Equirectangular approximation to distance on a great circle

Percentage Accurate: 60.2% → 99.9%

Time: 21.9s

Alternatives: 22

Speedup: 3.0×

• 31.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ R \cdot \sqrt{t\_0 \cdot t\_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))))
   (* R (sqrt (+ (* t_0 t_0) (* (- phi1 phi2) (- phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	return R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    code = r * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0));
	return R * Math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))
	return R * math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	return Float64(R * sqrt(Float64(Float64(t_0 * t_0) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	tmp = R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(R * N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Initial Program: 60.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ R \cdot \sqrt{t\_0 \cdot t\_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))))
   (* R (sqrt (+ (* t_0 t_0) (* (- phi1 phi2) (- phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	return R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    code = r * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0));
	return R * Math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))
	return R * math.sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0)))
	return Float64(R * sqrt(Float64(Float64(t_0 * t_0) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0));
	tmp = R * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(R * N[Sqrt[N[(N[(t$95$0 * t$95$0), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\mathsf{fma}\left(\sin \left(\frac{\phi_2}{2}\right), \sin \left(\frac{\phi_1}{2}\right) \cdot \left(\lambda_2 - \lambda_1\right), \cos \left(\frac{\phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right)\right), \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot
   (fma
    (sin (/ phi2 2.0))
    (* (sin (/ phi1 2.0)) (- lambda2 lambda1))
    (* (cos (/ phi2 2.0)) (* (- lambda1 lambda2) (cos (/ phi1 2.0)))))
   (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot(fma(sin((phi2 / 2.0)), (sin((phi1 / 2.0)) * (lambda2 - lambda1)), (cos((phi2 / 2.0)) * ((lambda1 - lambda2) * cos((phi1 / 2.0))))), (phi1 - phi2));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(fma(sin(Float64(phi2 / 2.0)), Float64(sin(Float64(phi1 / 2.0)) * Float64(lambda2 - lambda1)), Float64(cos(Float64(phi2 / 2.0)) * Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 / 2.0))))), Float64(phi1 - phi2)))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[Sin[N[(phi2 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision] + N[(N[Cos[N[(phi2 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.9%

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

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

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

   2. hypot-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. --lowering--.f64 94.4%

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

3. Simplified 94.4%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. cos-sum N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. fmm-def N/A

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

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

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

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

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. div-inv N/A

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

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \mathsf{cos.f64}\left(\left(\phi_1 \cdot \frac{1}{2}\right)\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \mathsf{cos.f64}\left(\left(\phi_1 \cdot \frac{1}{2}\right)\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   19. div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1}{2}\right)\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   20. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_1, 2\right)\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

6. Applied egg-rr 99.8%

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

   1. distribute-rgt-in N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-neg-in N/A

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

   4. associate-*l* N/A

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

   5. fma-define N/A

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

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

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

   \[\leadsto R \cdot \mathsf{hypot}\left(\mathsf{fma}\left(\sin \left(\frac{\phi_2}{2}\right), \sin \left(\frac{\phi_1}{2}\right) \cdot \left(\lambda_2 - \lambda_1\right), \cos \left(\frac{\phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right)\right), \phi_1 - \phi_2\right) \]
10. Add Preprocessing

Alternative 2: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(0 - \sin \left(\frac{\phi_2}{2}\right), \sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right), \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot
   (*
    (- lambda1 lambda2)
    (fma
     (- 0.0 (sin (/ phi2 2.0)))
     (sin (/ phi1 2.0))
     (* (cos (/ phi2 2.0)) (cos (/ phi1 2.0)))))
   (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot(((lambda1 - lambda2) * fma((0.0 - sin((phi2 / 2.0))), sin((phi1 / 2.0)), (cos((phi2 / 2.0)) * cos((phi1 / 2.0))))), (phi1 - phi2));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * fma(Float64(0.0 - sin(Float64(phi2 / 2.0))), sin(Float64(phi1 / 2.0)), Float64(cos(Float64(phi2 / 2.0)) * cos(Float64(phi1 / 2.0))))), Float64(phi1 - phi2)))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[(0.0 - N[Sin[N[(phi2 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi2 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.9%

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

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

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

   2. hypot-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. --lowering--.f64 94.4%

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

3. Simplified 94.4%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. cos-sum N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. fmm-def N/A

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

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

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

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

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. div-inv N/A

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

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \mathsf{cos.f64}\left(\left(\phi_1 \cdot \frac{1}{2}\right)\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \mathsf{cos.f64}\left(\left(\phi_1 \cdot \frac{1}{2}\right)\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   19. div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1}{2}\right)\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   20. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_1, 2\right)\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

6. Applied egg-rr 99.8%

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

   1. +-commutative N/A

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

   2. distribute-lft-neg-in N/A

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

   3. fma-define N/A

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

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

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

   5. neg-sub0 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\left(0 - \sin \left(\frac{\phi_2}{2}\right)\right), \sin \left(\frac{\phi_1}{2}\right), \left(\cos \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

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

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

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

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

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

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right)\right), \sin \left(\frac{\phi_1}{2}\right), \left(\cos \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

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

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right)\right), \mathsf{sin.f64}\left(\left(\frac{\phi_1}{2}\right)\right), \left(\cos \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\phi_1, 2\right)\right), \left(\cos \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\phi_1, 2\right)\right), \left(\cos \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. *-commutative N/A

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

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

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

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

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

   17. /-lowering-/.f64 N/A

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

   18. *-commutative N/A

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

   19. metadata-eval N/A

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

   20. div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{/.f64}\left(\phi_1, 2\right)\right), \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_1, 2\right)\right), \cos \left(\frac{\phi_2}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

8. Applied egg-rr 99.8%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{fma}\left(0 - \sin \left(\frac{\phi_2}{2}\right), \sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)\right)}, \phi_1 - \phi_2\right) \]

9. Final simplification 99.8%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(0 - \sin \left(\frac{\phi_2}{2}\right), \sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right), \phi_1 - \phi_2\right) \]
10. Add Preprocessing

Alternative 3: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{\phi_2}{2}\right), \cos \left(\frac{\phi_1}{2}\right), \sin \left(\frac{\phi_2}{2}\right) \cdot \left(0 - \sin \left(\frac{\phi_1}{2}\right)\right)\right), \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot
   (*
    (- lambda1 lambda2)
    (fma
     (cos (/ phi2 2.0))
     (cos (/ phi1 2.0))
     (* (sin (/ phi2 2.0)) (- 0.0 (sin (/ phi1 2.0))))))
   (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot(((lambda1 - lambda2) * fma(cos((phi2 / 2.0)), cos((phi1 / 2.0)), (sin((phi2 / 2.0)) * (0.0 - sin((phi1 / 2.0)))))), (phi1 - phi2));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * fma(cos(Float64(phi2 / 2.0)), cos(Float64(phi1 / 2.0)), Float64(sin(Float64(phi2 / 2.0)) * Float64(0.0 - sin(Float64(phi1 / 2.0)))))), Float64(phi1 - phi2)))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[Cos[N[(phi2 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[(phi2 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(0.0 - N[Sin[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.9%

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

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

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

   2. hypot-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. --lowering--.f64 94.4%

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

3. Simplified 94.4%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. cos-sum N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. fmm-def N/A

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

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

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

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

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. div-inv N/A

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

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \mathsf{cos.f64}\left(\left(\phi_1 \cdot \frac{1}{2}\right)\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \mathsf{cos.f64}\left(\left(\phi_1 \cdot \frac{1}{2}\right)\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   19. div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1}{2}\right)\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   20. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_1, 2\right)\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\cos \left(\frac{\phi_2}{2}\right), \cos \left(\frac{\phi_1}{2}\right), \sin \left(\frac{\phi_2}{2}\right) \cdot \left(0 - \sin \left(\frac{\phi_1}{2}\right)\right)\right), \phi_1 - \phi_2\right) \]
8. Add Preprocessing

Alternative 4: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\cos \left(\frac{\phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right) + \sin \left(\frac{\phi_2}{2}\right) \cdot \left(\sin \left(\frac{\phi_1}{2}\right) \cdot \left(\lambda_2 - \lambda_1\right)\right), \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot
   (+
    (* (cos (/ phi2 2.0)) (* (- lambda1 lambda2) (cos (/ phi1 2.0))))
    (* (sin (/ phi2 2.0)) (* (sin (/ phi1 2.0)) (- lambda2 lambda1))))
   (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot(((cos((phi2 / 2.0)) * ((lambda1 - lambda2) * cos((phi1 / 2.0)))) + (sin((phi2 / 2.0)) * (sin((phi1 / 2.0)) * (lambda2 - lambda1)))), (phi1 - phi2));
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.hypot(((Math.cos((phi2 / 2.0)) * ((lambda1 - lambda2) * Math.cos((phi1 / 2.0)))) + (Math.sin((phi2 / 2.0)) * (Math.sin((phi1 / 2.0)) * (lambda2 - lambda1)))), (phi1 - phi2));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.hypot(((math.cos((phi2 / 2.0)) * ((lambda1 - lambda2) * math.cos((phi1 / 2.0)))) + (math.sin((phi2 / 2.0)) * (math.sin((phi1 / 2.0)) * (lambda2 - lambda1)))), (phi1 - phi2))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(Float64(cos(Float64(phi2 / 2.0)) * Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 / 2.0)))) + Float64(sin(Float64(phi2 / 2.0)) * Float64(sin(Float64(phi1 / 2.0)) * Float64(lambda2 - lambda1)))), Float64(phi1 - phi2)))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot(((cos((phi2 / 2.0)) * ((lambda1 - lambda2) * cos((phi1 / 2.0)))) + (sin((phi2 / 2.0)) * (sin((phi1 / 2.0)) * (lambda2 - lambda1)))), (phi1 - phi2));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(N[Cos[N[(phi2 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[N[(phi2 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision] * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.9%

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

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

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

   2. hypot-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. --lowering--.f64 94.4%

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

3. Simplified 94.4%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. cos-sum N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. fmm-def N/A

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

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

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

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

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

   13. *-commutative N/A

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

   14. metadata-eval N/A

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

   15. div-inv N/A

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

   16. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

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

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \mathsf{cos.f64}\left(\left(\phi_1 \cdot \frac{1}{2}\right)\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \mathsf{cos.f64}\left(\left(\phi_1 \cdot \frac{1}{2}\right)\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   19. div-inv N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1}{2}\right)\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   20. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_1, 2\right)\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

6. Applied egg-rr 99.8%

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

   1. distribute-rgt-in N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-neg-in N/A

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

   4. associate-*l* N/A

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

   5. fma-define N/A

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

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

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

8. Applied egg-rr 99.8%

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

   1. +-commutative N/A

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

   2. distribute-rgt-neg-out N/A

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

   3. unsub-neg N/A

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

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

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

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

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

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

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

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

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

   8. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

    \[\leadsto R \cdot \mathsf{hypot}\left(\cos \left(\frac{\phi_2}{2}\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1}{2}\right)\right) + \sin \left(\frac{\phi_2}{2}\right) \cdot \left(\sin \left(\frac{\phi_1}{2}\right) \cdot \left(\lambda_2 - \lambda_1\right)\right), \phi_1 - \phi_2\right) \]
12. Add Preprocessing

Alternative 5: 96.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 10^{+182}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 1e+182)
   (*
    R
    (hypot (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0))) (- phi1 phi2)))
   (*
    R
    (hypot
     (*
      lambda2
      (-
       (* (sin (* phi1 0.5)) (sin (* phi2 0.5)))
       (* (cos (* phi1 0.5)) (cos (* phi2 0.5)))))
     (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1e+182) {
		tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	} else {
		tmp = R * hypot((lambda2 * ((sin((phi1 * 0.5)) * sin((phi2 * 0.5))) - (cos((phi1 * 0.5)) * cos((phi2 * 0.5))))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1e+182) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot((lambda2 * ((Math.sin((phi1 * 0.5)) * Math.sin((phi2 * 0.5))) - (Math.cos((phi1 * 0.5)) * Math.cos((phi2 * 0.5))))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 1e+182:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2))
	else:
		tmp = R * math.hypot((lambda2 * ((math.sin((phi1 * 0.5)) * math.sin((phi2 * 0.5))) - (math.cos((phi1 * 0.5)) * math.cos((phi2 * 0.5))))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1e+182)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi2 + phi1) / 2.0))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(lambda2 * Float64(Float64(sin(Float64(phi1 * 0.5)) * sin(Float64(phi2 * 0.5))) - Float64(cos(Float64(phi1 * 0.5)) * cos(Float64(phi2 * 0.5))))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 1e+182)
		tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	else
		tmp = R * hypot((lambda2 * ((sin((phi1 * 0.5)) * sin((phi2 * 0.5))) - (cos((phi1 * 0.5)) * cos((phi2 * 0.5))))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 1e+182], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(lambda2 * N[(N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.0000000000000001e182

   1. Initial program 56.0%

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

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

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

      2. hypot-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. --lowering--.f64 95.0%

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

   3. Simplified 95.0%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   if 1.0000000000000001e182 < lambda2

   1. Initial program 45.4%

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

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

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

      2. hypot-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. --lowering--.f64 89.0%

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

   3. Simplified 89.0%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. metadata-eval N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. cos-sum N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. fmm-def N/A

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

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

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

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

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

      13. *-commutative N/A

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

      14. metadata-eval N/A

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

      15. div-inv N/A

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

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \cos \left(\phi_1 \cdot \frac{1}{2}\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \mathsf{cos.f64}\left(\left(\phi_1 \cdot \frac{1}{2}\right)\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \mathsf{cos.f64}\left(\left(\phi_1 \cdot \frac{1}{2}\right)\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      19. div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1}{2}\right)\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{fma.f64}\left(\mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_2, 2\right)\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\phi_1, 2\right)\right), \left(\mathsf{neg}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   6. Applied egg-rr 99.6%

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

      1. distribute-rgt-in N/A

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

      2. +-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. associate-*l* N/A

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

      5. fma-define N/A

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

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

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in lambda1 around 0

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

       1. mul-1-neg N/A

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

       2. distribute-rgt-neg-in N/A

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

       3. mul-1-neg N/A

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

       4. distribute-lft-in N/A

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

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

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

       6. +-commutative N/A

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

       7. mul-1-neg N/A

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

       8. unsub-neg N/A

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

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

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

   11. Simplified 87.1%

       \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\lambda_2 \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)}, \phi_1 - \phi_2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 10^{+182}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right) - \sin \left(\frac{\phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right)\right), \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot
   (*
    (- lambda1 lambda2)
    (-
     (* (cos (/ phi2 2.0)) (cos (/ phi1 2.0)))
     (* (sin (/ phi2 2.0)) (sin (/ phi1 2.0)))))
   (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot(((lambda1 - lambda2) * ((cos((phi2 / 2.0)) * cos((phi1 / 2.0))) - (sin((phi2 / 2.0)) * sin((phi1 / 2.0))))), (phi1 - phi2));
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.hypot(((lambda1 - lambda2) * ((Math.cos((phi2 / 2.0)) * Math.cos((phi1 / 2.0))) - (Math.sin((phi2 / 2.0)) * Math.sin((phi1 / 2.0))))), (phi1 - phi2));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.hypot(((lambda1 - lambda2) * ((math.cos((phi2 / 2.0)) * math.cos((phi1 / 2.0))) - (math.sin((phi2 / 2.0)) * math.sin((phi1 / 2.0))))), (phi1 - phi2))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * Float64(Float64(cos(Float64(phi2 / 2.0)) * cos(Float64(phi1 / 2.0))) - Float64(sin(Float64(phi2 / 2.0)) * sin(Float64(phi1 / 2.0))))), Float64(phi1 - phi2)))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot(((lambda1 - lambda2) * ((cos((phi2 / 2.0)) * cos((phi1 / 2.0))) - (sin((phi2 / 2.0)) * sin((phi1 / 2.0))))), (phi1 - phi2));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[(N[Cos[N[(phi2 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(phi2 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.9%

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

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

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

   2. hypot-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. --lowering--.f64 94.4%

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

3. Simplified 94.4%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. clear-num N/A

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

   2. associate-/r/ N/A

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

   3. metadata-eval N/A

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

   4. +-commutative N/A

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

   5. distribute-rgt-in N/A

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

   6. cos-sum N/A

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

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

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

   8. *-commutative N/A

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

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

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

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

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

   11. *-commutative N/A

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

   12. metadata-eval N/A

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

   13. div-inv N/A

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

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

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

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

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

   16. metadata-eval N/A

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

   17. div-inv N/A

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

   18. /-lowering-/.f64 N/A

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

   19. *-commutative N/A

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

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

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

6. Applied egg-rr 99.8%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\frac{\phi_2}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right) - \sin \left(\frac{\phi_2}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right)\right)}, \phi_1 - \phi_2\right) \]
7. Add Preprocessing

Alternative 7: 93.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.052:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2}{2}\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 0.052)
   (* R (hypot (* (- lambda1 lambda2) (cos (/ phi1 2.0))) (- phi1 phi2)))
   (* R (hypot (* (- lambda1 lambda2) (cos (/ phi2 2.0))) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.052) {
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi1 / 2.0))), (phi1 - phi2));
	} else {
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi2 / 2.0))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.052) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((phi1 / 2.0))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((phi2 / 2.0))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 0.052:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((phi1 / 2.0))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((phi2 / 2.0))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 0.052)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 / 2.0))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 / 2.0))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 0.052)
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi1 / 2.0))), (phi1 - phi2));
	else
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi2 / 2.0))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 0.052], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.0519999999999999976

   1. Initial program 59.1%

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

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

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

      2. hypot-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. --lowering--.f64 96.6%

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

   3. Simplified 96.6%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around inf

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\color{blue}{\phi_1}, 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 92.4%

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

   if 0.0519999999999999976 < phi2

   1. Initial program 41.6%

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

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

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

      2. hypot-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. --lowering--.f64 87.6%

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

   3. Simplified 87.6%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\color{blue}{\phi_2}, 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 87.4%

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

Alternative 8: 95.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0))) (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.hypot(((lambda1 - lambda2) * Math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.hypot(((lambda1 - lambda2) * math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi2 + phi1) / 2.0))), Float64(phi1 - phi2)))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.9%

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

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

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

   2. hypot-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. --lowering--.f64 94.4%

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

3. Simplified 94.4%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
4. Add Preprocessing

5. Final simplification 94.4%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right) \]
6. Add Preprocessing

Alternative 9: 90.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1}{2}\right), \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R (hypot (* (- lambda1 lambda2) (cos (/ phi1 2.0))) (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot(((lambda1 - lambda2) * cos((phi1 / 2.0))), (phi1 - phi2));
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.hypot(((lambda1 - lambda2) * Math.cos((phi1 / 2.0))), (phi1 - phi2));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.hypot(((lambda1 - lambda2) * math.cos((phi1 / 2.0))), (phi1 - phi2))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 / 2.0))), Float64(phi1 - phi2)))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot(((lambda1 - lambda2) * cos((phi1 / 2.0))), (phi1 - phi2));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.9%

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

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

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

   2. hypot-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. --lowering--.f64 94.4%

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

3. Simplified 94.4%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
4. Add Preprocessing

5. Taylor expanded in phi1 around inf

   \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\color{blue}{\phi_1}, 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 88.4%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\color{blue}{\phi_1}}{2}\right), \phi_1 - \phi_2\right) \]
8. Add Preprocessing

Alternative 10: 73.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.15 \cdot 10^{+23}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1, \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2.15e+23)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (hypot lambda1 (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.15e+23) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * hypot(lambda1, (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.15e+23) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * Math.hypot(lambda1, (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2.15e+23:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	else:
		tmp = R * math.hypot(lambda1, (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2.15e+23)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R * hypot(lambda1, Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 2.15e+23)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = R * hypot(lambda1, (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2.15e+23], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[lambda1 ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.1499999999999999e23

   1. Initial program 58.9%

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

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

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

      2. hypot-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. --lowering--.f64 96.3%

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

   3. Simplified 96.3%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

      10. *-lowering-*.f64 82.9%

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

   7. Simplified 82.9%

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

   8. Taylor expanded in phi2 around 0

      \[\leadsto \color{blue}{R \cdot \sqrt{{\phi_1}^{2} + {\left(\lambda_1 - \lambda_2\right)}^{2}}} \]
   9. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{{\phi_1}^{2} + {\left(\lambda_1 - \lambda_2\right)}^{2}}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\sqrt{\phi_1 \cdot \phi_1 + {\left(\lambda_1 - \lambda_2\right)}^{2}}\right)\right) \]

      3. unpow2 N/A

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

      4. hypot-define N/A

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

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

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

      6. --lowering--.f64 69.8%

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

   10. Simplified 69.8%

       \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)} \]

   if 2.1499999999999999e23 < phi2

   1. Initial program 40.5%

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

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

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

      2. hypot-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. --lowering--.f64 87.7%

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

   3. Simplified 87.7%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

      10. *-lowering-*.f64 75.7%

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

   7. Simplified 75.7%

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

   8. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   9. Step-by-step derivation

      1. --lowering--.f64 77.5%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

   10. Simplified 77.5%

       \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\lambda_1 - \lambda_2}, \phi_1 - \phi_2\right) \]

   11. Taylor expanded in lambda2 around 0

       \[\leadsto \color{blue}{R \cdot \sqrt{{\lambda_1}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{{\lambda_1}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \left(\sqrt{\lambda_1 \cdot \lambda_1 + {\left(\phi_1 - \phi_2\right)}^{2}}\right)\right) \]

       3. unpow2 N/A

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

       4. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\lambda_1, \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\lambda_1, \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

       6. --lowering--.f64 76.1%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\lambda_1, \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   13. Simplified 76.1%

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

Alternative 11: 74.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 5.5 \cdot 10^{+150}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 5.5e+150)
   (* R (hypot lambda1 (- phi1 phi2)))
   (* R (* lambda2 (- 1.0 (/ lambda1 lambda2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 5.5e+150) {
		tmp = R * hypot(lambda1, (phi1 - phi2));
	} else {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 5.5e+150) {
		tmp = R * Math.hypot(lambda1, (phi1 - phi2));
	} else {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 5.5e+150:
		tmp = R * math.hypot(lambda1, (phi1 - phi2))
	else:
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 5.5e+150)
		tmp = Float64(R * hypot(lambda1, Float64(phi1 - phi2)));
	else
		tmp = Float64(R * Float64(lambda2 * Float64(1.0 - Float64(lambda1 / lambda2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 5.5e+150)
		tmp = R * hypot(lambda1, (phi1 - phi2));
	else
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 5.5e+150], N[(R * N[Sqrt[lambda1 ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[(lambda2 * N[(1.0 - N[(lambda1 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 5.50000000000000017e150

   1. Initial program 56.6%

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

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

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

      2. hypot-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. --lowering--.f64 94.9%

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

   3. Simplified 94.9%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

      10. *-lowering-*.f64 82.5%

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

   7. Simplified 82.5%

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

   8. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   9. Step-by-step derivation

      1. --lowering--.f64 85.7%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

   10. Simplified 85.7%

       \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\lambda_1 - \lambda_2}, \phi_1 - \phi_2\right) \]

   11. Taylor expanded in lambda2 around 0

       \[\leadsto \color{blue}{R \cdot \sqrt{{\lambda_1}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]
   12. Step-by-step derivation

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

          \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{{\lambda_1}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \left(\sqrt{\lambda_1 \cdot \lambda_1 + {\left(\phi_1 - \phi_2\right)}^{2}}\right)\right) \]

       3. unpow2 N/A

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

       4. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\lambda_1, \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

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

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\lambda_1, \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

       6. --lowering--.f64 73.7%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\lambda_1, \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   13. Simplified 73.7%

       \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\lambda_1, \phi_1 - \phi_2\right)} \]

   if 5.50000000000000017e150 < lambda2

   1. Initial program 41.2%

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

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

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

      2. hypot-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. --lowering--.f64 90.1%

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

   3. Simplified 90.1%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

      10. *-lowering-*.f64 72.0%

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

   7. Simplified 72.0%

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

   8. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   9. Step-by-step derivation

      1. --lowering--.f64 62.4%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

   10. Simplified 62.4%

       \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\lambda_1 - \lambda_2}, \phi_1 - \phi_2\right) \]

   11. Taylor expanded in lambda2 around inf

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

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

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

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

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{\lambda_1}{\lambda_2}\right)}\right)\right)\right) \]

       5. /-lowering-/.f64 40.1%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\lambda_1, \color{blue}{\lambda_2}\right)\right)\right)\right) \]

   13. Simplified 40.1%

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

Alternative 12: 85.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.9%

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

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

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

   2. hypot-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

   9. --lowering--.f64 94.4%

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

3. Simplified 94.4%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
4. Add Preprocessing

5. Taylor expanded in phi1 around 0

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

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

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

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

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

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

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

   4. *-commutative N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

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

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

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

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

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

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

   10. *-lowering-*.f64 81.3%

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

7. Simplified 81.3%

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

8. Taylor expanded in phi2 around 0

   \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
9. Step-by-step derivation

   1. --lowering--.f64 83.1%

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

10. Simplified 83.1%

    \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\lambda_1 - \lambda_2}, \phi_1 - \phi_2\right) \]
11. Add Preprocessing

Alternative 13: 32.5% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3 \cdot 10^{+17}:\\ \;\;\;\;\phi_1 \cdot \left(\frac{R \cdot \phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_1 \leq 2.55 \cdot 10^{-204}:\\ \;\;\;\;\lambda_1 \cdot \left(\frac{R \cdot \lambda_2 + \frac{-0.5 \cdot \left(R \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\right)\right)}{\lambda_1}}{\lambda_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -3e+17)
   (* phi1 (- (/ (* R phi2) phi1) R))
   (if (<= phi1 2.55e-204)
     (*
      lambda1
      (-
       (/
        (+
         (* R lambda2)
         (/ (* -0.5 (* R (* (- phi1 phi2) (- phi1 phi2)))) lambda1))
        lambda1)
       R))
     (* R phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3e+17) {
		tmp = phi1 * (((R * phi2) / phi1) - R);
	} else if (phi1 <= 2.55e-204) {
		tmp = lambda1 * ((((R * lambda2) + ((-0.5 * (R * ((phi1 - phi2) * (phi1 - phi2)))) / lambda1)) / lambda1) - R);
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-3d+17)) then
        tmp = phi1 * (((r * phi2) / phi1) - r)
    else if (phi1 <= 2.55d-204) then
        tmp = lambda1 * ((((r * lambda2) + (((-0.5d0) * (r * ((phi1 - phi2) * (phi1 - phi2)))) / lambda1)) / lambda1) - r)
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3e+17) {
		tmp = phi1 * (((R * phi2) / phi1) - R);
	} else if (phi1 <= 2.55e-204) {
		tmp = lambda1 * ((((R * lambda2) + ((-0.5 * (R * ((phi1 - phi2) * (phi1 - phi2)))) / lambda1)) / lambda1) - R);
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -3e+17:
		tmp = phi1 * (((R * phi2) / phi1) - R)
	elif phi1 <= 2.55e-204:
		tmp = lambda1 * ((((R * lambda2) + ((-0.5 * (R * ((phi1 - phi2) * (phi1 - phi2)))) / lambda1)) / lambda1) - R)
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -3e+17)
		tmp = Float64(phi1 * Float64(Float64(Float64(R * phi2) / phi1) - R));
	elseif (phi1 <= 2.55e-204)
		tmp = Float64(lambda1 * Float64(Float64(Float64(Float64(R * lambda2) + Float64(Float64(-0.5 * Float64(R * Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))) / lambda1)) / lambda1) - R));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -3e+17)
		tmp = phi1 * (((R * phi2) / phi1) - R);
	elseif (phi1 <= 2.55e-204)
		tmp = lambda1 * ((((R * lambda2) + ((-0.5 * (R * ((phi1 - phi2) * (phi1 - phi2)))) / lambda1)) / lambda1) - R);
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -3e+17], N[(phi1 * N[(N[(N[(R * phi2), $MachinePrecision] / phi1), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 2.55e-204], N[(lambda1 * N[(N[(N[(N[(R * lambda2), $MachinePrecision] + N[(N[(-0.5 * N[(R * N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lambda1), $MachinePrecision]), $MachinePrecision] / lambda1), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3e17

   1. Initial program 55.6%

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

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

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

      2. hypot-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. --lowering--.f64 89.1%

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

   3. Simplified 89.1%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right) \]

      2. neg-sub0 N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \color{blue}{\left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \left(R + \left(\mathsf{neg}\left(\frac{R \cdot \phi_2}{\phi_1}\right)\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \left(R - \color{blue}{\frac{R \cdot \phi_2}{\phi_1}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \mathsf{\_.f64}\left(R, \color{blue}{\left(\frac{R \cdot \phi_2}{\phi_1}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\left(R \cdot \phi_2\right), \color{blue}{\phi_1}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 60.2%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\mathsf{*.f64}\left(R, \phi_2\right), \phi_1\right)\right)\right)\right) \]

   7. Simplified 60.2%

      \[\leadsto \color{blue}{0 - \phi_1 \cdot \left(R - \frac{R \cdot \phi_2}{\phi_1}\right)} \]

   if -3e17 < phi1 < 2.55000000000000014e-204

   1. Initial program 60.0%

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

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

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

      2. hypot-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. --lowering--.f64 99.2%

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

   3. Simplified 99.2%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

      10. *-lowering-*.f64 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   9. Step-by-step derivation

      1. --lowering--.f64 86.1%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

   10. Simplified 86.1%

       \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\lambda_1 - \lambda_2}, \phi_1 - \phi_2\right) \]

   11. Taylor expanded in lambda1 around -inf

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. mul-1-neg N/A

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

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

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

   13. Simplified 25.8%

       \[\leadsto \color{blue}{\left(R - \frac{\lambda_2 \cdot R + \frac{-0.5 \cdot \left(R \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\right)\right)}{\lambda_1}}{\lambda_1}\right) \cdot \left(0 - \lambda_1\right)} \]

   if 2.55000000000000014e-204 < phi1

   1. Initial program 50.9%

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

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

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

      2. hypot-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. --lowering--.f64 93.7%

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

   3. Simplified 93.7%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 12.5%

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\phi_2}\right) \]

   7. Simplified 12.5%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3 \cdot 10^{+17}:\\ \;\;\;\;\phi_1 \cdot \left(\frac{R \cdot \phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_1 \leq 2.55 \cdot 10^{-204}:\\ \;\;\;\;\lambda_1 \cdot \left(\frac{R \cdot \lambda_2 + \frac{-0.5 \cdot \left(R \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\right)\right)}{\lambda_1}}{\lambda_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 32.7% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.2 \cdot 10^{+17}:\\ \;\;\;\;\phi_1 \cdot \left(\frac{R \cdot \phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_1 \leq 3.3 \cdot 10^{-204}:\\ \;\;\;\;R \cdot \left(\lambda_1 \cdot \left(\frac{\lambda_2 + \frac{-0.5 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\right)}{\lambda_1}}{\lambda_1} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -4.2e+17)
   (* phi1 (- (/ (* R phi2) phi1) R))
   (if (<= phi1 3.3e-204)
     (*
      R
      (*
       lambda1
       (+
        (/
         (+ lambda2 (/ (* -0.5 (* (- phi1 phi2) (- phi1 phi2))) lambda1))
         lambda1)
        -1.0)))
     (* R phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -4.2e+17) {
		tmp = phi1 * (((R * phi2) / phi1) - R);
	} else if (phi1 <= 3.3e-204) {
		tmp = R * (lambda1 * (((lambda2 + ((-0.5 * ((phi1 - phi2) * (phi1 - phi2))) / lambda1)) / lambda1) + -1.0));
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-4.2d+17)) then
        tmp = phi1 * (((r * phi2) / phi1) - r)
    else if (phi1 <= 3.3d-204) then
        tmp = r * (lambda1 * (((lambda2 + (((-0.5d0) * ((phi1 - phi2) * (phi1 - phi2))) / lambda1)) / lambda1) + (-1.0d0)))
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -4.2e+17) {
		tmp = phi1 * (((R * phi2) / phi1) - R);
	} else if (phi1 <= 3.3e-204) {
		tmp = R * (lambda1 * (((lambda2 + ((-0.5 * ((phi1 - phi2) * (phi1 - phi2))) / lambda1)) / lambda1) + -1.0));
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -4.2e+17:
		tmp = phi1 * (((R * phi2) / phi1) - R)
	elif phi1 <= 3.3e-204:
		tmp = R * (lambda1 * (((lambda2 + ((-0.5 * ((phi1 - phi2) * (phi1 - phi2))) / lambda1)) / lambda1) + -1.0))
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -4.2e+17)
		tmp = Float64(phi1 * Float64(Float64(Float64(R * phi2) / phi1) - R));
	elseif (phi1 <= 3.3e-204)
		tmp = Float64(R * Float64(lambda1 * Float64(Float64(Float64(lambda2 + Float64(Float64(-0.5 * Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2))) / lambda1)) / lambda1) + -1.0)));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -4.2e+17)
		tmp = phi1 * (((R * phi2) / phi1) - R);
	elseif (phi1 <= 3.3e-204)
		tmp = R * (lambda1 * (((lambda2 + ((-0.5 * ((phi1 - phi2) * (phi1 - phi2))) / lambda1)) / lambda1) + -1.0));
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -4.2e+17], N[(phi1 * N[(N[(N[(R * phi2), $MachinePrecision] / phi1), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 3.3e-204], N[(R * N[(lambda1 * N[(N[(N[(lambda2 + N[(N[(-0.5 * N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / lambda1), $MachinePrecision]), $MachinePrecision] / lambda1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -4.2e17

   1. Initial program 55.6%

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

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

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

      2. hypot-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. --lowering--.f64 89.1%

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

   3. Simplified 89.1%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right) \]

      2. neg-sub0 N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \color{blue}{\left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \left(R + \left(\mathsf{neg}\left(\frac{R \cdot \phi_2}{\phi_1}\right)\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \left(R - \color{blue}{\frac{R \cdot \phi_2}{\phi_1}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \mathsf{\_.f64}\left(R, \color{blue}{\left(\frac{R \cdot \phi_2}{\phi_1}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\left(R \cdot \phi_2\right), \color{blue}{\phi_1}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 60.2%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\mathsf{*.f64}\left(R, \phi_2\right), \phi_1\right)\right)\right)\right) \]

   7. Simplified 60.2%

      \[\leadsto \color{blue}{0 - \phi_1 \cdot \left(R - \frac{R \cdot \phi_2}{\phi_1}\right)} \]

   if -4.2e17 < phi1 < 3.30000000000000009e-204

   1. Initial program 60.0%

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

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

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

      2. hypot-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. --lowering--.f64 99.2%

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

   3. Simplified 99.2%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around 0

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

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

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

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

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

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

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

      10. *-lowering-*.f64 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   9. Step-by-step derivation

      1. --lowering--.f64 86.1%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

   10. Simplified 86.1%

       \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\lambda_1 - \lambda_2}, \phi_1 - \phi_2\right) \]

   11. Taylor expanded in lambda1 around -inf

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

       1. associate-*r* N/A

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

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

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

       3. mul-1-neg N/A

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

       4. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\left(0 - \lambda_1\right), \left(\color{blue}{1} + -1 \cdot \frac{\lambda_2 + \frac{-1}{2} \cdot \frac{{\left(\phi_1 - \phi_2\right)}^{2}}{\lambda_1}}{\lambda_1}\right)\right)\right) \]

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

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

       6. mul-1-neg N/A

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

       7. unsub-neg N/A

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

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

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

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

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(0, \lambda_1\right), \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\left(\lambda_2 + \frac{-1}{2} \cdot \frac{{\left(\phi_1 - \phi_2\right)}^{2}}{\lambda_1}\right), \color{blue}{\lambda_1}\right)\right)\right)\right) \]

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

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

       11. associate-*r/ N/A

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

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

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

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

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

       14. unpow2 N/A

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

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

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

       16. --lowering--.f64 N/A

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

       17. --lowering--.f64 28.0%

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

   13. Simplified 28.0%

       \[\leadsto R \cdot \color{blue}{\left(\left(0 - \lambda_1\right) \cdot \left(1 - \frac{\lambda_2 + \frac{-0.5 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\right)}{\lambda_1}}{\lambda_1}\right)\right)} \]

   if 3.30000000000000009e-204 < phi1

   1. Initial program 50.9%

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

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

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

      2. hypot-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. --lowering--.f64 93.7%

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

   3. Simplified 93.7%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 12.5%

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\phi_2}\right) \]

   7. Simplified 12.5%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.2 \cdot 10^{+17}:\\ \;\;\;\;\phi_1 \cdot \left(\frac{R \cdot \phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_1 \leq 3.3 \cdot 10^{-204}:\\ \;\;\;\;R \cdot \left(\lambda_1 \cdot \left(\frac{\lambda_2 + \frac{-0.5 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\right)}{\lambda_1}}{\lambda_1} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 32.5% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.2 \cdot 10^{+17}:\\ \;\;\;\;\phi_1 \cdot \left(\frac{R \cdot \phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_1 \leq 3.5 \cdot 10^{-204}:\\ \;\;\;\;\lambda_2 \cdot \left(R - \frac{R \cdot \lambda_1}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -3.2e+17)
   (* phi1 (- (/ (* R phi2) phi1) R))
   (if (<= phi1 3.5e-204)
     (* lambda2 (- R (/ (* R lambda1) lambda2)))
     (* R phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.2e+17) {
		tmp = phi1 * (((R * phi2) / phi1) - R);
	} else if (phi1 <= 3.5e-204) {
		tmp = lambda2 * (R - ((R * lambda1) / lambda2));
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-3.2d+17)) then
        tmp = phi1 * (((r * phi2) / phi1) - r)
    else if (phi1 <= 3.5d-204) then
        tmp = lambda2 * (r - ((r * lambda1) / lambda2))
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.2e+17) {
		tmp = phi1 * (((R * phi2) / phi1) - R);
	} else if (phi1 <= 3.5e-204) {
		tmp = lambda2 * (R - ((R * lambda1) / lambda2));
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -3.2e+17:
		tmp = phi1 * (((R * phi2) / phi1) - R)
	elif phi1 <= 3.5e-204:
		tmp = lambda2 * (R - ((R * lambda1) / lambda2))
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -3.2e+17)
		tmp = Float64(phi1 * Float64(Float64(Float64(R * phi2) / phi1) - R));
	elseif (phi1 <= 3.5e-204)
		tmp = Float64(lambda2 * Float64(R - Float64(Float64(R * lambda1) / lambda2)));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -3.2e+17)
		tmp = phi1 * (((R * phi2) / phi1) - R);
	elseif (phi1 <= 3.5e-204)
		tmp = lambda2 * (R - ((R * lambda1) / lambda2));
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -3.2e+17], N[(phi1 * N[(N[(N[(R * phi2), $MachinePrecision] / phi1), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 3.5e-204], N[(lambda2 * N[(R - N[(N[(R * lambda1), $MachinePrecision] / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -3.2e17

   1. Initial program 55.6%

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

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

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

      2. hypot-define N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. --lowering--.f64 89.1%

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

   3. Simplified 89.1%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right) \]

      2. neg-sub0 N/A

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

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

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \color{blue}{\left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \left(R + \left(\mathsf{neg}\left(\frac{R \cdot \phi_2}{\phi_1}\right)\right)\right)\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \left(R - \color{blue}{\frac{R \cdot \phi_2}{\phi_1}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \mathsf{\_.f64}\left(R, \color{blue}{\left(\frac{R \cdot \phi_2}{\phi_1}\right)}\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\left(R \cdot \phi_2\right), \color{blue}{\phi_1}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 60.2%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\phi_1, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\mathsf{*.f64}\left(R, \phi_2\right), \phi_1\right)\right)\right)\right) \]

   7. Simplified 60.2%

      \[\leadsto \color{blue}{0 - \phi_1 \cdot \left(R - \frac{R \cdot \phi_2}{\phi_1}\right)} \]

   if -3.2e17 < phi1 < 3.50000000000000027e-204

   1. Initial program 60.0%

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

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

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

      2. hypot-define N/A

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

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 99.2%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right), \left(\frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \frac{-1}{2}\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\left(\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \phi_1\right) \cdot \frac{-1}{2}\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right), \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      10. *-lowering-*.f64 98.4%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\phi_1, \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. Simplified 98.4%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(0.5 \cdot \phi_2\right) + \sin \left(0.5 \cdot \phi_2\right) \cdot \left(\phi_1 \cdot -0.5\right)\right)}, \phi_1 - \phi_2\right) \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   9. Step-by-step derivation

      1. --lowering--.f64 86.1%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

   10. Simplified 86.1%

       \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\lambda_1 - \lambda_2}, \phi_1 - \phi_2\right) \]

   11. Taylor expanded in lambda2 around inf

       \[\leadsto \color{blue}{\lambda_2 \cdot \left(R + -1 \cdot \frac{R \cdot \lambda_1}{\lambda_2}\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\lambda_2, \color{blue}{\left(R + -1 \cdot \frac{R \cdot \lambda_1}{\lambda_2}\right)}\right) \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\lambda_2, \left(R + \left(\mathsf{neg}\left(\frac{R \cdot \lambda_1}{\lambda_2}\right)\right)\right)\right) \]

       3. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\lambda_2, \left(R - \color{blue}{\frac{R \cdot \lambda_1}{\lambda_2}}\right)\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(R, \color{blue}{\left(\frac{R \cdot \lambda_1}{\lambda_2}\right)}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\left(R \cdot \lambda_1\right), \color{blue}{\lambda_2}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\left(\lambda_1 \cdot R\right), \lambda_2\right)\right)\right) \]

       7. *-lowering-*.f64 26.3%

          \[\leadsto \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\lambda_1, R\right), \lambda_2\right)\right)\right) \]

   13. Simplified 26.3%

       \[\leadsto \color{blue}{\lambda_2 \cdot \left(R - \frac{\lambda_1 \cdot R}{\lambda_2}\right)} \]

   if 3.50000000000000027e-204 < phi1

   1. Initial program 50.9%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 93.7%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 93.7%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 12.5%

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\phi_2}\right) \]

   7. Simplified 12.5%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.2 \cdot 10^{+17}:\\ \;\;\;\;\phi_1 \cdot \left(\frac{R \cdot \phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_1 \leq 3.5 \cdot 10^{-204}:\\ \;\;\;\;\lambda_2 \cdot \left(R - \frac{R \cdot \lambda_1}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 33.3% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.2 \cdot 10^{-224}:\\ \;\;\;\;\phi_1 \cdot \left(0 - R\right)\\ \mathbf{elif}\;\phi_2 \leq 1.75 \cdot 10^{+25}:\\ \;\;\;\;\lambda_2 \cdot \left(R - \frac{R \cdot \lambda_1}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -1.2e-224)
   (* phi1 (- 0.0 R))
   (if (<= phi2 1.75e+25)
     (* lambda2 (- R (/ (* R lambda1) lambda2)))
     (* R (* phi2 (- 1.0 (/ phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -1.2e-224) {
		tmp = phi1 * (0.0 - R);
	} else if (phi2 <= 1.75e+25) {
		tmp = lambda2 * (R - ((R * lambda1) / lambda2));
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= (-1.2d-224)) then
        tmp = phi1 * (0.0d0 - r)
    else if (phi2 <= 1.75d+25) then
        tmp = lambda2 * (r - ((r * lambda1) / lambda2))
    else
        tmp = r * (phi2 * (1.0d0 - (phi1 / phi2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -1.2e-224) {
		tmp = phi1 * (0.0 - R);
	} else if (phi2 <= 1.75e+25) {
		tmp = lambda2 * (R - ((R * lambda1) / lambda2));
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= -1.2e-224:
		tmp = phi1 * (0.0 - R)
	elif phi2 <= 1.75e+25:
		tmp = lambda2 * (R - ((R * lambda1) / lambda2))
	else:
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -1.2e-224)
		tmp = Float64(phi1 * Float64(0.0 - R));
	elseif (phi2 <= 1.75e+25)
		tmp = Float64(lambda2 * Float64(R - Float64(Float64(R * lambda1) / lambda2)));
	else
		tmp = Float64(R * Float64(phi2 * Float64(1.0 - Float64(phi1 / phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= -1.2e-224)
		tmp = phi1 * (0.0 - R);
	elseif (phi2 <= 1.75e+25)
		tmp = lambda2 * (R - ((R * lambda1) / lambda2));
	else
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -1.2e-224], N[(phi1 * N[(0.0 - R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.75e+25], N[(lambda2 * N[(R - N[(N[(R * lambda1), $MachinePrecision] / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.20000000000000007e-224

   1. Initial program 59.5%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 95.0%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 95.0%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \phi_1\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(R \cdot \phi_1\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{R \cdot \phi_1} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(R \cdot \phi_1\right)}\right) \]

      4. *-lowering-*.f64 13.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(R, \color{blue}{\phi_1}\right)\right) \]

   7. Simplified 13.3%

      \[\leadsto \color{blue}{0 - R \cdot \phi_1} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(R \cdot \phi_1\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(R \cdot \phi_1\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\phi_1 \cdot R\right)\right) \]

      4. *-lowering-*.f64 13.3%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\phi_1, R\right)\right) \]

   9. Applied egg-rr 13.3%

      \[\leadsto \color{blue}{-\phi_1 \cdot R} \]

   if -1.20000000000000007e-224 < phi2 < 1.75e25

   1. Initial program 58.2%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 98.0%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right), \left(\frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \frac{-1}{2}\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\left(\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \phi_1\right) \cdot \frac{-1}{2}\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right), \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      10. *-lowering-*.f64 81.3%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\phi_1, \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. Simplified 81.3%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(0.5 \cdot \phi_2\right) + \sin \left(0.5 \cdot \phi_2\right) \cdot \left(\phi_1 \cdot -0.5\right)\right)}, \phi_1 - \phi_2\right) \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   9. Step-by-step derivation

      1. --lowering--.f64 86.6%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

   10. Simplified 86.6%

       \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\lambda_1 - \lambda_2}, \phi_1 - \phi_2\right) \]

   11. Taylor expanded in lambda2 around inf

       \[\leadsto \color{blue}{\lambda_2 \cdot \left(R + -1 \cdot \frac{R \cdot \lambda_1}{\lambda_2}\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\lambda_2, \color{blue}{\left(R + -1 \cdot \frac{R \cdot \lambda_1}{\lambda_2}\right)}\right) \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\lambda_2, \left(R + \left(\mathsf{neg}\left(\frac{R \cdot \lambda_1}{\lambda_2}\right)\right)\right)\right) \]

       3. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\lambda_2, \left(R - \color{blue}{\frac{R \cdot \lambda_1}{\lambda_2}}\right)\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(R, \color{blue}{\left(\frac{R \cdot \lambda_1}{\lambda_2}\right)}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\left(R \cdot \lambda_1\right), \color{blue}{\lambda_2}\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\left(\lambda_1 \cdot R\right), \lambda_2\right)\right)\right) \]

       7. *-lowering-*.f64 24.6%

          \[\leadsto \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\lambda_1, R\right), \lambda_2\right)\right)\right) \]

   13. Simplified 24.6%

       \[\leadsto \color{blue}{\lambda_2 \cdot \left(R - \frac{\lambda_1 \cdot R}{\lambda_2}\right)} \]

   if 1.75e25 < phi2

   1. Initial program 40.5%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 87.7%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 87.7%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi2 around inf

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\phi_2, \color{blue}{\left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)}\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\phi_2, \left(1 + \left(\mathsf{neg}\left(\frac{\phi_1}{\phi_2}\right)\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\phi_2, \left(1 - \color{blue}{\frac{\phi_1}{\phi_2}}\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{\phi_1}{\phi_2}\right)}\right)\right)\right) \]

      5. /-lowering-/.f64 65.1%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right)\right) \]

   7. Simplified 65.1%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.2 \cdot 10^{-224}:\\ \;\;\;\;\phi_1 \cdot \left(0 - R\right)\\ \mathbf{elif}\;\phi_2 \leq 1.75 \cdot 10^{+25}:\\ \;\;\;\;\lambda_2 \cdot \left(R - \frac{R \cdot \lambda_1}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 32.9% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.2 \cdot 10^{-219}:\\ \;\;\;\;\phi_1 \cdot \left(0 - R\right)\\ \mathbf{elif}\;\phi_2 \leq 2.15 \cdot 10^{+23}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -2.2e-219)
   (* phi1 (- 0.0 R))
   (if (<= phi2 2.15e+23)
     (* R (* lambda2 (- 1.0 (/ lambda1 lambda2))))
     (* R (* phi2 (- 1.0 (/ phi1 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -2.2e-219) {
		tmp = phi1 * (0.0 - R);
	} else if (phi2 <= 2.15e+23) {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= (-2.2d-219)) then
        tmp = phi1 * (0.0d0 - r)
    else if (phi2 <= 2.15d+23) then
        tmp = r * (lambda2 * (1.0d0 - (lambda1 / lambda2)))
    else
        tmp = r * (phi2 * (1.0d0 - (phi1 / phi2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -2.2e-219) {
		tmp = phi1 * (0.0 - R);
	} else if (phi2 <= 2.15e+23) {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= -2.2e-219:
		tmp = phi1 * (0.0 - R)
	elif phi2 <= 2.15e+23:
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)))
	else:
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -2.2e-219)
		tmp = Float64(phi1 * Float64(0.0 - R));
	elseif (phi2 <= 2.15e+23)
		tmp = Float64(R * Float64(lambda2 * Float64(1.0 - Float64(lambda1 / lambda2))));
	else
		tmp = Float64(R * Float64(phi2 * Float64(1.0 - Float64(phi1 / phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= -2.2e-219)
		tmp = phi1 * (0.0 - R);
	elseif (phi2 <= 2.15e+23)
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	else
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -2.2e-219], N[(phi1 * N[(0.0 - R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 2.15e+23], N[(R * N[(lambda2 * N[(1.0 - N[(lambda1 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -2.1999999999999999e-219

   1. Initial program 59.5%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 95.0%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 95.0%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \phi_1\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(R \cdot \phi_1\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{R \cdot \phi_1} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(R \cdot \phi_1\right)}\right) \]

      4. *-lowering-*.f64 13.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(R, \color{blue}{\phi_1}\right)\right) \]

   7. Simplified 13.3%

      \[\leadsto \color{blue}{0 - R \cdot \phi_1} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(R \cdot \phi_1\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(R \cdot \phi_1\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\phi_1 \cdot R\right)\right) \]

      4. *-lowering-*.f64 13.3%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\phi_1, R\right)\right) \]

   9. Applied egg-rr 13.3%

      \[\leadsto \color{blue}{-\phi_1 \cdot R} \]

   if -2.1999999999999999e-219 < phi2 < 2.1499999999999999e23

   1. Initial program 58.2%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 98.0%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right), \left(\frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \frac{-1}{2}\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\left(\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \phi_1\right) \cdot \frac{-1}{2}\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right), \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      10. *-lowering-*.f64 81.3%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\phi_1, \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. Simplified 81.3%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(0.5 \cdot \phi_2\right) + \sin \left(0.5 \cdot \phi_2\right) \cdot \left(\phi_1 \cdot -0.5\right)\right)}, \phi_1 - \phi_2\right) \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   9. Step-by-step derivation

      1. --lowering--.f64 86.6%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

   10. Simplified 86.6%

       \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\lambda_1 - \lambda_2}, \phi_1 - \phi_2\right) \]

   11. Taylor expanded in lambda2 around inf

       \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\lambda_2 \cdot \left(1 + -1 \cdot \frac{\lambda_1}{\lambda_2}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \color{blue}{\left(1 + -1 \cdot \frac{\lambda_1}{\lambda_2}\right)}\right)\right) \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \left(1 + \left(\mathsf{neg}\left(\frac{\lambda_1}{\lambda_2}\right)\right)\right)\right)\right) \]

       3. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \left(1 - \color{blue}{\frac{\lambda_1}{\lambda_2}}\right)\right)\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{\lambda_1}{\lambda_2}\right)}\right)\right)\right) \]

       5. /-lowering-/.f64 24.4%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\lambda_1, \color{blue}{\lambda_2}\right)\right)\right)\right) \]

   13. Simplified 24.4%

       \[\leadsto R \cdot \color{blue}{\left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)} \]

   if 2.1499999999999999e23 < phi2

   1. Initial program 40.5%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 87.7%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 87.7%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi2 around inf

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\phi_2, \color{blue}{\left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)}\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\phi_2, \left(1 + \left(\mathsf{neg}\left(\frac{\phi_1}{\phi_2}\right)\right)\right)\right)\right) \]

      3. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\phi_2, \left(1 - \color{blue}{\frac{\phi_1}{\phi_2}}\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{\phi_1}{\phi_2}\right)}\right)\right)\right) \]

      5. /-lowering-/.f64 65.1%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right)\right) \]

   7. Simplified 65.1%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.2 \cdot 10^{-219}:\\ \;\;\;\;\phi_1 \cdot \left(0 - R\right)\\ \mathbf{elif}\;\phi_2 \leq 2.15 \cdot 10^{+23}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 31.8% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -6.6 \cdot 10^{-220}:\\ \;\;\;\;\phi_1 \cdot \left(0 - R\right)\\ \mathbf{elif}\;\phi_2 \leq 1.9 \cdot 10^{+23}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -6.6e-220)
   (* phi1 (- 0.0 R))
   (if (<= phi2 1.9e+23)
     (* R (* lambda2 (- 1.0 (/ lambda1 lambda2))))
     (* R phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -6.6e-220) {
		tmp = phi1 * (0.0 - R);
	} else if (phi2 <= 1.9e+23) {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= (-6.6d-220)) then
        tmp = phi1 * (0.0d0 - r)
    else if (phi2 <= 1.9d+23) then
        tmp = r * (lambda2 * (1.0d0 - (lambda1 / lambda2)))
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -6.6e-220) {
		tmp = phi1 * (0.0 - R);
	} else if (phi2 <= 1.9e+23) {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= -6.6e-220:
		tmp = phi1 * (0.0 - R)
	elif phi2 <= 1.9e+23:
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)))
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= -6.6e-220)
		tmp = Float64(phi1 * Float64(0.0 - R));
	elseif (phi2 <= 1.9e+23)
		tmp = Float64(R * Float64(lambda2 * Float64(1.0 - Float64(lambda1 / lambda2))));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= -6.6e-220)
		tmp = phi1 * (0.0 - R);
	elseif (phi2 <= 1.9e+23)
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, -6.6e-220], N[(phi1 * N[(0.0 - R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.9e+23], N[(R * N[(lambda2 * N[(1.0 - N[(lambda1 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -6.59999999999999999e-220

   1. Initial program 59.5%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 95.0%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 95.0%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \phi_1\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(R \cdot \phi_1\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{R \cdot \phi_1} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(R \cdot \phi_1\right)}\right) \]

      4. *-lowering-*.f64 13.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(R, \color{blue}{\phi_1}\right)\right) \]

   7. Simplified 13.3%

      \[\leadsto \color{blue}{0 - R \cdot \phi_1} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(R \cdot \phi_1\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(R \cdot \phi_1\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\phi_1 \cdot R\right)\right) \]

      4. *-lowering-*.f64 13.3%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\phi_1, R\right)\right) \]

   9. Applied egg-rr 13.3%

      \[\leadsto \color{blue}{-\phi_1 \cdot R} \]

   if -6.59999999999999999e-220 < phi2 < 1.89999999999999987e23

   1. Initial program 58.2%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 98.0%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 98.0%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right), \left(\frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \frac{-1}{2}\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\left(\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \phi_1\right) \cdot \frac{-1}{2}\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right), \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      10. *-lowering-*.f64 81.3%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\phi_1, \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. Simplified 81.3%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(0.5 \cdot \phi_2\right) + \sin \left(0.5 \cdot \phi_2\right) \cdot \left(\phi_1 \cdot -0.5\right)\right)}, \phi_1 - \phi_2\right) \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   9. Step-by-step derivation

      1. --lowering--.f64 86.6%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

   10. Simplified 86.6%

       \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\lambda_1 - \lambda_2}, \phi_1 - \phi_2\right) \]

   11. Taylor expanded in lambda2 around inf

       \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\lambda_2 \cdot \left(1 + -1 \cdot \frac{\lambda_1}{\lambda_2}\right)\right)}\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \color{blue}{\left(1 + -1 \cdot \frac{\lambda_1}{\lambda_2}\right)}\right)\right) \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \left(1 + \left(\mathsf{neg}\left(\frac{\lambda_1}{\lambda_2}\right)\right)\right)\right)\right) \]

       3. unsub-neg N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \left(1 - \color{blue}{\frac{\lambda_1}{\lambda_2}}\right)\right)\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(1, \color{blue}{\left(\frac{\lambda_1}{\lambda_2}\right)}\right)\right)\right) \]

       5. /-lowering-/.f64 24.4%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\lambda_2, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\lambda_1, \color{blue}{\lambda_2}\right)\right)\right)\right) \]

   13. Simplified 24.4%

       \[\leadsto R \cdot \color{blue}{\left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)} \]

   if 1.89999999999999987e23 < phi2

   1. Initial program 40.5%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 87.7%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 87.7%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 63.5%

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\phi_2}\right) \]

   7. Simplified 63.5%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -6.6 \cdot 10^{-220}:\\ \;\;\;\;\phi_1 \cdot \left(0 - R\right)\\ \mathbf{elif}\;\phi_2 \leq 1.9 \cdot 10^{+23}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 29.1% accurate, 21.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.65 \cdot 10^{+45}:\\ \;\;\;\;\phi_1 \cdot \left(0 - R\right)\\ \mathbf{elif}\;\phi_1 \leq -3.4 \cdot 10^{-145}:\\ \;\;\;\;0 - R \cdot \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -2.65e+45)
   (* phi1 (- 0.0 R))
   (if (<= phi1 -3.4e-145) (- 0.0 (* R lambda1)) (* R phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.65e+45) {
		tmp = phi1 * (0.0 - R);
	} else if (phi1 <= -3.4e-145) {
		tmp = 0.0 - (R * lambda1);
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-2.65d+45)) then
        tmp = phi1 * (0.0d0 - r)
    else if (phi1 <= (-3.4d-145)) then
        tmp = 0.0d0 - (r * lambda1)
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.65e+45) {
		tmp = phi1 * (0.0 - R);
	} else if (phi1 <= -3.4e-145) {
		tmp = 0.0 - (R * lambda1);
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -2.65e+45:
		tmp = phi1 * (0.0 - R)
	elif phi1 <= -3.4e-145:
		tmp = 0.0 - (R * lambda1)
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -2.65e+45)
		tmp = Float64(phi1 * Float64(0.0 - R));
	elseif (phi1 <= -3.4e-145)
		tmp = Float64(0.0 - Float64(R * lambda1));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -2.65e+45)
		tmp = phi1 * (0.0 - R);
	elseif (phi1 <= -3.4e-145)
		tmp = 0.0 - (R * lambda1);
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -2.65e+45], N[(phi1 * N[(0.0 - R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, -3.4e-145], N[(0.0 - N[(R * lambda1), $MachinePrecision]), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.64999999999999996e45

   1. Initial program 55.5%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 88.7%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 88.7%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \phi_1\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(R \cdot \phi_1\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{R \cdot \phi_1} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(R \cdot \phi_1\right)}\right) \]

      4. *-lowering-*.f64 56.3%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(R, \color{blue}{\phi_1}\right)\right) \]

   7. Simplified 56.3%

      \[\leadsto \color{blue}{0 - R \cdot \phi_1} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(R \cdot \phi_1\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(R \cdot \phi_1\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\phi_1 \cdot R\right)\right) \]

      4. *-lowering-*.f64 56.3%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\phi_1, R\right)\right) \]

   9. Applied egg-rr 56.3%

      \[\leadsto \color{blue}{-\phi_1 \cdot R} \]

   if -2.64999999999999996e45 < phi1 < -3.3999999999999999e-145

   1. Initial program 63.7%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 96.9%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 96.9%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right), \left(\frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \frac{-1}{2}\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\left(\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \phi_1\right) \cdot \frac{-1}{2}\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right), \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      10. *-lowering-*.f64 92.6%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\phi_1, \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. Simplified 92.6%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(0.5 \cdot \phi_2\right) + \sin \left(0.5 \cdot \phi_2\right) \cdot \left(\phi_1 \cdot -0.5\right)\right)}, \phi_1 - \phi_2\right) \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   9. Step-by-step derivation

      1. --lowering--.f64 84.6%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

   10. Simplified 84.6%

       \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\lambda_1 - \lambda_2}, \phi_1 - \phi_2\right) \]

   11. Taylor expanded in lambda1 around -inf

       \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \lambda_1\right)} \]
   12. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(R \cdot \lambda_1\right) \]

       2. neg-sub0 N/A

          \[\leadsto 0 - \color{blue}{R \cdot \lambda_1} \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(R \cdot \lambda_1\right)}\right) \]

       4. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(0, \left(\lambda_1 \cdot \color{blue}{R}\right)\right) \]

       5. *-lowering-*.f64 10.9%

          \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(\lambda_1, \color{blue}{R}\right)\right) \]

   13. Simplified 10.9%

       \[\leadsto \color{blue}{0 - \lambda_1 \cdot R} \]

   if -3.3999999999999999e-145 < phi1

   1. Initial program 52.5%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 95.6%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 95.6%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 15.8%

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\phi_2}\right) \]

   7. Simplified 15.8%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 23.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.65 \cdot 10^{+45}:\\ \;\;\;\;\phi_1 \cdot \left(0 - R\right)\\ \mathbf{elif}\;\phi_1 \leq -3.4 \cdot 10^{-145}:\\ \;\;\;\;0 - R \cdot \lambda_1\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 29.2% accurate, 32.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.5 \cdot 10^{+39}:\\ \;\;\;\;\phi_1 \cdot \left(0 - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 3.5e+39) (* phi1 (- 0.0 R)) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.5e+39) {
		tmp = phi1 * (0.0 - R);
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 3.5d+39) then
        tmp = phi1 * (0.0d0 - r)
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.5e+39) {
		tmp = phi1 * (0.0 - R);
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 3.5e+39:
		tmp = phi1 * (0.0 - R)
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 3.5e+39)
		tmp = Float64(phi1 * Float64(0.0 - R));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 3.5e+39)
		tmp = phi1 * (0.0 - R);
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 3.5e+39], N[(phi1 * N[(0.0 - R), $MachinePrecision]), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.5000000000000002e39

   1. Initial program 59.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 96.3%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 96.3%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \phi_1\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(R \cdot \phi_1\right) \]

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{R \cdot \phi_1} \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(R \cdot \phi_1\right)}\right) \]

      4. *-lowering-*.f64 16.4%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(R, \color{blue}{\phi_1}\right)\right) \]

   7. Simplified 16.4%

      \[\leadsto \color{blue}{0 - R \cdot \phi_1} \]
   8. Step-by-step derivation

      1. sub0-neg N/A

         \[\leadsto \mathsf{neg}\left(R \cdot \phi_1\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(R \cdot \phi_1\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\phi_1 \cdot R\right)\right) \]

      4. *-lowering-*.f64 16.4%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\phi_1, R\right)\right) \]

   9. Applied egg-rr 16.4%

      \[\leadsto \color{blue}{-\phi_1 \cdot R} \]

   if 3.5000000000000002e39 < phi2

   1. Initial program 38.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 87.2%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 87.2%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 63.8%

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\phi_2}\right) \]

   7. Simplified 63.8%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.5 \cdot 10^{+39}:\\ \;\;\;\;\phi_1 \cdot \left(0 - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 26.4% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.2 \cdot 10^{+23}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2.2e+23) (* R lambda2) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.2e+23) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 2.2d+23) then
        tmp = r * lambda2
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.2e+23) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2.2e+23:
		tmp = R * lambda2
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2.2e+23)
		tmp = Float64(R * lambda2);
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 2.2e+23)
		tmp = R * lambda2;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2.2e+23], N[(R * lambda2), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.20000000000000008e23

   1. Initial program 58.9%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 96.3%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 96.3%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi1 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) + \frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right), \left(\frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      2. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\frac{-1}{2} \cdot \left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\left(\phi_1 \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right) \cdot \frac{-1}{2}\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\left(\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \phi_1\right) \cdot \frac{-1}{2}\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right), \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      8. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(\frac{1}{2} \cdot \phi_2\right)\right), \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \left(\phi_1 \cdot \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

      10. *-lowering-*.f64 82.9%

          \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{+.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \phi_2\right)\right), \mathsf{*.f64}\left(\phi_1, \frac{-1}{2}\right)\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]

   7. Simplified 82.9%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(0.5 \cdot \phi_2\right) + \sin \left(0.5 \cdot \phi_2\right) \cdot \left(\phi_1 \cdot -0.5\right)\right)}, \phi_1 - \phi_2\right) \]

   8. Taylor expanded in phi2 around 0

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right)}, \mathsf{\_.f64}\left(\phi_1, \phi_2\right)\right)\right) \]
   9. Step-by-step derivation

      1. --lowering--.f64 84.6%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{\_.f64}\left(\color{blue}{\phi_1}, \phi_2\right)\right)\right) \]

   10. Simplified 84.6%

       \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\lambda_1 - \lambda_2}, \phi_1 - \phi_2\right) \]

   11. Taylor expanded in lambda2 around inf

       \[\leadsto \color{blue}{R \cdot \lambda_2} \]
   12. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \lambda_2 \cdot \color{blue}{R} \]

       2. *-lowering-*.f64 14.5%

          \[\leadsto \mathsf{*.f64}\left(\lambda_2, \color{blue}{R}\right) \]

   13. Simplified 14.5%

       \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

   if 2.20000000000000008e23 < phi2

   1. Initial program 40.5%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

      2. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

      3. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      6. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

      9. --lowering--.f64 87.7%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

   3. Simplified 87.7%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in phi2 around inf

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 63.5%

         \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\phi_2}\right) \]

   7. Simplified 63.5%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.2 \cdot 10^{+23}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 18.2% accurate, 109.7× speedup

Math

\[\begin{array}{l} \\ R \cdot \phi_2 \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2) :precision binary64 (* R phi2))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * phi2;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r * phi2
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * phi2;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * phi2

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * phi2)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * phi2;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * phi2), $MachinePrecision]

Derivation

1. Initial program 54.9%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
2. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\right)}\right) \]

   2. hypot-define N/A

      \[\leadsto \mathsf{*.f64}\left(R, \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \color{blue}{\phi_1 - \phi_2}\right)\right)\right) \]

   3. hypot-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \color{blue}{\left(\phi_1 - \phi_2\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\left(\lambda_1 - \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\color{blue}{\phi_1} - \phi_2\right)\right)\right) \]

   5. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   6. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\left(\frac{\phi_1 + \phi_2}{2}\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\left(\phi_1 + \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   8. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \left(\phi_1 - \phi_2\right)\right)\right) \]

   9. --lowering--.f64 94.4%

      \[\leadsto \mathsf{*.f64}\left(R, \mathsf{hypot.f64}\left(\mathsf{*.f64}\left(\mathsf{\_.f64}\left(\lambda_1, \lambda_2\right), \mathsf{cos.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\phi_1, \phi_2\right), 2\right)\right)\right), \mathsf{\_.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right) \]

3. Simplified 94.4%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
4. Add Preprocessing

5. Taylor expanded in phi2 around inf

   \[\leadsto \color{blue}{R \cdot \phi_2} \]
6. Step-by-step derivation

   1. *-lowering-*.f64 16.4%

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\phi_2}\right) \]

7. Simplified 16.4%

   \[\leadsto \color{blue}{R \cdot \phi_2} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024164 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  :precision binary64
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))))))