Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.6% → 99.8%

Time: 36.1s

Alternatives: 13

Speedup: 3.0×

• 30.1% 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: 59.6% 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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\frac{\mathsf{fma}\left(0 - \sin \left(\phi_2 \cdot 0.5\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}{\frac{1}{\lambda_1 - \lambda_2}}, \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot
   (/
    (fma
     (- 0.0 (sin (* phi2 0.5)))
     (sin (* 0.5 phi1))
     (* (cos (* phi2 0.5)) (cos (* 0.5 phi1))))
    (/ 1.0 (- lambda1 lambda2)))
   (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot((fma((0.0 - sin((phi2 * 0.5))), sin((0.5 * phi1)), (cos((phi2 * 0.5)) * cos((0.5 * phi1)))) / (1.0 / (lambda1 - lambda2))), (phi1 - phi2));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(fma(Float64(0.0 - sin(Float64(phi2 * 0.5))), sin(Float64(0.5 * phi1)), Float64(cos(Float64(phi2 * 0.5)) * cos(Float64(0.5 * phi1)))) / Float64(1.0 / Float64(lambda1 - lambda2))), Float64(phi1 - phi2)))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(N[(0.0 - N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.6%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
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 97.6%

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

3. Simplified 97.6%

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

   1. *-commutative N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   9. clear-num N/A

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

   10. flip-- N/A

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

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

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

   12. --lowering--.f64 97.5%

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

6. Applied egg-rr 97.5%

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

   1. div-inv N/A

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. cos-sum N/A

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

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

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

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

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

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

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

   10. *-commutative N/A

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

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

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

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

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

   13. *-commutative N/A

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

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

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

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

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

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

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

   17. *-commutative N/A

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

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

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

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

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

   20. *-commutative N/A

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

   21. *-lowering-*.f64 99.8%

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

8. Applied egg-rr 99.8%

   \[\leadsto R \cdot \mathsf{hypot}\left(\frac{\color{blue}{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)}}{\frac{1}{\lambda_1 - \lambda_2}}, \phi_1 - \phi_2\right) \]
9. Step-by-step derivation

   1. sub-neg N/A

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

   2. +-commutative N/A

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

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

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

   4. fma-define N/A

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

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

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

   6. neg-sub0 N/A

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

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

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

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

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

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

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

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

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

   11. *-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

   17. *-commutative N/A

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

   18. *-lowering-*.f64 99.8%

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

10. Applied egg-rr 99.8%

    \[\leadsto R \cdot \mathsf{hypot}\left(\frac{\color{blue}{\mathsf{fma}\left(0 - \sin \left(\phi_2 \cdot 0.5\right), \sin \left(0.5 \cdot \phi_1\right), \cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}}{\frac{1}{\lambda_1 - \lambda_2}}, \phi_1 - \phi_2\right) \]
11. Add Preprocessing

Alternative 2: 99.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\frac{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)}{\frac{1}{\lambda_1 - \lambda_2}}, \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot
   (/
    (-
     (* (cos (* phi2 0.5)) (cos (* 0.5 phi1)))
     (* (sin (* phi2 0.5)) (sin (* 0.5 phi1))))
    (/ 1.0 (- lambda1 lambda2)))
   (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot((((cos((phi2 * 0.5)) * cos((0.5 * phi1))) - (sin((phi2 * 0.5)) * sin((0.5 * phi1)))) / (1.0 / (lambda1 - lambda2))), (phi1 - phi2));
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.hypot((((Math.cos((phi2 * 0.5)) * Math.cos((0.5 * phi1))) - (Math.sin((phi2 * 0.5)) * Math.sin((0.5 * phi1)))) / (1.0 / (lambda1 - lambda2))), (phi1 - phi2));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.hypot((((math.cos((phi2 * 0.5)) * math.cos((0.5 * phi1))) - (math.sin((phi2 * 0.5)) * math.sin((0.5 * phi1)))) / (1.0 / (lambda1 - lambda2))), (phi1 - phi2))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(Float64(Float64(cos(Float64(phi2 * 0.5)) * cos(Float64(0.5 * phi1))) - Float64(sin(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1)))) / Float64(1.0 / Float64(lambda1 - lambda2))), Float64(phi1 - phi2)))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot((((cos((phi2 * 0.5)) * cos((0.5 * phi1))) - (sin((phi2 * 0.5)) * sin((0.5 * phi1)))) / (1.0 / (lambda1 - lambda2))), (phi1 - phi2));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.6%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
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 97.6%

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

3. Simplified 97.6%

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

   1. *-commutative N/A

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

   2. flip-- N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

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

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

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

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

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

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

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

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

   9. clear-num N/A

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

   10. flip-- N/A

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

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

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

   12. --lowering--.f64 97.5%

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

6. Applied egg-rr 97.5%

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

   1. div-inv N/A

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. +-commutative N/A

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

   5. distribute-lft-in N/A

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

   6. cos-sum N/A

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

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

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

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

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

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

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

   10. *-commutative N/A

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

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

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

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

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

   13. *-commutative N/A

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

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

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

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

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

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

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

   17. *-commutative N/A

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

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

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

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

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

   20. *-commutative N/A

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

   21. *-lowering-*.f64 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

   \[\leadsto R \cdot \mathsf{hypot}\left(\frac{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)}{\frac{1}{\lambda_1 - \lambda_2}}, \phi_1 - \phi_2\right) \]
10. Add Preprocessing

Alternative 3: 92.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1850000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1850000.0)
   (* R (hypot (* (cos (* 0.5 phi1)) (- lambda1 lambda2)) (- phi1 phi2)))
   (* R (hypot (* (cos (* phi2 0.5)) (- lambda1 lambda2)) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1850000.0) {
		tmp = R * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
	} else {
		tmp = R * hypot((cos((phi2 * 0.5)) * (lambda1 - lambda2)), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1850000.0) {
		tmp = R * Math.hypot((Math.cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot((Math.cos((phi2 * 0.5)) * (lambda1 - lambda2)), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1850000.0:
		tmp = R * math.hypot((math.cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2))
	else:
		tmp = R * math.hypot((math.cos((phi2 * 0.5)) * (lambda1 - lambda2)), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1850000.0)
		tmp = Float64(R * hypot(Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2)), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(cos(Float64(phi2 * 0.5)) * Float64(lambda1 - lambda2)), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1850000.0)
		tmp = R * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
	else
		tmp = R * hypot((cos((phi2 * 0.5)) * (lambda1 - lambda2)), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1850000.0], N[(R * N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.85e6

   1. Initial program 50.0%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. *-lowering-*.f64 96.9%

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

   7. Simplified 96.9%

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

   if -1.85e6 < phi1

   1. Initial program 66.4%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. *-lowering-*.f64 93.8%

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

   7. Simplified 93.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)}, \phi_1 - \phi_2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1850000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.3 \cdot 10^{+39}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.3e+39)
   (* R (hypot (* (cos (* 0.5 phi1)) lambda1) (- phi1 phi2)))
   (* R (hypot phi2 (- lambda1 lambda2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.3e+39) {
		tmp = R * hypot((cos((0.5 * phi1)) * lambda1), (phi1 - phi2));
	} else {
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.3e+39) {
		tmp = R * Math.hypot((Math.cos((0.5 * phi1)) * lambda1), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1.3e+39:
		tmp = R * math.hypot((math.cos((0.5 * phi1)) * lambda1), (phi1 - phi2))
	else:
		tmp = R * math.hypot(phi2, (lambda1 - lambda2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.3e+39)
		tmp = Float64(R * hypot(Float64(cos(Float64(0.5 * phi1)) * lambda1), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(phi2, Float64(lambda1 - lambda2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1.3e+39)
		tmp = R * hypot((cos((0.5 * phi1)) * lambda1), (phi1 - phi2));
	else
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.3e+39], N[(R * N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * lambda1), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.3e39

   1. Initial program 48.2%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 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 phi2 around 0

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. *-lowering-*.f64 96.7%

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

   7. Simplified 96.7%

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

   8. Taylor expanded in lambda1 around inf

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

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

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

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

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

      3. *-lowering-*.f64 90.0%

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

   10. Simplified 90.0%

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

   if -1.3e39 < phi1

   1. Initial program 66.5%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. *-lowering-*.f64 91.1%

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

   7. Simplified 91.1%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto \color{blue}{R \cdot \sqrt{{\phi_2}^{2} + {\left(\lambda_1 - \lambda_2\right)}^{2}}} \]
   9. Step-by-step derivation

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

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. hypot-define N/A

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

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

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

      6. --lowering--.f64 73.2%

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

   10. Simplified 73.2%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.3 \cdot 10^{+39}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.8% 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 62.6%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
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 97.6%

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

3. Simplified 97.6%

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

5. Final simplification 97.6%

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

Alternative 6: 90.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R (hypot (* (cos (* 0.5 phi1)) (- lambda1 lambda2)) (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.hypot((Math.cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.hypot((math.cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2)), Float64(phi1 - phi2)))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), (phi1 - phi2));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.6%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
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 97.6%

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

3. Simplified 97.6%

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

5. Taylor expanded in phi2 around 0

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

   1. *-commutative N/A

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

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

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

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

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

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

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

   5. *-lowering-*.f64 92.3%

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

7. Simplified 92.3%

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

8. Final simplification 92.3%

   \[\leadsto R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right) \]
9. Add Preprocessing

Alternative 7: 71.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.5 \cdot 10^{+84}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -5.5e+84)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (hypot phi2 (- lambda1 lambda2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -5.5e+84) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -5.5e+84) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * Math.hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -5.5e+84:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	else:
		tmp = R * math.hypot(phi2, (lambda1 - lambda2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -5.5e+84)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R * hypot(phi2, Float64(lambda1 - lambda2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -5.5e+84)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -5.5e+84], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -5.5000000000000004e84

   1. Initial program 44.2%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 97.7%

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

   3. Simplified 97.7%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   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 70.5%

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

   7. Simplified 70.5%

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

   8. Taylor expanded in phi2 around 0

      \[\leadsto \color{blue}{R \cdot \sqrt{{\phi_1}^{2} + {\left(\lambda_1 - \lambda_2\right)}^{2}}} \]
   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 76.6%

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

   10. Simplified 76.6%

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

   if -5.5000000000000004e84 < phi1

   1. Initial program 66.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. *-lowering-*.f64 91.1%

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

   7. Simplified 91.1%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto \color{blue}{R \cdot \sqrt{{\phi_2}^{2} + {\left(\lambda_1 - \lambda_2\right)}^{2}}} \]
   9. Step-by-step derivation

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

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. hypot-define N/A

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

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

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

      6. --lowering--.f64 72.8%

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

   10. Simplified 72.8%

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

Alternative 8: 70.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 800000000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 800000000.0)
   (* R (hypot phi1 (- 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 <= 800000000.0) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 800000000.0) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 800000000.0:
		tmp = R * math.hypot(phi1, (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 <= 800000000.0)
		tmp = Float64(R * hypot(phi1, 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 <= 800000000.0)
		tmp = R * hypot(phi1, (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, 800000000.0], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 8e8

   1. Initial program 63.8%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 83.9%

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

   7. Simplified 83.9%

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

   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 75.9%

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

   10. Simplified 75.9%

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

   if 8e8 < phi2

   1. Initial program 59.0%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 96.4%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in phi2 around inf

      \[\leadsto \mathsf{*.f64}\left(R, \color{blue}{\left(\phi_2 \cdot \left(1 + -1 \cdot \frac{\phi_1}{\phi_2}\right)\right)}\right) \]
   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 64.5%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right)\right) \]

   7. Simplified 64.5%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 33.8% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.4 \cdot 10^{+203}:\\ \;\;\;\;\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 1.4e+203) (* phi2 (- R (/ (* R phi1) phi2))) (* R lambda2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.4e+203) {
		tmp = phi2 * (R - ((R * phi1) / phi2));
	} else {
		tmp = R * lambda2;
	}
	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 (lambda2 <= 1.4d+203) then
        tmp = phi2 * (r - ((r * phi1) / phi2))
    else
        tmp = r * lambda2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.4e+203) {
		tmp = phi2 * (R - ((R * phi1) / phi2));
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 1.4e+203:
		tmp = phi2 * (R - ((R * phi1) / phi2))
	else:
		tmp = R * lambda2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1.4e+203)
		tmp = Float64(phi2 * Float64(R - Float64(Float64(R * phi1) / phi2)));
	else
		tmp = Float64(R * lambda2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 1.4e+203)
		tmp = phi2 * (R - ((R * phi1) / phi2));
	else
		tmp = R * lambda2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 1.4e+203], N[(phi2 * N[(R - N[(N[(R * phi1), $MachinePrecision] / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.39999999999999995e203

   1. Initial program 63.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 97.7%

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

   3. Simplified 97.7%

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

   5. Taylor expanded in phi2 around inf

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

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

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

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

      6. *-lowering-*.f64 30.3%

         \[\leadsto \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(R, \mathsf{/.f64}\left(\mathsf{*.f64}\left(R, \phi_1\right), \phi_2\right)\right)\right) \]

   7. Simplified 30.3%

      \[\leadsto \color{blue}{\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)} \]

   if 1.39999999999999995e203 < lambda2

   1. Initial program 55.1%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 96.4%

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

   3. Simplified 96.4%

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

   5. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. *-lowering-*.f64 83.5%

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

   7. Simplified 83.5%

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

   8. Taylor expanded in lambda2 around inf

      \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

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

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

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

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

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

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

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

      5. *-lowering-*.f64 61.8%

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

   10. Simplified 61.8%

       \[\leadsto \color{blue}{\left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)} \]

   11. Taylor expanded in phi1 around 0

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

       1. *-lowering-*.f64 76.3%

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

   13. Simplified 76.3%

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

Alternative 10: 33.5% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 5.5 \cdot 10^{+131}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 5.5e+131) (* R (* phi2 (- 1.0 (/ phi1 phi2)))) (* R lambda2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 5.5e+131) {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	} else {
		tmp = R * lambda2;
	}
	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 (lambda2 <= 5.5d+131) then
        tmp = r * (phi2 * (1.0d0 - (phi1 / phi2)))
    else
        tmp = r * lambda2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 5.5e+131) {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 5.5e+131:
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)))
	else:
		tmp = R * lambda2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 5.5e+131)
		tmp = Float64(R * Float64(phi2 * Float64(1.0 - Float64(phi1 / phi2))));
	else
		tmp = Float64(R * lambda2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 5.5e+131)
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	else
		tmp = R * lambda2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 5.5e+131], N[(R * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 5.49999999999999971e131

   1. Initial program 63.8%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in 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 26.4%

         \[\leadsto \mathsf{*.f64}\left(R, \mathsf{*.f64}\left(\phi_2, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(\phi_1, \color{blue}{\phi_2}\right)\right)\right)\right) \]

   7. Simplified 26.4%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)} \]

   if 5.49999999999999971e131 < lambda2

   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 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. *-lowering-*.f64 83.6%

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

   7. Simplified 83.6%

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

   8. Taylor expanded in lambda2 around inf

      \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

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

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

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

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

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

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

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

      5. *-lowering-*.f64 54.7%

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

   10. Simplified 54.7%

       \[\leadsto \color{blue}{\left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)} \]

   11. Taylor expanded in phi1 around 0

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

       1. *-lowering-*.f64 64.6%

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

   13. Simplified 64.6%

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

Alternative 11: 28.9% accurate, 25.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.02 \cdot 10^{-137}:\\ \;\;\;\;0 - R \cdot \phi_1\\ \mathbf{elif}\;\phi_2 \leq 0.0031:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.02e-137)
   (- 0.0 (* R phi1))
   (if (<= phi2 0.0031) (* R lambda2) (* R phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.02e-137) {
		tmp = 0.0 - (R * phi1);
	} else if (phi2 <= 0.0031) {
		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 <= 1.02d-137) then
        tmp = 0.0d0 - (r * phi1)
    else if (phi2 <= 0.0031d0) 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 <= 1.02e-137) {
		tmp = 0.0 - (R * phi1);
	} else if (phi2 <= 0.0031) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.02e-137:
		tmp = 0.0 - (R * phi1)
	elif phi2 <= 0.0031:
		tmp = R * lambda2
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.02e-137)
		tmp = Float64(0.0 - Float64(R * phi1));
	elseif (phi2 <= 0.0031)
		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 <= 1.02e-137)
		tmp = 0.0 - (R * phi1);
	elseif (phi2 <= 0.0031)
		tmp = R * lambda2;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.02e-137], N[(0.0 - N[(R * phi1), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 0.0031], N[(R * lambda2), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 1.02e-137

   1. Initial program 61.9%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in phi1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \phi_1\right)} \]
   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 18.1%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(R, \color{blue}{\phi_1}\right)\right) \]

   7. Simplified 18.1%

      \[\leadsto \color{blue}{0 - R \cdot \phi_1} \]
   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. *-lowering-*.f64 18.1%

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

   9. Applied egg-rr 18.1%

      \[\leadsto \color{blue}{-R \cdot \phi_1} \]

   if 1.02e-137 < phi2 < 0.00309999999999999989

   1. Initial program 76.2%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. *-lowering-*.f64 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in lambda2 around inf

      \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

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

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

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

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

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

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

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

      5. *-lowering-*.f64 24.0%

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

   10. Simplified 24.0%

       \[\leadsto \color{blue}{\left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)} \]

   11. Taylor expanded in phi1 around 0

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

       1. *-lowering-*.f64 32.7%

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

   13. Simplified 32.7%

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

   if 0.00309999999999999989 < phi2

   1. Initial program 59.4%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 95.4%

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

   3. Simplified 95.4%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   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 59.1%

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

   7. Simplified 59.1%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.02 \cdot 10^{-137}:\\ \;\;\;\;0 - R \cdot \phi_1\\ \mathbf{elif}\;\phi_2 \leq 0.0031:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 26.3% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.32 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2.32e-5) (* R lambda2) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.32e-5) {
		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 <= 2.32d-5) 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 <= 2.32e-5) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2.32e-5:
		tmp = R * lambda2
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2.32e-5)
		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 <= 2.32e-5)
		tmp = R * lambda2;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2.32e-5], N[(R * lambda2), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.3200000000000001e-5

   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 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

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

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

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

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

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

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

      5. *-lowering-*.f64 94.4%

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

   7. Simplified 94.4%

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

   8. Taylor expanded in lambda2 around inf

      \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

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

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

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

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

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

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

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

      5. *-lowering-*.f64 22.9%

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

   10. Simplified 22.9%

       \[\leadsto \color{blue}{\left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)} \]

   11. Taylor expanded in phi1 around 0

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

       1. *-lowering-*.f64 19.3%

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

   13. Simplified 19.3%

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

   if 2.3200000000000001e-5 < phi2

   1. Initial program 59.4%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 95.4%

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

   3. Simplified 95.4%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   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 59.1%

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

   7. Simplified 59.1%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 13.9% accurate, 109.7× speedup

Math

\[\begin{array}{l} \\ R \cdot \lambda_2 \end{array} \]

FPCore

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

C

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

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 * lambda2
end function

Java

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

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * lambda2

Julia

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

MATLAB

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * lambda2), $MachinePrecision]

Derivation

1. Initial program 62.6%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
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 97.6%

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

3. Simplified 97.6%

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

5. Taylor expanded in phi2 around 0

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

   1. *-commutative N/A

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

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

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

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

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

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

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

   5. *-lowering-*.f64 92.3%

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

7. Simplified 92.3%

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

8. Taylor expanded in lambda2 around inf

   \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right)} \]
9. Step-by-step derivation

   1. associate-*r* N/A

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

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

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

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

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

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

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

   5. *-lowering-*.f64 19.2%

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

10. Simplified 19.2%

    \[\leadsto \color{blue}{\left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)} \]

11. Taylor expanded in phi1 around 0

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

    1. *-lowering-*.f64 17.3%

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

13. Simplified 17.3%

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

Reproduce

herbie shell --seed 2024141 
(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))))))