around 0

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \phi_1 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} \cdot R \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   2. lower-*.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_1 - \lambda_2\right) \cdot \cos \phi_1\right)} \cdot R \]

   3. sub-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

   4. remove-double-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_1\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   5. mul-1-neg N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\left(\mathsf{neg}\left(\color{blue}{-1 \cdot \lambda_1}\right)\right) + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   6. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \lambda_1 + \lambda_2\right)\right)\right)} \cdot \cos \phi_1\right) \cdot R \]

   7. +-commutative N/A

      \[\leadsto \cos^{-1} \left(\cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_2 + -1 \cdot \lambda_1\right)}\right)\right) \cdot \cos \phi_1\right) \cdot R \]

   8. cos-neg N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

   9. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \left(\color{blue}{\cos \left(\lambda_2 + -1 \cdot \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

   10. mul-1-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 + \color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)}\right) \cdot \cos \phi_1\right) \cdot R \]

   11. unsub-neg N/A

       \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

   12. lower--.f64 N/A

       \[\leadsto \cos^{-1} \left(\cos \color{blue}{\left(\lambda_2 - \lambda_1\right)} \cdot \cos \phi_1\right) \cdot R \]

   13. lower-cos.f64 42.9

       \[\leadsto \cos^{-1} \left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \color{blue}{\cos \phi_1}\right) \cdot R \]

5. Simplified 42.9%

   \[\leadsto \cos^{-1} \color{blue}{\left(\cos \left(\lambda_2 - \lambda_1\right) \cdot \cos \phi_1\right)} \cdot R \]

6. Taylor expanded in phi1 around 0

   \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_2 - \lambda_1\right)} \cdot R \]
7. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_2 + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right)} \cdot R \]

   2. remove-double-neg N/A

      \[\leadsto \cos^{-1} \cos \left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\lambda_2\right)\right)\right)\right)} + \left(\mathsf{neg}\left(\lambda_1\right)\right)\right) \cdot R \]

   3. distribute-neg-in N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(\lambda_2\right)\right) + \lambda_1\right)\right)\right)} \cdot R \]

   4. +-commutative N/A

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\color{blue}{\left(\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)\right)}\right)\right) \cdot R \]

   5. neg-mul-1 N/A

      \[\leadsto \cos^{-1} \cos \left(\mathsf{neg}\left(\left(\lambda_1 + \color{blue}{-1 \cdot \lambda_2}\right)\right)\right) \cdot R \]

   6. cos-neg N/A

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

   7. lower-cos.f64 N/A

      \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 + -1 \cdot \lambda_2\right)} \cdot R \]

   8. neg-mul-1 N/A

      \[\leadsto \cos^{-1} \cos \left(\lambda_1 + \color{blue}{\left(\mathsf{neg}\left(\lambda_2\right)\right)}\right) \cdot R \]

   9. sub-neg N/A

      \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

   10. lower--.f64 24.8

       \[\leadsto \cos^{-1} \cos \color{blue}{\left(\lambda_1 - \lambda_2\right)} \cdot R \]

8. Simplified 24.8%

   \[\leadsto \cos^{-1} \color{blue}{\cos \left(\lambda_1 - \lambda_2\right)} \cdot R \]

9. Taylor expanded in lambda1 around inf

   \[\leadsto \color{blue}{R \cdot \lambda_1} \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \color{blue}{\lambda_1 \cdot R} \]

    2. lower-*.f64 5.0

       \[\leadsto \color{blue}{\lambda_1 \cdot R} \]

11. Simplified 5.0%

    \[\leadsto \color{blue}{\lambda_1 \cdot R} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(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))