Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.4% → 99.4%

Time: 32.1s

Alternatives: 15

Speedup: 3.0×

• 30.5% 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: 59.4% 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: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \left(\sqrt{R\_m} \cdot \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)\right), \phi_1 - \phi_2\right) \cdot \sqrt{R\_m}\right)\right) \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 1 R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (*
   (sqrt R_m)
   (*
    (hypot
     (*
      (- lambda1 lambda2)
      (-
       (* (cos (* phi2 0.5)) (cos (* 0.5 phi1)))
       (expm1 (log1p (* (sin (* phi2 0.5)) (sin (* 0.5 phi1)))))))
     (- phi1 phi2))
    (sqrt R_m)))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (sqrt(R_m) * (hypot(((lambda1 - lambda2) * ((cos((phi2 * 0.5)) * cos((0.5 * phi1))) - expm1(log1p((sin((phi2 * 0.5)) * sin((0.5 * phi1))))))), (phi1 - phi2)) * sqrt(R_m)));
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (Math.sqrt(R_m) * (Math.hypot(((lambda1 - lambda2) * ((Math.cos((phi2 * 0.5)) * Math.cos((0.5 * phi1))) - Math.expm1(Math.log1p((Math.sin((phi2 * 0.5)) * Math.sin((0.5 * phi1))))))), (phi1 - phi2)) * Math.sqrt(R_m)));
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	return R_s * (math.sqrt(R_m) * (math.hypot(((lambda1 - lambda2) * ((math.cos((phi2 * 0.5)) * math.cos((0.5 * phi1))) - math.expm1(math.log1p((math.sin((phi2 * 0.5)) * math.sin((0.5 * phi1))))))), (phi1 - phi2)) * math.sqrt(R_m)))

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	return Float64(R_s * Float64(sqrt(R_m) * Float64(hypot(Float64(Float64(lambda1 - lambda2) * Float64(Float64(cos(Float64(phi2 * 0.5)) * cos(Float64(0.5 * phi1))) - expm1(log1p(Float64(sin(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1))))))), Float64(phi1 - phi2)) * sqrt(R_m))))
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * N[(N[Sqrt[R$95$m], $MachinePrecision] * N[(N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(Exp[N[Log[1 + N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision] * N[Sqrt[R$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.7%

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

   1. hypot-define 97.9%

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

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. Step-by-step derivation

   1. expm1-log1p-u 58.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)\right)\right)} \]

   2. *-commutative 58.1%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1 - \phi_2\right)\right)\right) \]

   3. div-inv 58.1%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\right)\right) \]

   4. metadata-eval 58.1%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\right)\right) \]

6. Applied egg-rr 58.1%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\right)\right)} \]
7. Step-by-step derivation

   1. expm1-log1p-u 97.9%

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

   2. *-commutative 97.9%

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

   3. add-sqr-sqrt 50.5%

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

   4. associate-*r* 50.5%

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

   5. *-commutative 50.5%

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

8. Applied egg-rr 50.5%

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

   1. *-commutative 50.5%

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

   2. +-commutative 50.5%

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

   3. distribute-rgt-in 50.5%

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

   4. *-commutative 50.5%

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

   5. cos-sum 51.7%

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

   6. *-commutative 51.7%

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

   7. *-commutative 51.7%

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

10. Applied egg-rr 51.7%

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

    1. expm1-log1p-u 51.7%

       \[\leadsto \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}\right), \phi_1 - \phi_2\right) \cdot \sqrt{R}\right) \cdot \sqrt{R} \]

    2. expm1-undefine 51.6%

       \[\leadsto \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)} - 1\right)}\right), \phi_1 - \phi_2\right) \cdot \sqrt{R}\right) \cdot \sqrt{R} \]

    3. *-commutative 51.6%

       \[\leadsto \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \left(e^{\mathsf{log1p}\left(\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \color{blue}{\left(0.5 \cdot \phi_1\right)}\right)} - 1\right)\right), \phi_1 - \phi_2\right) \cdot \sqrt{R}\right) \cdot \sqrt{R} \]

12. Applied egg-rr 51.6%

    \[\leadsto \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)} - 1\right)}\right), \phi_1 - \phi_2\right) \cdot \sqrt{R}\right) \cdot \sqrt{R} \]
13. Step-by-step derivation

    1. expm1-define 51.7%

       \[\leadsto \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}\right), \phi_1 - \phi_2\right) \cdot \sqrt{R}\right) \cdot \sqrt{R} \]

14. Simplified 51.7%

    \[\leadsto \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}\right), \phi_1 - \phi_2\right) \cdot \sqrt{R}\right) \cdot \sqrt{R} \]

15. Final simplification 51.7%

    \[\leadsto \sqrt{R} \cdot \left(\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)\right), \phi_1 - \phi_2\right) \cdot \sqrt{R}\right) \]
16. Add Preprocessing

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \left(\sqrt{R\_m} \cdot \left(\sqrt{R\_m} \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right), \phi_1 - \phi_2\right)\right)\right) \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 1 R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (*
   (sqrt R_m)
   (*
    (sqrt R_m)
    (hypot
     (*
      (- lambda1 lambda2)
      (-
       (* (cos (* phi2 0.5)) (cos (* 0.5 phi1)))
       (* (sin (* phi2 0.5)) (sin (* 0.5 phi1)))))
     (- phi1 phi2))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (sqrt(R_m) * (sqrt(R_m) * hypot(((lambda1 - lambda2) * ((cos((phi2 * 0.5)) * cos((0.5 * phi1))) - (sin((phi2 * 0.5)) * sin((0.5 * phi1))))), (phi1 - phi2))));
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (Math.sqrt(R_m) * (Math.sqrt(R_m) * Math.hypot(((lambda1 - lambda2) * ((Math.cos((phi2 * 0.5)) * Math.cos((0.5 * phi1))) - (Math.sin((phi2 * 0.5)) * Math.sin((0.5 * phi1))))), (phi1 - phi2))));
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	return R_s * (math.sqrt(R_m) * (math.sqrt(R_m) * math.hypot(((lambda1 - lambda2) * ((math.cos((phi2 * 0.5)) * math.cos((0.5 * phi1))) - (math.sin((phi2 * 0.5)) * math.sin((0.5 * phi1))))), (phi1 - phi2))))

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	return Float64(R_s * Float64(sqrt(R_m) * Float64(sqrt(R_m) * hypot(Float64(Float64(lambda1 - lambda2) * Float64(Float64(cos(Float64(phi2 * 0.5)) * cos(Float64(0.5 * phi1))) - Float64(sin(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1))))), Float64(phi1 - phi2)))))
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = R_s * (sqrt(R_m) * (sqrt(R_m) * hypot(((lambda1 - lambda2) * ((cos((phi2 * 0.5)) * cos((0.5 * phi1))) - (sin((phi2 * 0.5)) * sin((0.5 * phi1))))), (phi1 - phi2))));
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * N[(N[Sqrt[R$95$m], $MachinePrecision] * N[(N[Sqrt[R$95$m], $MachinePrecision] * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * 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]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.7%

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

   1. hypot-define 97.9%

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

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. Step-by-step derivation

   1. expm1-log1p-u 58.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)\right)\right)} \]

   2. *-commutative 58.1%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1 - \phi_2\right)\right)\right) \]

   3. div-inv 58.1%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\right)\right) \]

   4. metadata-eval 58.1%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\right)\right) \]

6. Applied egg-rr 58.1%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\right)\right)} \]
7. Step-by-step derivation

   1. expm1-log1p-u 97.9%

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

   2. *-commutative 97.9%

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

   3. add-sqr-sqrt 50.5%

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

   4. associate-*r* 50.5%

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

   5. *-commutative 50.5%

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

8. Applied egg-rr 50.5%

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

   1. *-commutative 50.5%

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

   2. +-commutative 50.5%

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

   3. distribute-rgt-in 50.5%

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

   4. *-commutative 50.5%

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

   5. cos-sum 51.7%

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

   6. *-commutative 51.7%

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

   7. *-commutative 51.7%

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

10. Applied egg-rr 51.7%

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

11. Final simplification 51.7%

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

Alternative 3: 93.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.78 \cdot 10^{-6}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 1 R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi2 1.78e-6)
    (* R_m (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi1))) (- phi1 phi2)))
    (* R_m (hypot (* (- lambda1 lambda2) (cos (* phi2 0.5))) (- phi1 phi2))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.78e-6) {
		tmp = R_m * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	} else {
		tmp = R_m * hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return R_s * tmp;
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.78e-6) {
		tmp = R_m * Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi1))), (phi1 - phi2));
	} else {
		tmp = R_m * Math.hypot(((lambda1 - lambda2) * Math.cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.78e-6:
		tmp = R_m * math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi1))), (phi1 - phi2))
	else:
		tmp = R_m * math.hypot(((lambda1 - lambda2) * math.cos((phi2 * 0.5))), (phi1 - phi2))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.78e-6)
		tmp = Float64(R_m * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R_m * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5))), Float64(phi1 - phi2)));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.78e-6)
		tmp = R_m * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	else
		tmp = R_m * hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), (phi1 - phi2));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi2, 1.78e-6], N[(R$95$m * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.78e-6

   1. Initial program 57.5%

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

      1. hypot-define 98.4%

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

   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 94.0%

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

   if 1.78e-6 < phi2

   1. Initial program 66.9%

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

      1. hypot-define 96.4%

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

   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 phi1 around 0 96.4%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.78 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -5 \cdot 10^{-215}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\lambda_2 \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 1 R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda1 -5e-215)
    (* R_m (hypot (- lambda1 lambda2) (- phi1 phi2)))
    (* R_m (hypot (* lambda2 (cos (* phi2 0.5))) (- phi1 phi2))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -5e-215) {
		tmp = R_m * hypot((lambda1 - lambda2), (phi1 - phi2));
	} else {
		tmp = R_m * hypot((lambda2 * cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return R_s * tmp;
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -5e-215) {
		tmp = R_m * Math.hypot((lambda1 - lambda2), (phi1 - phi2));
	} else {
		tmp = R_m * Math.hypot((lambda2 * Math.cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -5e-215:
		tmp = R_m * math.hypot((lambda1 - lambda2), (phi1 - phi2))
	else:
		tmp = R_m * math.hypot((lambda2 * math.cos((phi2 * 0.5))), (phi1 - phi2))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -5e-215)
		tmp = Float64(R_m * hypot(Float64(lambda1 - lambda2), Float64(phi1 - phi2)));
	else
		tmp = Float64(R_m * hypot(Float64(lambda2 * cos(Float64(phi2 * 0.5))), Float64(phi1 - phi2)));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -5e-215)
		tmp = R_m * hypot((lambda1 - lambda2), (phi1 - phi2));
	else
		tmp = R_m * hypot((lambda2 * cos((phi2 * 0.5))), (phi1 - phi2));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda1, -5e-215], N[(R$95$m * N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[Sqrt[N[(lambda2 * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -4.99999999999999956e-215

   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. hypot-define 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in phi1 around 0 91.3%

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

   6. Taylor expanded in phi2 around 0 84.7%

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

   if -4.99999999999999956e-215 < lambda1

   1. Initial program 63.0%

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

      1. hypot-define 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in phi1 around 0 94.4%

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

   6. Taylor expanded in lambda1 around 0 78.9%

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

      1. mul-1-neg 78.9%

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

   8. Simplified 78.9%

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

      1. pow1 78.9%

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

      2. add-sqr-sqrt 42.1%

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

      3. sqrt-unprod 71.5%

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

      4. sqr-neg 71.5%

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

      5. sqrt-unprod 36.7%

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

      6. add-sqr-sqrt 78.9%

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

      7. *-commutative 78.9%

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

   10. Applied egg-rr 78.9%

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

       1. unpow1 78.9%

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

   12. Simplified 78.9%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -5 \cdot 10^{-215}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \left(R\_m \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\right) \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 1 R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (*
   R_m
   (hypot (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0))) (- phi1 phi2)))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (R_m * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2)));
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (R_m * Math.hypot(((lambda1 - lambda2) * Math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2)));
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	return R_s * (R_m * math.hypot(((lambda1 - lambda2) * math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2)))

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	return Float64(R_s * Float64(R_m * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi2 + phi1) / 2.0))), Float64(phi1 - phi2))))
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = R_s * (R_m * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2)));
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * N[(R$95$m * 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]), $MachinePrecision]

Derivation

1. Initial program 59.7%

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

   1. hypot-define 97.9%

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

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. Final simplification 97.9%

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

Alternative 6: 90.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \left(R\_m \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\right) \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 1 R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (* R_m (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi1))) (- phi1 phi2)))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (R_m * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2)));
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (R_m * Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi1))), (phi1 - phi2)));
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	return R_s * (R_m * math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi1))), (phi1 - phi2)))

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	return Float64(R_s * Float64(R_m * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1))), Float64(phi1 - phi2))))
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = R_s * (R_m * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2)));
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * N[(R$95$m * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.7%

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

   1. hypot-define 97.9%

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

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 0 92.8%

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

6. Final simplification 92.8%

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

Alternative 7: 25.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -6.2 \cdot 10^{+205}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\_m\right)\\ \mathbf{elif}\;\lambda_1 \leq 8.6 \cdot 10^{-222}:\\ \;\;\;\;R\_m \cdot \left(\phi_1 \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 1 R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda1 -6.2e+205)
    (* lambda1 (- R_m))
    (if (<= lambda1 8.6e-222)
      (* R_m (* phi1 (+ (/ phi2 phi1) -1.0)))
      (* R_m (* lambda2 (cos (* 0.5 phi1))))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -6.2e+205) {
		tmp = lambda1 * -R_m;
	} else if (lambda1 <= 8.6e-222) {
		tmp = R_m * (phi1 * ((phi2 / phi1) + -1.0));
	} else {
		tmp = R_m * (lambda2 * cos((0.5 * phi1)));
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-6.2d+205)) then
        tmp = lambda1 * -r_m
    else if (lambda1 <= 8.6d-222) then
        tmp = r_m * (phi1 * ((phi2 / phi1) + (-1.0d0)))
    else
        tmp = r_m * (lambda2 * cos((0.5d0 * phi1)))
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -6.2e+205) {
		tmp = lambda1 * -R_m;
	} else if (lambda1 <= 8.6e-222) {
		tmp = R_m * (phi1 * ((phi2 / phi1) + -1.0));
	} else {
		tmp = R_m * (lambda2 * Math.cos((0.5 * phi1)));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -6.2e+205:
		tmp = lambda1 * -R_m
	elif lambda1 <= 8.6e-222:
		tmp = R_m * (phi1 * ((phi2 / phi1) + -1.0))
	else:
		tmp = R_m * (lambda2 * math.cos((0.5 * phi1)))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -6.2e+205)
		tmp = Float64(lambda1 * Float64(-R_m));
	elseif (lambda1 <= 8.6e-222)
		tmp = Float64(R_m * Float64(phi1 * Float64(Float64(phi2 / phi1) + -1.0)));
	else
		tmp = Float64(R_m * Float64(lambda2 * cos(Float64(0.5 * phi1))));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -6.2e+205)
		tmp = lambda1 * -R_m;
	elseif (lambda1 <= 8.6e-222)
		tmp = R_m * (phi1 * ((phi2 / phi1) + -1.0));
	else
		tmp = R_m * (lambda2 * cos((0.5 * phi1)));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda1, -6.2e+205], N[(lambda1 * (-R$95$m)), $MachinePrecision], If[LessEqual[lambda1, 8.6e-222], N[(R$95$m * N[(phi1 * N[(N[(phi2 / phi1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[(lambda2 * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -6.20000000000000035e205

   1. Initial program 49.7%

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

      1. hypot-define 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in phi1 around 0 77.7%

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

   6. Taylor expanded in lambda1 around -inf 52.5%

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\lambda_1 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 52.5%

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

   8. Simplified 52.5%

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

   9. Taylor expanded in phi2 around 0 69.1%

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

   if -6.20000000000000035e205 < lambda1 < 8.59999999999999983e-222

   1. Initial program 63.6%

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

      1. hypot-define 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in phi1 around 0 95.2%

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

   6. Taylor expanded in phi1 around -inf 38.6%

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

      1. mul-1-neg 38.6%

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

      2. mul-1-neg 38.6%

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

   8. Simplified 38.6%

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

   if 8.59999999999999983e-222 < lambda1

   1. Initial program 57.2%

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

      1. hypot-define 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in lambda2 around inf 21.0%

      \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 21.0%

         \[\leadsto \color{blue}{\left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)} \]

      2. +-commutative 21.0%

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \color{blue}{\left(\phi_2 + \phi_1\right)}\right) \]

   7. Simplified 21.0%

      \[\leadsto \color{blue}{\left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)} \]

   8. Taylor expanded in phi2 around 0 21.2%

      \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \phi_1\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -6.2 \cdot 10^{+205}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\lambda_1 \leq 8.6 \cdot 10^{-222}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 27.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.9 \cdot 10^{+206}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\_m\right)\\ \mathbf{elif}\;\lambda_1 \leq 2.3 \cdot 10^{-82}:\\ \;\;\;\;R\_m \cdot \left(\phi_1 \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \left(\lambda_2 \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 1 R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda1 -1.9e+206)
    (* lambda1 (- R_m))
    (if (<= lambda1 2.3e-82)
      (* R_m (* phi1 (+ (/ phi2 phi1) -1.0)))
      (* R_m (* lambda2 (cos (* phi2 0.5))))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.9e+206) {
		tmp = lambda1 * -R_m;
	} else if (lambda1 <= 2.3e-82) {
		tmp = R_m * (phi1 * ((phi2 / phi1) + -1.0));
	} else {
		tmp = R_m * (lambda2 * cos((phi2 * 0.5)));
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-1.9d+206)) then
        tmp = lambda1 * -r_m
    else if (lambda1 <= 2.3d-82) then
        tmp = r_m * (phi1 * ((phi2 / phi1) + (-1.0d0)))
    else
        tmp = r_m * (lambda2 * cos((phi2 * 0.5d0)))
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.9e+206) {
		tmp = lambda1 * -R_m;
	} else if (lambda1 <= 2.3e-82) {
		tmp = R_m * (phi1 * ((phi2 / phi1) + -1.0));
	} else {
		tmp = R_m * (lambda2 * Math.cos((phi2 * 0.5)));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -1.9e+206:
		tmp = lambda1 * -R_m
	elif lambda1 <= 2.3e-82:
		tmp = R_m * (phi1 * ((phi2 / phi1) + -1.0))
	else:
		tmp = R_m * (lambda2 * math.cos((phi2 * 0.5)))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.9e+206)
		tmp = Float64(lambda1 * Float64(-R_m));
	elseif (lambda1 <= 2.3e-82)
		tmp = Float64(R_m * Float64(phi1 * Float64(Float64(phi2 / phi1) + -1.0)));
	else
		tmp = Float64(R_m * Float64(lambda2 * cos(Float64(phi2 * 0.5))));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -1.9e+206)
		tmp = lambda1 * -R_m;
	elseif (lambda1 <= 2.3e-82)
		tmp = R_m * (phi1 * ((phi2 / phi1) + -1.0));
	else
		tmp = R_m * (lambda2 * cos((phi2 * 0.5)));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda1, -1.9e+206], N[(lambda1 * (-R$95$m)), $MachinePrecision], If[LessEqual[lambda1, 2.3e-82], N[(R$95$m * N[(phi1 * N[(N[(phi2 / phi1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[(lambda2 * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -1.8999999999999999e206

   1. Initial program 49.7%

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

      1. hypot-define 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in phi1 around 0 77.7%

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

   6. Taylor expanded in lambda1 around -inf 52.5%

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\lambda_1 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 52.5%

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

   8. Simplified 52.5%

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

   9. Taylor expanded in phi2 around 0 69.1%

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

   if -1.8999999999999999e206 < lambda1 < 2.29999999999999997e-82

   1. Initial program 61.8%

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

      1. hypot-define 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in phi1 around 0 95.6%

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

   6. Taylor expanded in phi1 around -inf 38.3%

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

      1. mul-1-neg 38.3%

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

      2. mul-1-neg 38.3%

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

   8. Simplified 38.3%

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

   if 2.29999999999999997e-82 < lambda1

   1. Initial program 58.3%

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

      1. hypot-define 98.0%

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

   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 lambda2 around inf 17.7%

      \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 17.7%

         \[\leadsto \color{blue}{\left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)} \]

      2. +-commutative 17.7%

         \[\leadsto \left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \color{blue}{\left(\phi_2 + \phi_1\right)}\right) \]

   7. Simplified 17.7%

      \[\leadsto \color{blue}{\left(R \cdot \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)} \]

   8. Taylor expanded in phi1 around 0 15.7%

      \[\leadsto \color{blue}{R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.9 \cdot 10^{+206}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\lambda_1 \leq 2.3 \cdot 10^{-82}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -6 \cdot 10^{+206}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\_m\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(-\lambda_2, \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 1 R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda1 -6e+206)
    (* lambda1 (- R_m))
    (* R_m (hypot (- lambda2) (- phi1 phi2))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -6e+206) {
		tmp = lambda1 * -R_m;
	} else {
		tmp = R_m * hypot(-lambda2, (phi1 - phi2));
	}
	return R_s * tmp;
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -6e+206) {
		tmp = lambda1 * -R_m;
	} else {
		tmp = R_m * Math.hypot(-lambda2, (phi1 - phi2));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -6e+206:
		tmp = lambda1 * -R_m
	else:
		tmp = R_m * math.hypot(-lambda2, (phi1 - phi2))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -6e+206)
		tmp = Float64(lambda1 * Float64(-R_m));
	else
		tmp = Float64(R_m * hypot(Float64(-lambda2), Float64(phi1 - phi2)));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -6e+206)
		tmp = lambda1 * -R_m;
	else
		tmp = R_m * hypot(-lambda2, (phi1 - phi2));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda1, -6e+206], N[(lambda1 * (-R$95$m)), $MachinePrecision], N[(R$95$m * N[Sqrt[(-lambda2) ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -6.0000000000000002e206

   1. Initial program 47.2%

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

      1. hypot-define 92.0%

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

   3. Simplified 92.0%

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

   5. Taylor expanded in phi1 around 0 76.6%

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

   6. Taylor expanded in lambda1 around -inf 50.1%

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\lambda_1 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 50.1%

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

   8. Simplified 50.1%

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

   9. Taylor expanded in phi2 around 0 67.6%

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

   if -6.0000000000000002e206 < lambda1

   1. Initial program 60.7%

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

      1. hypot-define 98.4%

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

   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 phi1 around 0 94.5%

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

   6. Taylor expanded in lambda1 around 0 80.1%

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

      1. mul-1-neg 80.1%

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

   8. Simplified 80.1%

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

   9. Taylor expanded in phi2 around 0 77.5%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -6 \cdot 10^{+206}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(-\lambda_2, \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 85.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \left(R\_m \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\right) \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 1 R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R_s (* R_m (hypot (- lambda1 lambda2) (- phi1 phi2)))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (R_m * hypot((lambda1 - lambda2), (phi1 - phi2)));
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (R_m * Math.hypot((lambda1 - lambda2), (phi1 - phi2)));
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	return R_s * (R_m * math.hypot((lambda1 - lambda2), (phi1 - phi2)))

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	return Float64(R_s * Float64(R_m * hypot(Float64(lambda1 - lambda2), Float64(phi1 - phi2))))
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = R_s * (R_m * hypot((lambda1 - lambda2), (phi1 - phi2)));
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * N[(R$95$m * N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.7%

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

   1. hypot-define 97.9%

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

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 phi1 around 0 93.1%

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

6. Taylor expanded in phi2 around 0 87.9%

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

7. Final simplification 87.9%

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

Alternative 11: 31.3% accurate, 23.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -6 \cdot 10^{+206}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R\_m - R\_m \cdot \frac{\phi_1}{\phi_2}\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 1 R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda1 -6e+206)
    (* lambda1 (- R_m))
    (* phi2 (- R_m (* R_m (/ phi1 phi2)))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -6e+206) {
		tmp = lambda1 * -R_m;
	} else {
		tmp = phi2 * (R_m - (R_m * (phi1 / phi2)));
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-6d+206)) then
        tmp = lambda1 * -r_m
    else
        tmp = phi2 * (r_m - (r_m * (phi1 / phi2)))
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -6e+206) {
		tmp = lambda1 * -R_m;
	} else {
		tmp = phi2 * (R_m - (R_m * (phi1 / phi2)));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -6e+206:
		tmp = lambda1 * -R_m
	else:
		tmp = phi2 * (R_m - (R_m * (phi1 / phi2)))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -6e+206)
		tmp = Float64(lambda1 * Float64(-R_m));
	else
		tmp = Float64(phi2 * Float64(R_m - Float64(R_m * Float64(phi1 / phi2))));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -6e+206)
		tmp = lambda1 * -R_m;
	else
		tmp = phi2 * (R_m - (R_m * (phi1 / phi2)));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda1, -6e+206], N[(lambda1 * (-R$95$m)), $MachinePrecision], N[(phi2 * N[(R$95$m - N[(R$95$m * N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -6.0000000000000002e206

   1. Initial program 47.2%

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

      1. hypot-define 92.0%

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

   3. Simplified 92.0%

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

   5. Taylor expanded in phi1 around 0 76.6%

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

   6. Taylor expanded in lambda1 around -inf 50.1%

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\lambda_1 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 50.1%

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

   8. Simplified 50.1%

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

   9. Taylor expanded in phi2 around 0 67.6%

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

   if -6.0000000000000002e206 < lambda1

   1. Initial program 60.7%

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

      1. hypot-define 98.4%

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

   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 inf 34.8%

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

      1. mul-1-neg 34.8%

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

      2. unsub-neg 34.8%

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

      3. associate-/l* 32.3%

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

   7. Simplified 32.3%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -6 \cdot 10^{+206}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R - R \cdot \frac{\phi_1}{\phi_2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 30.4% accurate, 23.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -6.2 \cdot 10^{+205}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\_m\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \left(\phi_1 \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 1 R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda1 -6.2e+205)
    (* lambda1 (- R_m))
    (* R_m (* phi1 (+ (/ phi2 phi1) -1.0))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -6.2e+205) {
		tmp = lambda1 * -R_m;
	} else {
		tmp = R_m * (phi1 * ((phi2 / phi1) + -1.0));
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-6.2d+205)) then
        tmp = lambda1 * -r_m
    else
        tmp = r_m * (phi1 * ((phi2 / phi1) + (-1.0d0)))
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -6.2e+205) {
		tmp = lambda1 * -R_m;
	} else {
		tmp = R_m * (phi1 * ((phi2 / phi1) + -1.0));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -6.2e+205:
		tmp = lambda1 * -R_m
	else:
		tmp = R_m * (phi1 * ((phi2 / phi1) + -1.0))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -6.2e+205)
		tmp = Float64(lambda1 * Float64(-R_m));
	else
		tmp = Float64(R_m * Float64(phi1 * Float64(Float64(phi2 / phi1) + -1.0)));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -6.2e+205)
		tmp = lambda1 * -R_m;
	else
		tmp = R_m * (phi1 * ((phi2 / phi1) + -1.0));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda1, -6.2e+205], N[(lambda1 * (-R$95$m)), $MachinePrecision], N[(R$95$m * N[(phi1 * N[(N[(phi2 / phi1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -6.20000000000000035e205

   1. Initial program 49.7%

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

      1. hypot-define 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in phi1 around 0 77.7%

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

   6. Taylor expanded in lambda1 around -inf 52.5%

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\lambda_1 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 52.5%

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

   8. Simplified 52.5%

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

   9. Taylor expanded in phi2 around 0 69.1%

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

   if -6.20000000000000035e205 < lambda1

   1. Initial program 60.6%

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

      1. hypot-define 98.4%

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

   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 phi1 around 0 94.5%

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

   6. Taylor expanded in phi1 around -inf 34.1%

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

      1. mul-1-neg 34.1%

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

      2. mul-1-neg 34.1%

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

   8. Simplified 34.1%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -6.2 \cdot 10^{+205}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 26.5% accurate, 36.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.8 \cdot 10^{-50}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\_m\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 1 R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R_s (if (<= phi2 3.8e-50) (* lambda1 (- R_m)) (* phi2 R_m))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.8e-50) {
		tmp = lambda1 * -R_m;
	} else {
		tmp = phi2 * R_m;
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 3.8d-50) then
        tmp = lambda1 * -r_m
    else
        tmp = phi2 * r_m
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.8e-50) {
		tmp = lambda1 * -R_m;
	} else {
		tmp = phi2 * R_m;
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 3.8e-50:
		tmp = lambda1 * -R_m
	else:
		tmp = phi2 * R_m
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 3.8e-50)
		tmp = Float64(lambda1 * Float64(-R_m));
	else
		tmp = Float64(phi2 * R_m);
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 3.8e-50)
		tmp = lambda1 * -R_m;
	else
		tmp = phi2 * R_m;
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi2, 3.8e-50], N[(lambda1 * (-R$95$m)), $MachinePrecision], N[(phi2 * R$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.7999999999999999e-50

   1. Initial program 56.5%

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

      1. hypot-define 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in phi1 around 0 92.6%

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

   6. Taylor expanded in lambda1 around -inf 16.6%

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\lambda_1 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 16.6%

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

   8. Simplified 16.6%

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

   9. Taylor expanded in phi2 around 0 16.1%

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

   if 3.7999999999999999e-50 < phi2

   1. Initial program 68.6%

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

      1. hypot-define 96.1%

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

   3. Simplified 96.1%

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

   5. Taylor expanded in phi2 around inf 57.6%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   6. Step-by-step derivation

      1. *-commutative 57.6%

         \[\leadsto \color{blue}{\phi_2 \cdot R} \]

   7. Simplified 57.6%

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

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.8 \cdot 10^{-50}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 28.4% accurate, 36.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -27:\\ \;\;\;\;\phi_1 \cdot \left(-R\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\_m\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 1 R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R_s (if (<= phi1 -27.0) (* phi1 (- R_m)) (* phi2 R_m))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -27.0) {
		tmp = phi1 * -R_m;
	} else {
		tmp = phi2 * R_m;
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-27.0d0)) then
        tmp = phi1 * -r_m
    else
        tmp = phi2 * r_m
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -27.0) {
		tmp = phi1 * -R_m;
	} else {
		tmp = phi2 * R_m;
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -27.0:
		tmp = phi1 * -R_m
	else:
		tmp = phi2 * R_m
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -27.0)
		tmp = Float64(phi1 * Float64(-R_m));
	else
		tmp = Float64(phi2 * R_m);
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -27.0)
		tmp = phi1 * -R_m;
	else
		tmp = phi2 * R_m;
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi1, -27.0], N[(phi1 * (-R$95$m)), $MachinePrecision], N[(phi2 * R$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi1 < -27

   1. Initial program 55.6%

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

      1. hypot-define 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in phi1 around -inf 71.3%

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

      1. mul-1-neg 71.3%

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

      2. *-commutative 71.3%

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

      3. distribute-lft-neg-in 71.3%

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

   7. Simplified 71.3%

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

   if -27 < phi1

   1. Initial program 61.1%

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

      1. hypot-define 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in phi2 around inf 21.5%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   6. Step-by-step derivation

      1. *-commutative 21.5%

         \[\leadsto \color{blue}{\phi_2 \cdot R} \]

   7. Simplified 21.5%

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

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -27:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 17.7% accurate, 109.7× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \left(\phi_2 \cdot R\_m\right) \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 1 R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R_s (* phi2 R_m)))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (phi2 * R_m);
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r_s * (phi2 * r_m)
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (phi2 * R_m);
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	return R_s * (phi2 * R_m)

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	return Float64(R_s * Float64(phi2 * R_m))
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = R_s * (phi2 * R_m);
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * N[(phi2 * R$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 59.7%

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

   1. hypot-define 97.9%

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

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 19.0%

   \[\leadsto \color{blue}{R \cdot \phi_2} \]
6. Step-by-step derivation

   1. *-commutative 19.0%

      \[\leadsto \color{blue}{\phi_2 \cdot R} \]

7. Simplified 19.0%

   \[\leadsto \color{blue}{\phi_2 \cdot R} \]

8. Final simplification 19.0%

   \[\leadsto \phi_2 \cdot R \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024053 
(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))))))