Equirectangular approximation to distance on a great circle

Percentage Accurate: 60.5% → 99.4%

Time: 41.6s

Alternatives: 15

Speedup: 1.6×

• 30.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 60.5% 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, 0.5× speedup

Math

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

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (*
   (sqrt R_m)
   (*
    (hypot
     (*
      (- lambda1 lambda2)
      (-
       (* (cos (* 0.5 phi1)) (cos (* phi2 0.5)))
       (* (sin (* 0.5 phi1)) (sin (* phi2 0.5)))))
     (- 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((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((0.5 * phi1)) * sin((phi2 * 0.5))))), (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((0.5 * phi1)) * Math.cos((phi2 * 0.5))) - (Math.sin((0.5 * phi1)) * Math.sin((phi2 * 0.5))))), (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((0.5 * phi1)) * math.cos((phi2 * 0.5))) - (math.sin((0.5 * phi1)) * math.sin((phi2 * 0.5))))), (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(0.5 * phi1)) * cos(Float64(phi2 * 0.5))) - Float64(sin(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5))))), Float64(phi1 - phi2)) * sqrt(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 * (sqrt(R_m) * (hypot(((lambda1 - lambda2) * ((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((0.5 * phi1)) * sin((phi2 * 0.5))))), (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[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision] * N[Sqrt[R$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 95.7%

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

3. Simplified 95.7%

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

   1. add-sqr-sqrt 45.1%

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

   2. pow2 45.1%

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

   3. *-commutative 45.1%

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

   4. div-inv 45.1%

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

   5. metadata-eval 45.1%

      \[\leadsto {\left(\sqrt{R \cdot \mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)}^{2} \]

6. Applied egg-rr 45.1%

   \[\leadsto \color{blue}{{\left(\sqrt{R \cdot \mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)}^{2}} \]
7. Step-by-step derivation

   1. unpow2 45.1%

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

   2. add-sqr-sqrt 95.7%

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

   3. *-commutative 95.7%

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

   4. add-sqr-sqrt 45.1%

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

   5. associate-*r* 45.1%

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

   6. *-commutative 45.1%

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

8. Applied egg-rr 45.1%

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

   1. *-commutative 45.1%

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

   2. +-commutative 45.1%

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

   3. distribute-rgt-in 45.1%

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

   4. cos-sum 47.4%

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

10. Applied egg-rr 47.4%

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

11. Final simplification 47.4%

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

Alternative 2: 87.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\\ t_1 := \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -5.8 \cdot 10^{+174}:\\ \;\;\;\;{\left(\sqrt{R\_m \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(t\_0 - t\_1\right), \phi_1 - \phi_2\right)}\right)}^{2}\\ \mathbf{elif}\;\lambda_1 \leq -2.4 \cdot 10^{-254}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{R\_m \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(t\_1 - t\_0\right), \phi_1 - \phi_2\right)}\right)}^{2}\\ \end{array} \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos (* 0.5 phi1)) (cos (* phi2 0.5))))
        (t_1 (* (sin (* 0.5 phi1)) (sin (* phi2 0.5)))))
   (*
    R_s
    (if (<= lambda1 -5.8e+174)
      (pow (sqrt (* R_m (hypot (* lambda1 (- t_0 t_1)) (- phi1 phi2)))) 2.0)
      (if (<= lambda1 -2.4e-254)
        (*
         R_m
         (hypot
          (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0)))
          (- phi1 phi2)))
        (pow
         (sqrt (* R_m (hypot (* lambda2 (- t_1 t_0)) (- phi1 phi2))))
         2.0))))))

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 t_0 = cos((0.5 * phi1)) * cos((phi2 * 0.5));
	double t_1 = sin((0.5 * phi1)) * sin((phi2 * 0.5));
	double tmp;
	if (lambda1 <= -5.8e+174) {
		tmp = pow(sqrt((R_m * hypot((lambda1 * (t_0 - t_1)), (phi1 - phi2)))), 2.0);
	} else if (lambda1 <= -2.4e-254) {
		tmp = R_m * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	} else {
		tmp = pow(sqrt((R_m * hypot((lambda2 * (t_1 - t_0)), (phi1 - phi2)))), 2.0);
	}
	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 t_0 = Math.cos((0.5 * phi1)) * Math.cos((phi2 * 0.5));
	double t_1 = Math.sin((0.5 * phi1)) * Math.sin((phi2 * 0.5));
	double tmp;
	if (lambda1 <= -5.8e+174) {
		tmp = Math.pow(Math.sqrt((R_m * Math.hypot((lambda1 * (t_0 - t_1)), (phi1 - phi2)))), 2.0);
	} else if (lambda1 <= -2.4e-254) {
		tmp = R_m * Math.hypot(((lambda1 - lambda2) * Math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	} else {
		tmp = Math.pow(Math.sqrt((R_m * Math.hypot((lambda2 * (t_1 - t_0)), (phi1 - phi2)))), 2.0);
	}
	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):
	t_0 = math.cos((0.5 * phi1)) * math.cos((phi2 * 0.5))
	t_1 = math.sin((0.5 * phi1)) * math.sin((phi2 * 0.5))
	tmp = 0
	if lambda1 <= -5.8e+174:
		tmp = math.pow(math.sqrt((R_m * math.hypot((lambda1 * (t_0 - t_1)), (phi1 - phi2)))), 2.0)
	elif lambda1 <= -2.4e-254:
		tmp = R_m * math.hypot(((lambda1 - lambda2) * math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2))
	else:
		tmp = math.pow(math.sqrt((R_m * math.hypot((lambda2 * (t_1 - t_0)), (phi1 - phi2)))), 2.0)
	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)
	t_0 = Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5)))
	t_1 = Float64(sin(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5)))
	tmp = 0.0
	if (lambda1 <= -5.8e+174)
		tmp = sqrt(Float64(R_m * hypot(Float64(lambda1 * Float64(t_0 - t_1)), Float64(phi1 - phi2)))) ^ 2.0;
	elseif (lambda1 <= -2.4e-254)
		tmp = Float64(R_m * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi2 + phi1) / 2.0))), Float64(phi1 - phi2)));
	else
		tmp = sqrt(Float64(R_m * hypot(Float64(lambda2 * Float64(t_1 - t_0)), Float64(phi1 - phi2)))) ^ 2.0;
	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)
	t_0 = cos((0.5 * phi1)) * cos((phi2 * 0.5));
	t_1 = sin((0.5 * phi1)) * sin((phi2 * 0.5));
	tmp = 0.0;
	if (lambda1 <= -5.8e+174)
		tmp = sqrt((R_m * hypot((lambda1 * (t_0 - t_1)), (phi1 - phi2)))) ^ 2.0;
	elseif (lambda1 <= -2.4e-254)
		tmp = R_m * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	else
		tmp = sqrt((R_m * hypot((lambda2 * (t_1 - t_0)), (phi1 - phi2)))) ^ 2.0;
	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_] := Block[{t$95$0 = N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[lambda1, -5.8e+174], N[Power[N[Sqrt[N[(R$95$m * N[Sqrt[N[(lambda1 * N[(t$95$0 - t$95$1), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], If[LessEqual[lambda1, -2.4e-254], 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], N[Power[N[Sqrt[N[(R$95$m * N[Sqrt[N[(lambda2 * N[(t$95$1 - t$95$0), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -5.7999999999999999e174

   1. Initial program 38.7%

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

      1. hypot-define 90.6%

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

   3. Simplified 90.6%

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

      1. add-sqr-sqrt 57.1%

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

      2. pow2 57.1%

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

      3. *-commutative 57.1%

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

      4. div-inv 57.1%

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

      5. metadata-eval 57.1%

         \[\leadsto {\left(\sqrt{R \cdot \mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)}^{2} \]

   6. Applied egg-rr 57.1%

      \[\leadsto \color{blue}{{\left(\sqrt{R \cdot \mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative 57.1%

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

      2. +-commutative 57.1%

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

      3. distribute-rgt-in 57.1%

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

      4. cos-sum 61.1%

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

   8. Applied egg-rr 61.1%

      \[\leadsto {\left(\sqrt{R \cdot \mathsf{hypot}\left(\color{blue}{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)}^{2} \]

   9. Taylor expanded in lambda1 around inf 61.1%

      \[\leadsto {\left(\sqrt{R \cdot \mathsf{hypot}\left(\color{blue}{\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}, \phi_1 - \phi_2\right)}\right)}^{2} \]

   if -5.7999999999999999e174 < lambda1 < -2.40000000000000002e-254

   1. Initial program 60.2%

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

      1. hypot-define 99.0%

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

   3. Simplified 99.0%

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

   if -2.40000000000000002e-254 < lambda1

   1. Initial program 58.2%

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

      1. hypot-define 94.8%

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

   3. Simplified 94.8%

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

      1. add-sqr-sqrt 42.4%

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

      2. pow2 42.4%

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

      3. *-commutative 42.4%

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

      4. div-inv 42.4%

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

      5. metadata-eval 42.4%

         \[\leadsto {\left(\sqrt{R \cdot \mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)}^{2} \]

   6. Applied egg-rr 42.4%

      \[\leadsto \color{blue}{{\left(\sqrt{R \cdot \mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative 42.4%

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

      2. +-commutative 42.4%

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

      3. distribute-rgt-in 42.4%

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

      4. cos-sum 45.8%

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

   8. Applied egg-rr 45.8%

      \[\leadsto {\left(\sqrt{R \cdot \mathsf{hypot}\left(\color{blue}{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)}^{2} \]

   9. Taylor expanded in lambda1 around 0 39.8%

      \[\leadsto {\left(\sqrt{R \cdot \mathsf{hypot}\left(\color{blue}{-1 \cdot \left(\lambda_2 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)\right)}, \phi_1 - \phi_2\right)}\right)}^{2} \]
   10. Step-by-step derivation

       1. mul-1-neg 39.8%

          \[\leadsto {\left(\sqrt{R \cdot \mathsf{hypot}\left(\color{blue}{-\lambda_2 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}, \phi_1 - \phi_2\right)}\right)}^{2} \]

   11. Simplified 39.8%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -5.8 \cdot 10^{+174}:\\ \;\;\;\;{\left(\sqrt{R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)}\right)}^{2}\\ \mathbf{elif}\;\lambda_1 \leq -2.4 \cdot 10^{-254}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.95 \cdot 10^{+172}:\\ \;\;\;\;{\left(\sqrt{R\_m \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda1 -1.95e+172)
    (pow
     (sqrt
      (*
       R_m
       (hypot
        (*
         lambda1
         (-
          (* (cos (* 0.5 phi1)) (cos (* phi2 0.5)))
          (* (sin (* 0.5 phi1)) (sin (* phi2 0.5)))))
        (- phi1 phi2))))
     2.0)
    (*
     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) {
	double tmp;
	if (lambda1 <= -1.95e+172) {
		tmp = pow(sqrt((R_m * hypot((lambda1 * ((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((0.5 * phi1)) * sin((phi2 * 0.5))))), (phi1 - phi2)))), 2.0);
	} else {
		tmp = R_m * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (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 <= -1.95e+172) {
		tmp = Math.pow(Math.sqrt((R_m * Math.hypot((lambda1 * ((Math.cos((0.5 * phi1)) * Math.cos((phi2 * 0.5))) - (Math.sin((0.5 * phi1)) * Math.sin((phi2 * 0.5))))), (phi1 - phi2)))), 2.0);
	} else {
		tmp = R_m * Math.hypot(((lambda1 - lambda2) * Math.cos(((phi2 + phi1) / 2.0))), (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 <= -1.95e+172:
		tmp = math.pow(math.sqrt((R_m * math.hypot((lambda1 * ((math.cos((0.5 * phi1)) * math.cos((phi2 * 0.5))) - (math.sin((0.5 * phi1)) * math.sin((phi2 * 0.5))))), (phi1 - phi2)))), 2.0)
	else:
		tmp = R_m * math.hypot(((lambda1 - lambda2) * math.cos(((phi2 + phi1) / 2.0))), (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 <= -1.95e+172)
		tmp = sqrt(Float64(R_m * hypot(Float64(lambda1 * Float64(Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5))) - Float64(sin(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5))))), Float64(phi1 - phi2)))) ^ 2.0;
	else
		tmp = Float64(R_m * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi2 + phi1) / 2.0))), 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 <= -1.95e+172)
		tmp = sqrt((R_m * hypot((lambda1 * ((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((0.5 * phi1)) * sin((phi2 * 0.5))))), (phi1 - phi2)))) ^ 2.0;
	else
		tmp = R_m * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (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, -1.95e+172], N[Power[N[Sqrt[N[(R$95$m * N[Sqrt[N[(lambda1 * N[(N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision], 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. Split input into 2 regimes

2. if lambda1 < -1.94999999999999984e172

   1. Initial program 38.7%

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

      1. hypot-define 90.6%

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

   3. Simplified 90.6%

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

      1. add-sqr-sqrt 57.1%

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

      2. pow2 57.1%

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

      3. *-commutative 57.1%

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

      4. div-inv 57.1%

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

      5. metadata-eval 57.1%

         \[\leadsto {\left(\sqrt{R \cdot \mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)}^{2} \]

   6. Applied egg-rr 57.1%

      \[\leadsto \color{blue}{{\left(\sqrt{R \cdot \mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative 57.1%

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

      2. +-commutative 57.1%

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

      3. distribute-rgt-in 57.1%

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

      4. cos-sum 61.1%

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

   8. Applied egg-rr 61.1%

      \[\leadsto {\left(\sqrt{R \cdot \mathsf{hypot}\left(\color{blue}{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)}^{2} \]

   9. Taylor expanded in lambda1 around inf 61.1%

      \[\leadsto {\left(\sqrt{R \cdot \mathsf{hypot}\left(\color{blue}{\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}, \phi_1 - \phi_2\right)}\right)}^{2} \]

   if -1.94999999999999984e172 < lambda1

   1. Initial program 59.0%

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

      1. 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
3. Recombined 2 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.95 \cdot 10^{+172}:\\ \;\;\;\;{\left(\sqrt{R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \left(R\_m \cdot {\left(\sqrt[3]{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)}\right)}^{3}\right) \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (*
   R_m
   (pow
    (cbrt
     (hypot
      (*
       (- lambda1 lambda2)
       (-
        (* (cos (* 0.5 phi1)) (cos (* phi2 0.5)))
        (* (sin (* 0.5 phi1)) (sin (* phi2 0.5)))))
      (- phi1 phi2)))
    3.0))))

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) {
	return R_s * (R_m * pow(cbrt(hypot(((lambda1 - lambda2) * ((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((0.5 * phi1)) * sin((phi2 * 0.5))))), (phi1 - phi2))), 3.0));
}

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.pow(Math.cbrt(Math.hypot(((lambda1 - lambda2) * ((Math.cos((0.5 * phi1)) * Math.cos((phi2 * 0.5))) - (Math.sin((0.5 * phi1)) * Math.sin((phi2 * 0.5))))), (phi1 - phi2))), 3.0));
}

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 * (cbrt(hypot(Float64(Float64(lambda1 - lambda2) * Float64(Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5))) - Float64(sin(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5))))), Float64(phi1 - phi2))) ^ 3.0)))
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[Power[N[Power[N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 95.7%

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

3. Simplified 95.7%

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

   1. expm1-log1p-u 89.8%

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

   2. *-commutative 89.8%

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

   3. div-inv 89.8%

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

   4. metadata-eval 89.8%

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

6. Applied egg-rr 89.8%

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

   1. expm1-log1p-u 95.7%

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

   2. add-cube-cbrt 94.5%

      \[\leadsto R \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)} \cdot \sqrt[3]{\mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right) \cdot \sqrt[3]{\mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)} \]

   3. pow3 94.5%

      \[\leadsto R \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)}^{3}} \]

   4. *-commutative 94.5%

      \[\leadsto R \cdot {\left(\sqrt[3]{\mathsf{hypot}\left(\color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}, \phi_1 - \phi_2\right)}\right)}^{3} \]

8. Applied egg-rr 94.5%

   \[\leadsto R \cdot \color{blue}{{\left(\sqrt[3]{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right), \phi_1 - \phi_2\right)}\right)}^{3}} \]
9. Step-by-step derivation

   1. *-commutative 45.1%

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

   2. +-commutative 45.1%

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

   3. distribute-rgt-in 45.1%

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

   4. cos-sum 47.4%

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

10. Applied egg-rr 98.7%

    \[\leadsto R \cdot {\left(\sqrt[3]{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}, \phi_1 - \phi_2\right)}\right)}^{3} \]

11. Final simplification 98.7%

    \[\leadsto R \cdot {\left(\sqrt[3]{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)}\right)}^{3} \]
12. Add Preprocessing

Alternative 5: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right) \cdot R\_m}\right)}^{2} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (pow
   (sqrt
    (*
     (hypot
      (*
       (- lambda1 lambda2)
       (-
        (* (cos (* 0.5 phi1)) (cos (* phi2 0.5)))
        (* (sin (* 0.5 phi1)) (sin (* phi2 0.5)))))
      (- phi1 phi2))
     R_m))
   2.0)))

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) {
	return R_s * pow(sqrt((hypot(((lambda1 - lambda2) * ((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((0.5 * phi1)) * sin((phi2 * 0.5))))), (phi1 - phi2)) * R_m)), 2.0);
}

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.pow(Math.sqrt((Math.hypot(((lambda1 - lambda2) * ((Math.cos((0.5 * phi1)) * Math.cos((phi2 * 0.5))) - (Math.sin((0.5 * phi1)) * Math.sin((phi2 * 0.5))))), (phi1 - phi2)) * R_m)), 2.0);
}

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.pow(math.sqrt((math.hypot(((lambda1 - lambda2) * ((math.cos((0.5 * phi1)) * math.cos((phi2 * 0.5))) - (math.sin((0.5 * phi1)) * math.sin((phi2 * 0.5))))), (phi1 - phi2)) * R_m)), 2.0)

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 * (sqrt(Float64(hypot(Float64(Float64(lambda1 - lambda2) * Float64(Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5))) - Float64(sin(Float64(0.5 * phi1)) * sin(Float64(phi2 * 0.5))))), Float64(phi1 - phi2)) * R_m)) ^ 2.0))
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((hypot(((lambda1 - lambda2) * ((cos((0.5 * phi1)) * cos((phi2 * 0.5))) - (sin((0.5 * phi1)) * sin((phi2 * 0.5))))), (phi1 - phi2)) * R_m)) ^ 2.0);
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[Power[N[Sqrt[N[(N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision] * R$95$m), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

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 95.7%

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

3. Simplified 95.7%

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

   1. add-sqr-sqrt 45.1%

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

   2. pow2 45.1%

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

   3. *-commutative 45.1%

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

   4. div-inv 45.1%

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

   5. metadata-eval 45.1%

      \[\leadsto {\left(\sqrt{R \cdot \mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)}^{2} \]

6. Applied egg-rr 45.1%

   \[\leadsto \color{blue}{{\left(\sqrt{R \cdot \mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)}^{2}} \]
7. Step-by-step derivation

   1. *-commutative 45.1%

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

   2. +-commutative 45.1%

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

   3. distribute-rgt-in 45.1%

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

   4. cos-sum 47.4%

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

8. Applied egg-rr 47.4%

   \[\leadsto {\left(\sqrt{R \cdot \mathsf{hypot}\left(\color{blue}{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)}\right)}^{2} \]

9. Final simplification 47.4%

   \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right) \cdot R}\right)}^{2} \]
10. Add Preprocessing

Alternative 6: 92.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.7 \cdot 10^{-38}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi2 4.7e-38)
    (* 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 <= 4.7e-38) {
		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 <= 4.7e-38) {
		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 <= 4.7e-38:
		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 <= 4.7e-38)
		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 <= 4.7e-38)
		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, 4.7e-38], 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 < 4.69999999999999998e-38

   1. Initial program 58.4%

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

      1. 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 phi2 around 0 94.4%

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

   if 4.69999999999999998e-38 < phi2

   1. Initial program 51.9%

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

      1. hypot-define 89.0%

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

   3. Simplified 89.0%

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

   5. Taylor expanded in phi1 around 0 88.4%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.7 \cdot 10^{-38}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 9.2 \cdot 10^{+144}:\\ \;\;\;\;\left|R\_m \cdot \left(\phi_2 + \phi_1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\lambda_2 \cdot \left(R\_m \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda2 9.2e+144)
    (fabs (* R_m (+ phi2 phi1)))
    (fabs (* lambda2 (* R_m (cos (* 0.5 (+ phi2 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 (lambda2 <= 9.2e+144) {
		tmp = fabs((R_m * (phi2 + phi1)));
	} else {
		tmp = fabs((lambda2 * (R_m * cos((0.5 * (phi2 + 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 (lambda2 <= 9.2d+144) then
        tmp = abs((r_m * (phi2 + phi1)))
    else
        tmp = abs((lambda2 * (r_m * cos((0.5d0 * (phi2 + 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 (lambda2 <= 9.2e+144) {
		tmp = Math.abs((R_m * (phi2 + phi1)));
	} else {
		tmp = Math.abs((lambda2 * (R_m * Math.cos((0.5 * (phi2 + 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 lambda2 <= 9.2e+144:
		tmp = math.fabs((R_m * (phi2 + phi1)))
	else:
		tmp = math.fabs((lambda2 * (R_m * math.cos((0.5 * (phi2 + 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 (lambda2 <= 9.2e+144)
		tmp = abs(Float64(R_m * Float64(phi2 + phi1)));
	else
		tmp = abs(Float64(lambda2 * Float64(R_m * cos(Float64(0.5 * Float64(phi2 + 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 (lambda2 <= 9.2e+144)
		tmp = abs((R_m * (phi2 + phi1)));
	else
		tmp = abs((lambda2 * (R_m * cos((0.5 * (phi2 + 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[lambda2, 9.2e+144], N[Abs[N[(R$95$m * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(lambda2 * N[(R$95$m * N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 9.2000000000000006e144

   1. Initial program 61.2%

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

      1. hypot-define 96.7%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in phi2 around inf 30.1%

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

      1. associate-*r/ 30.1%

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

      2. mul-1-neg 30.1%

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

      3. *-commutative 30.1%

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

   7. Simplified 30.1%

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

      1. +-commutative 30.1%

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

      2. distribute-lft-in 29.1%

         \[\leadsto \color{blue}{\phi_2 \cdot \frac{-\phi_1 \cdot R}{\phi_2} + \phi_2 \cdot R} \]

      3. add-sqr-sqrt 14.9%

         \[\leadsto \phi_2 \cdot \frac{\color{blue}{\sqrt{-\phi_1 \cdot R} \cdot \sqrt{-\phi_1 \cdot R}}}{\phi_2} + \phi_2 \cdot R \]

      4. sqrt-unprod 23.1%

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

      5. sqr-neg 23.1%

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

      6. sqrt-unprod 13.4%

         \[\leadsto \phi_2 \cdot \frac{\color{blue}{\sqrt{\phi_1 \cdot R} \cdot \sqrt{\phi_1 \cdot R}}}{\phi_2} + \phi_2 \cdot R \]

      7. add-sqr-sqrt 28.2%

         \[\leadsto \phi_2 \cdot \frac{\color{blue}{\phi_1 \cdot R}}{\phi_2} + \phi_2 \cdot R \]

   9. Applied egg-rr 28.2%

      \[\leadsto \color{blue}{\phi_2 \cdot \frac{\phi_1 \cdot R}{\phi_2} + \phi_2 \cdot R} \]

   10. Taylor expanded in phi2 around 0 27.9%

       \[\leadsto \color{blue}{R \cdot \phi_1 + R \cdot \phi_2} \]
   11. Step-by-step derivation

       1. distribute-lft-in 28.8%

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

   12. Simplified 28.8%

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

       1. add-sqr-sqrt 12.3%

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

       2. sqrt-unprod 24.7%

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

       3. pow2 24.7%

          \[\leadsto \sqrt{\color{blue}{{\left(R \cdot \left(\phi_1 + \phi_2\right)\right)}^{2}}} \]

       4. *-commutative 24.7%

          \[\leadsto \sqrt{{\color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot R\right)}}^{2}} \]

   14. Applied egg-rr 24.7%

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

       1. unpow2 24.7%

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

       2. rem-sqrt-square 30.3%

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

   16. Simplified 30.3%

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

   if 9.2000000000000006e144 < lambda2

   1. Initial program 33.5%

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

      1. hypot-define 90.8%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in lambda2 around inf 57.4%

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

      1. *-commutative 57.4%

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

      2. *-commutative 57.4%

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

      3. *-commutative 57.4%

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

      4. *-commutative 57.4%

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

      5. +-commutative 57.4%

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

   7. Simplified 57.4%

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

      1. add-sqr-sqrt 35.5%

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

      2. sqrt-unprod 32.5%

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

      3. pow2 32.5%

         \[\leadsto \sqrt{\color{blue}{{\left(\left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot \lambda_2\right) \cdot R\right)}^{2}}} \]

      4. *-commutative 32.5%

         \[\leadsto \sqrt{{\left(\color{blue}{\left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)} \cdot R\right)}^{2}} \]

      5. associate-*l* 32.4%

         \[\leadsto \sqrt{{\color{blue}{\left(\lambda_2 \cdot \left(\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right) \cdot R\right)\right)}}^{2}} \]

      6. +-commutative 32.4%

         \[\leadsto \sqrt{{\left(\lambda_2 \cdot \left(\cos \left(0.5 \cdot \color{blue}{\left(\phi_1 + \phi_2\right)}\right) \cdot R\right)\right)}^{2}} \]

      7. *-commutative 32.4%

         \[\leadsto \sqrt{{\left(\lambda_2 \cdot \left(\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)} \cdot R\right)\right)}^{2}} \]

   9. Applied egg-rr 32.4%

      \[\leadsto \color{blue}{\sqrt{{\left(\lambda_2 \cdot \left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot R\right)\right)}^{2}}} \]
   10. Step-by-step derivation

       1. unpow2 32.4%

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

       2. rem-sqrt-square 43.4%

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

   11. Simplified 43.4%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 9.2 \cdot 10^{+144}:\\ \;\;\;\;\left|R \cdot \left(\phi_2 + \phi_1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\lambda_2 \cdot \left(R \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 #s(literal 1 binary64) 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 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 95.7%

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

3. Simplified 95.7%

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

5. Final simplification 95.7%

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

Alternative 9: 90.8% accurate, 1.6× speedup

Math

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

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) 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 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 95.7%

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

3. Simplified 95.7%

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

5. Taylor expanded in phi2 around 0 90.0%

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

6. Final simplification 90.0%

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

Alternative 10: 58.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 9 \cdot 10^{+144}:\\ \;\;\;\;\left|R\_m \cdot \left(\phi_2 + \phi_1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_2 \cdot R\_m\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda2 9e+144)
    (fabs (* R_m (+ phi2 phi1)))
    (* (cos (* 0.5 phi1)) (* lambda2 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 (lambda2 <= 9e+144) {
		tmp = fabs((R_m * (phi2 + phi1)));
	} else {
		tmp = cos((0.5 * phi1)) * (lambda2 * 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 (lambda2 <= 9d+144) then
        tmp = abs((r_m * (phi2 + phi1)))
    else
        tmp = cos((0.5d0 * phi1)) * (lambda2 * 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 (lambda2 <= 9e+144) {
		tmp = Math.abs((R_m * (phi2 + phi1)));
	} else {
		tmp = Math.cos((0.5 * phi1)) * (lambda2 * 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 lambda2 <= 9e+144:
		tmp = math.fabs((R_m * (phi2 + phi1)))
	else:
		tmp = math.cos((0.5 * phi1)) * (lambda2 * 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 (lambda2 <= 9e+144)
		tmp = abs(Float64(R_m * Float64(phi2 + phi1)));
	else
		tmp = Float64(cos(Float64(0.5 * phi1)) * Float64(lambda2 * 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 (lambda2 <= 9e+144)
		tmp = abs((R_m * (phi2 + phi1)));
	else
		tmp = cos((0.5 * phi1)) * (lambda2 * R_m);
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda2, 9e+144], N[Abs[N[(R$95$m * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda2 * R$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 8.99999999999999935e144

   1. Initial program 61.2%

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

      1. hypot-define 96.7%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in phi2 around inf 30.1%

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

      1. associate-*r/ 30.1%

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

      2. mul-1-neg 30.1%

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

      3. *-commutative 30.1%

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

   7. Simplified 30.1%

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

      1. +-commutative 30.1%

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

      2. distribute-lft-in 29.1%

         \[\leadsto \color{blue}{\phi_2 \cdot \frac{-\phi_1 \cdot R}{\phi_2} + \phi_2 \cdot R} \]

      3. add-sqr-sqrt 14.9%

         \[\leadsto \phi_2 \cdot \frac{\color{blue}{\sqrt{-\phi_1 \cdot R} \cdot \sqrt{-\phi_1 \cdot R}}}{\phi_2} + \phi_2 \cdot R \]

      4. sqrt-unprod 23.1%

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

      5. sqr-neg 23.1%

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

      6. sqrt-unprod 13.4%

         \[\leadsto \phi_2 \cdot \frac{\color{blue}{\sqrt{\phi_1 \cdot R} \cdot \sqrt{\phi_1 \cdot R}}}{\phi_2} + \phi_2 \cdot R \]

      7. add-sqr-sqrt 28.2%

         \[\leadsto \phi_2 \cdot \frac{\color{blue}{\phi_1 \cdot R}}{\phi_2} + \phi_2 \cdot R \]

   9. Applied egg-rr 28.2%

      \[\leadsto \color{blue}{\phi_2 \cdot \frac{\phi_1 \cdot R}{\phi_2} + \phi_2 \cdot R} \]

   10. Taylor expanded in phi2 around 0 27.9%

       \[\leadsto \color{blue}{R \cdot \phi_1 + R \cdot \phi_2} \]
   11. Step-by-step derivation

       1. distribute-lft-in 28.8%

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

   12. Simplified 28.8%

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

       1. add-sqr-sqrt 12.3%

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

       2. sqrt-unprod 24.7%

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

       3. pow2 24.7%

          \[\leadsto \sqrt{\color{blue}{{\left(R \cdot \left(\phi_1 + \phi_2\right)\right)}^{2}}} \]

       4. *-commutative 24.7%

          \[\leadsto \sqrt{{\color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot R\right)}}^{2}} \]

   14. Applied egg-rr 24.7%

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

       1. unpow2 24.7%

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

       2. rem-sqrt-square 30.3%

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

   16. Simplified 30.3%

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

   if 8.99999999999999935e144 < lambda2

   1. Initial program 33.5%

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

      1. hypot-define 90.8%

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

   3. Simplified 90.8%

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

   5. Taylor expanded in lambda2 around inf 57.4%

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

      1. *-commutative 57.4%

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

      2. *-commutative 57.4%

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

      3. *-commutative 57.4%

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

      4. *-commutative 57.4%

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

      5. +-commutative 57.4%

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

   7. Simplified 57.4%

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

   8. Taylor expanded in phi2 around 0 50.1%

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

      1. associate-*r* 50.1%

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

   10. Simplified 50.1%

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

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 9 \cdot 10^{+144}:\\ \;\;\;\;\left|R \cdot \left(\phi_2 + \phi_1\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_2 \cdot R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.2% accurate, 23.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;R\_m \leq 5 \cdot 10^{+66}:\\ \;\;\;\;\phi_2 \cdot R\_m - \phi_1 \cdot R\_m\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot \left(R\_m \cdot \frac{\phi_2}{\phi_1} - R\_m\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= R_m 5e+66)
    (- (* phi2 R_m) (* phi1 R_m))
    (* phi1 (- (* R_m (/ phi2 phi1)) 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 (R_m <= 5e+66) {
		tmp = (phi2 * R_m) - (phi1 * R_m);
	} else {
		tmp = phi1 * ((R_m * (phi2 / phi1)) - 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 (r_m <= 5d+66) then
        tmp = (phi2 * r_m) - (phi1 * r_m)
    else
        tmp = phi1 * ((r_m * (phi2 / phi1)) - 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 (R_m <= 5e+66) {
		tmp = (phi2 * R_m) - (phi1 * R_m);
	} else {
		tmp = phi1 * ((R_m * (phi2 / phi1)) - 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 R_m <= 5e+66:
		tmp = (phi2 * R_m) - (phi1 * R_m)
	else:
		tmp = phi1 * ((R_m * (phi2 / phi1)) - 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 (R_m <= 5e+66)
		tmp = Float64(Float64(phi2 * R_m) - Float64(phi1 * R_m));
	else
		tmp = Float64(phi1 * Float64(Float64(R_m * Float64(phi2 / phi1)) - 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 (R_m <= 5e+66)
		tmp = (phi2 * R_m) - (phi1 * R_m);
	else
		tmp = phi1 * ((R_m * (phi2 / phi1)) - 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[R$95$m, 5e+66], N[(N[(phi2 * R$95$m), $MachinePrecision] - N[(phi1 * R$95$m), $MachinePrecision]), $MachinePrecision], N[(phi1 * N[(N[(R$95$m * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision] - R$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if R < 4.99999999999999991e66

   1. Initial program 48.9%

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

      1. hypot-define 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in phi2 around inf 27.8%

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

      1. associate-*r/ 27.8%

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

      2. mul-1-neg 27.8%

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

      3. *-commutative 27.8%

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

   7. Simplified 27.8%

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

   8. Taylor expanded in phi2 around 0 29.1%

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

      1. +-commutative 29.1%

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

      2. mul-1-neg 29.1%

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

      3. *-commutative 29.1%

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

      4. unsub-neg 29.1%

         \[\leadsto \color{blue}{R \cdot \phi_2 - \phi_1 \cdot R} \]

      5. *-commutative 29.1%

         \[\leadsto \color{blue}{\phi_2 \cdot R} - \phi_1 \cdot R \]

   10. Simplified 29.1%

       \[\leadsto \color{blue}{\phi_2 \cdot R - \phi_1 \cdot R} \]

   if 4.99999999999999991e66 < R

   1. Initial program 97.9%

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

      1. hypot-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in phi2 around inf 28.4%

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

      1. associate-*r/ 28.4%

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

      2. mul-1-neg 28.4%

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

      3. *-commutative 28.4%

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

   7. Simplified 28.4%

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

   8. Taylor expanded in phi1 around inf 30.7%

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

      1. neg-mul-1 30.7%

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

      2. +-commutative 30.7%

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

      3. unsub-neg 30.7%

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

      4. associate-/l* 33.2%

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

   10. Simplified 33.2%

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

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;R \leq 5 \cdot 10^{+66}:\\ \;\;\;\;\phi_2 \cdot R - \phi_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 31.0% accurate, 23.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;R\_m \leq 7.6 \cdot 10^{+66}:\\ \;\;\;\;\phi_2 \cdot R\_m - \phi_1 \cdot R\_m\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot \left(\phi_2 \cdot \frac{R\_m}{\phi_1} - R\_m\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= R_m 7.6e+66)
    (- (* phi2 R_m) (* phi1 R_m))
    (* phi1 (- (* phi2 (/ R_m phi1)) 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 (R_m <= 7.6e+66) {
		tmp = (phi2 * R_m) - (phi1 * R_m);
	} else {
		tmp = phi1 * ((phi2 * (R_m / phi1)) - 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 (r_m <= 7.6d+66) then
        tmp = (phi2 * r_m) - (phi1 * r_m)
    else
        tmp = phi1 * ((phi2 * (r_m / phi1)) - 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 (R_m <= 7.6e+66) {
		tmp = (phi2 * R_m) - (phi1 * R_m);
	} else {
		tmp = phi1 * ((phi2 * (R_m / phi1)) - 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 R_m <= 7.6e+66:
		tmp = (phi2 * R_m) - (phi1 * R_m)
	else:
		tmp = phi1 * ((phi2 * (R_m / phi1)) - 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 (R_m <= 7.6e+66)
		tmp = Float64(Float64(phi2 * R_m) - Float64(phi1 * R_m));
	else
		tmp = Float64(phi1 * Float64(Float64(phi2 * Float64(R_m / phi1)) - 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 (R_m <= 7.6e+66)
		tmp = (phi2 * R_m) - (phi1 * R_m);
	else
		tmp = phi1 * ((phi2 * (R_m / phi1)) - 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[R$95$m, 7.6e+66], N[(N[(phi2 * R$95$m), $MachinePrecision] - N[(phi1 * R$95$m), $MachinePrecision]), $MachinePrecision], N[(phi1 * N[(N[(phi2 * N[(R$95$m / phi1), $MachinePrecision]), $MachinePrecision] - R$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if R < 7.6000000000000004e66

   1. Initial program 48.9%

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

      1. hypot-define 94.9%

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

   3. Simplified 94.9%

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

   5. Taylor expanded in phi2 around inf 27.8%

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

      1. associate-*r/ 27.8%

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

      2. mul-1-neg 27.8%

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

      3. *-commutative 27.8%

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

   7. Simplified 27.8%

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

   8. Taylor expanded in phi2 around 0 29.1%

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

      1. +-commutative 29.1%

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

      2. mul-1-neg 29.1%

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

      3. *-commutative 29.1%

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

      4. unsub-neg 29.1%

         \[\leadsto \color{blue}{R \cdot \phi_2 - \phi_1 \cdot R} \]

      5. *-commutative 29.1%

         \[\leadsto \color{blue}{\phi_2 \cdot R} - \phi_1 \cdot R \]

   10. Simplified 29.1%

       \[\leadsto \color{blue}{\phi_2 \cdot R - \phi_1 \cdot R} \]

   if 7.6000000000000004e66 < R

   1. Initial program 97.9%

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

      1. hypot-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in phi1 around -inf 30.7%

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

      1. mul-1-neg 30.7%

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

      2. distribute-rgt-neg-in 30.7%

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

      3. mul-1-neg 30.7%

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

      4. unsub-neg 30.7%

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

      5. *-commutative 30.7%

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

      6. associate-/l* 35.7%

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

   7. Simplified 35.7%

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

      1. distribute-rgt-neg-out 35.7%

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

      2. neg-sub0 35.7%

         \[\leadsto \color{blue}{0 - \phi_1 \cdot \left(R - \phi_2 \cdot \frac{R}{\phi_1}\right)} \]

      3. associate-*r/ 30.7%

         \[\leadsto 0 - \phi_1 \cdot \left(R - \color{blue}{\frac{\phi_2 \cdot R}{\phi_1}}\right) \]

   9. Applied egg-rr 30.7%

      \[\leadsto \color{blue}{0 - \phi_1 \cdot \left(R - \frac{\phi_2 \cdot R}{\phi_1}\right)} \]
   10. Step-by-step derivation

       1. neg-sub0 30.7%

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

       2. distribute-rgt-neg-in 30.7%

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

       3. sub-neg 30.7%

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

       4. associate-/l* 35.7%

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

       5. distribute-rgt-neg-out 35.7%

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

       6. distribute-neg-out 35.7%

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

       7. +-commutative 35.7%

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

       8. unsub-neg 35.7%

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

       9. distribute-rgt-neg-in 35.7%

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

       10. remove-double-neg 35.7%

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

   11. Simplified 35.7%

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

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;R \leq 7.6 \cdot 10^{+66}:\\ \;\;\;\;\phi_2 \cdot R - \phi_1 \cdot R\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot \left(\phi_2 \cdot \frac{R}{\phi_1} - R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 28.3% accurate, 36.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.9 \cdot 10^{+62}:\\ \;\;\;\;\phi_1 \cdot \left(-R\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\_m\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R_s (if (<= phi1 -2.9e+62) (* 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 <= -2.9e+62) {
		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 <= (-2.9d+62)) 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 <= -2.9e+62) {
		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 <= -2.9e+62:
		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 <= -2.9e+62)
		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 <= -2.9e+62)
		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, -2.9e+62], N[(phi1 * (-R$95$m)), $MachinePrecision], N[(phi2 * R$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.89999999999999984e62

   1. Initial program 33.7%

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

      1. hypot-define 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in phi1 around -inf 63.7%

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

      1. mul-1-neg 63.7%

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

      2. *-commutative 63.7%

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

      3. distribute-rgt-neg-in 63.7%

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

   7. Simplified 63.7%

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

   if -2.89999999999999984e62 < phi1

   1. Initial program 63.3%

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

      1. hypot-define 96.8%

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

   3. Simplified 96.8%

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

   5. Taylor expanded in phi2 around inf 17.6%

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

      1. *-commutative 17.6%

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

   7. Simplified 17.6%

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

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.9 \cdot 10^{+62}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 28.5% accurate, 47.0× speedup

Math

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

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R_s (- (* phi2 R_m) (* phi1 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) - (phi1 * 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) - (phi1 * 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) - (phi1 * 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) - (phi1 * 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(Float64(phi2 * R_m) - Float64(phi1 * 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) - (phi1 * 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[(phi2 * R$95$m), $MachinePrecision] - N[(phi1 * R$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 95.7%

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

3. Simplified 95.7%

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

5. Taylor expanded in phi2 around inf 27.9%

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

   1. associate-*r/ 27.9%

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

   2. mul-1-neg 27.9%

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

   3. *-commutative 27.9%

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

7. Simplified 27.9%

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

8. Taylor expanded in phi2 around 0 28.6%

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

   1. +-commutative 28.6%

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

   2. mul-1-neg 28.6%

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

   3. *-commutative 28.6%

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

   4. unsub-neg 28.6%

      \[\leadsto \color{blue}{R \cdot \phi_2 - \phi_1 \cdot R} \]

   5. *-commutative 28.6%

      \[\leadsto \color{blue}{\phi_2 \cdot R} - \phi_1 \cdot R \]

10. Simplified 28.6%

    \[\leadsto \color{blue}{\phi_2 \cdot R - \phi_1 \cdot R} \]

11. Final simplification 28.6%

    \[\leadsto \phi_2 \cdot R - \phi_1 \cdot R \]
12. Add Preprocessing

Alternative 15: 17.3% 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 #s(literal 1 binary64) 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 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 95.7%

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

3. Simplified 95.7%

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

5. Taylor expanded in phi2 around inf 16.5%

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

   1. *-commutative 16.5%

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

7. Simplified 16.5%

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

8. Final simplification 16.5%

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

Reproduce

herbie shell --seed 2024073 
(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))))))