Equirectangular approximation to distance on a great circle

Percentage Accurate: 60.1% → 99.4%

Time: 26.1s

Alternatives: 19

Speedup: 3.0×

• 31.4% 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: 60.1% 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

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

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
    (*
     R_m
     (hypot
      (*
       (- lambda1 lambda2)
       (-
        (* (cos (* 0.5 phi2)) (cos (* phi1 0.5)))
        (* (sin (* phi1 0.5)) (sin (* 0.5 phi2)))))
      (- 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) {
	return R_s * pow(sqrt((R_m * hypot(((lambda1 - lambda2) * ((cos((0.5 * phi2)) * cos((phi1 * 0.5))) - (sin((phi1 * 0.5)) * sin((0.5 * phi2))))), (phi1 - phi2)))), 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((R_m * Math.hypot(((lambda1 - lambda2) * ((Math.cos((0.5 * phi2)) * Math.cos((phi1 * 0.5))) - (Math.sin((phi1 * 0.5)) * Math.sin((0.5 * phi2))))), (phi1 - phi2)))), 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((R_m * math.hypot(((lambda1 - lambda2) * ((math.cos((0.5 * phi2)) * math.cos((phi1 * 0.5))) - (math.sin((phi1 * 0.5)) * math.sin((0.5 * phi2))))), (phi1 - phi2)))), 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(R_m * hypot(Float64(Float64(lambda1 - lambda2) * Float64(Float64(cos(Float64(0.5 * phi2)) * cos(Float64(phi1 * 0.5))) - Float64(sin(Float64(phi1 * 0.5)) * sin(Float64(0.5 * phi2))))), Float64(phi1 - phi2)))) ^ 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((R_m * hypot(((lambda1 - lambda2) * ((cos((0.5 * phi2)) * cos((phi1 * 0.5))) - (sin((phi1 * 0.5)) * sin((0.5 * phi2))))), (phi1 - phi2)))) ^ 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[(R$95$m * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.4%

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

   1. hypot-define 96.5%

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

3. Simplified 96.5%

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

   1. add-sqr-sqrt 52.1%

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

   2. pow2 52.1%

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

6. Applied egg-rr 52.1%

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

   1. *-commutative 52.1%

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

   2. distribute-rgt-in 52.1%

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

   3. *-commutative 52.1%

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

   4. cos-sum 53.6%

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

   5. *-commutative 53.6%

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

   6. *-commutative 53.6%

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

8. Applied egg-rr 53.6%

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

9. Final simplification 53.6%

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

Alternative 2: 93.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.25:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

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 (<= phi1 -1.25)
    (* R_m (hypot (* (- lambda1 lambda2) (cos (* phi1 0.5))) (- phi1 phi2)))
    (* R_m (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi2))) (- 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 (phi1 <= -1.25) {
		tmp = R_m * hypot(((lambda1 - lambda2) * cos((phi1 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R_m * hypot(((lambda1 - lambda2) * cos((0.5 * phi2))), (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 (phi1 <= -1.25) {
		tmp = R_m * Math.hypot(((lambda1 - lambda2) * Math.cos((phi1 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R_m * Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi2))), (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 phi1 <= -1.25:
		tmp = R_m * math.hypot(((lambda1 - lambda2) * math.cos((phi1 * 0.5))), (phi1 - phi2))
	else:
		tmp = R_m * math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi2))), (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 (phi1 <= -1.25)
		tmp = Float64(R_m * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R_m * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi2))), 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 (phi1 <= -1.25)
		tmp = R_m * hypot(((lambda1 - lambda2) * cos((phi1 * 0.5))), (phi1 - phi2));
	else
		tmp = R_m * hypot(((lambda1 - lambda2) * cos((0.5 * phi2))), (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[phi1, -1.25], N[(R$95$m * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $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[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.25

   1. Initial program 51.7%

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

      1. hypot-define 90.5%

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

   3. Simplified 90.5%

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

   5. Taylor expanded in phi2 around 0 90.6%

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

   if -1.25 < phi1

   1. Initial program 56.6%

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

      1. hypot-define 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in phi1 around 0 92.8%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.25:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ \begin{array}{l} t_0 := \cos \left(\phi_1 \cdot 0.5\right)\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 3.2 \cdot 10^{+50}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\lambda_1 \cdot t\_0, \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(t\_0 \cdot \left(-\lambda_2\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \end{array} \]

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 (* phi1 0.5))))
   (*
    R_s
    (if (<= lambda2 3.2e+50)
      (* R_m (hypot (* lambda1 t_0) (- phi1 phi2)))
      (* R_m (hypot (* t_0 (- lambda2)) (- phi1 phi2)))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((phi1 * 0.5));
	double tmp;
	if (lambda2 <= 3.2e+50) {
		tmp = R_m * hypot((lambda1 * t_0), (phi1 - phi2));
	} else {
		tmp = R_m * hypot((t_0 * -lambda2), (phi1 - phi2));
	}
	return R_s * tmp;
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((phi1 * 0.5));
	double tmp;
	if (lambda2 <= 3.2e+50) {
		tmp = R_m * Math.hypot((lambda1 * t_0), (phi1 - phi2));
	} else {
		tmp = R_m * Math.hypot((t_0 * -lambda2), (phi1 - phi2));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((phi1 * 0.5))
	tmp = 0
	if lambda2 <= 3.2e+50:
		tmp = R_m * math.hypot((lambda1 * t_0), (phi1 - phi2))
	else:
		tmp = R_m * math.hypot((t_0 * -lambda2), (phi1 - phi2))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(phi1 * 0.5))
	tmp = 0.0
	if (lambda2 <= 3.2e+50)
		tmp = Float64(R_m * hypot(Float64(lambda1 * t_0), Float64(phi1 - phi2)));
	else
		tmp = Float64(R_m * hypot(Float64(t_0 * Float64(-lambda2)), Float64(phi1 - phi2)));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	t_0 = cos((phi1 * 0.5));
	tmp = 0.0;
	if (lambda2 <= 3.2e+50)
		tmp = R_m * hypot((lambda1 * t_0), (phi1 - phi2));
	else
		tmp = R_m * hypot((t_0 * -lambda2), (phi1 - phi2));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(R$95$s * If[LessEqual[lambda2, 3.2e+50], N[(R$95$m * N[Sqrt[N[(lambda1 * t$95$0), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[Sqrt[N[(t$95$0 * (-lambda2)), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 3.19999999999999983e50

   1. Initial program 57.1%

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

      1. hypot-define 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in phi2 around 0 88.6%

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

   6. Taylor expanded in lambda1 around inf 77.1%

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

   if 3.19999999999999983e50 < lambda2

   1. Initial program 50.3%

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

      1. hypot-define 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in phi2 around 0 81.2%

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

   6. Taylor expanded in lambda1 around 0 75.4%

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

      1. mul-1-neg 75.4%

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

      2. distribute-lft-neg-in 75.4%

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

   8. Simplified 75.4%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 3.2 \cdot 10^{+50}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(\phi_1 \cdot 0.5\right) \cdot \left(-\lambda_2\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 4 \cdot 10^{-90}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

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 (<= lambda2 4e-90)
    (* R_m (hypot (* lambda1 (cos (* phi1 0.5))) (- phi1 phi2)))
    (* R_m (hypot (- lambda1 lambda2) (- phi1 phi2))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 4e-90) {
		tmp = R_m * hypot((lambda1 * cos((phi1 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R_m * hypot((lambda1 - lambda2), (phi1 - phi2));
	}
	return R_s * tmp;
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 4e-90) {
		tmp = R_m * Math.hypot((lambda1 * Math.cos((phi1 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R_m * Math.hypot((lambda1 - lambda2), (phi1 - phi2));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 4e-90:
		tmp = R_m * math.hypot((lambda1 * math.cos((phi1 * 0.5))), (phi1 - phi2))
	else:
		tmp = R_m * math.hypot((lambda1 - lambda2), (phi1 - phi2))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 4e-90)
		tmp = Float64(R_m * hypot(Float64(lambda1 * cos(Float64(phi1 * 0.5))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R_m * hypot(Float64(lambda1 - lambda2), Float64(phi1 - phi2)));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 4e-90)
		tmp = R_m * hypot((lambda1 * cos((phi1 * 0.5))), (phi1 - phi2));
	else
		tmp = R_m * hypot((lambda1 - lambda2), (phi1 - phi2));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda2, 4e-90], N[(R$95$m * N[Sqrt[N[(lambda1 * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 3.99999999999999998e-90

   1. Initial program 57.9%

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

      1. hypot-define 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in phi2 around 0 88.2%

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

   6. Taylor expanded in lambda1 around inf 75.3%

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

   if 3.99999999999999998e-90 < lambda2

   1. Initial program 50.9%

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

      1. hypot-define 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in phi2 around 0 84.4%

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

   6. Taylor expanded in phi1 around 0 74.0%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 4 \cdot 10^{-90}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.1% accurate, 1.5× speedup

Math

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

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 (/ (+ phi1 phi2) 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(((phi1 + phi2) / 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(((phi1 + phi2) / 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(((phi1 + phi2) / 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(phi1 + phi2) / 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(((phi1 + phi2) / 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[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.4%

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

   1. hypot-define 96.5%

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

3. Simplified 96.5%

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

5. Final simplification 96.5%

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

Alternative 6: 90.6% accurate, 1.6× speedup

Math

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

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 (* phi1 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) {
	return R_s * (R_m * hypot(((lambda1 - lambda2) * cos((phi1 * 0.5))), (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((phi1 * 0.5))), (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((phi1 * 0.5))), (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(phi1 * 0.5))), 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((phi1 * 0.5))), (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[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.4%

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

   1. hypot-define 96.5%

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

3. Simplified 96.5%

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

5. Taylor expanded in phi2 around 0 86.8%

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

6. Final simplification 86.8%

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

Alternative 7: 71.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.02 \cdot 10^{+46}:\\ \;\;\;\;\phi_1 \cdot \left(R\_m \cdot \frac{\phi_2}{\phi_1} - R\_m\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

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 (<= phi1 -1.02e+46)
    (* phi1 (- (* R_m (/ phi2 phi1)) R_m))
    (* R_m (hypot phi2 (- lambda1 lambda2))))))

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 <= -1.02e+46) {
		tmp = phi1 * ((R_m * (phi2 / phi1)) - R_m);
	} else {
		tmp = R_m * hypot(phi2, (lambda1 - lambda2));
	}
	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 (phi1 <= -1.02e+46) {
		tmp = phi1 * ((R_m * (phi2 / phi1)) - R_m);
	} else {
		tmp = R_m * Math.hypot(phi2, (lambda1 - lambda2));
	}
	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 <= -1.02e+46:
		tmp = phi1 * ((R_m * (phi2 / phi1)) - R_m)
	else:
		tmp = R_m * math.hypot(phi2, (lambda1 - lambda2))
	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 <= -1.02e+46)
		tmp = Float64(phi1 * Float64(Float64(R_m * Float64(phi2 / phi1)) - R_m));
	else
		tmp = Float64(R_m * hypot(phi2, Float64(lambda1 - lambda2)));
	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 <= -1.02e+46)
		tmp = phi1 * ((R_m * (phi2 / phi1)) - R_m);
	else
		tmp = R_m * hypot(phi2, (lambda1 - lambda2));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi1, -1.02e+46], N[(phi1 * N[(N[(R$95$m * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision] - R$95$m), $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[Sqrt[phi2 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.0199999999999999e46

   1. Initial program 51.6%

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

      1. hypot-define 93.2%

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

   3. Simplified 93.2%

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

   5. Taylor expanded in phi2 around inf 71.7%

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

      1. associate-*r/ 71.7%

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

      2. mul-1-neg 71.7%

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

      3. *-commutative 71.7%

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

   7. Simplified 71.7%

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

   8. Taylor expanded in phi1 around inf 73.5%

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

      1. neg-mul-1 73.5%

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

      2. +-commutative 73.5%

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

      3. unsub-neg 73.5%

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

      4. associate-/l* 75.5%

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

   10. Simplified 75.5%

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

   if -1.0199999999999999e46 < phi1

   1. Initial program 56.3%

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

      1. hypot-define 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in phi2 around 0 85.3%

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

   6. Taylor expanded in phi1 around 0 44.1%

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

      1. unpow2 44.1%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_2 \cdot \phi_2} + {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      2. unpow2 44.1%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}} \]

      3. hypot-define 66.1%

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

   8. Simplified 66.1%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.02 \cdot 10^{+46}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.4%

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

   1. hypot-define 96.5%

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

3. Simplified 96.5%

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

5. Taylor expanded in phi2 around 0 86.8%

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

6. Taylor expanded in phi1 around 0 78.9%

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

7. Final simplification 78.9%

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

Alternative 9: 27.6% accurate, 9.6× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ \begin{array}{l} t_0 := R\_m \cdot \left(-\lambda_1\right)\\ t_1 := \phi_1 \cdot \left(-R\_m\right)\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq -4.1 \cdot 10^{-212}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 1.6 \cdot 10^{-220}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 2.9 \cdot 10^{-140}:\\ \;\;\;\;R\_m \cdot \lambda_2\\ \mathbf{elif}\;\phi_2 \leq 1.18 \cdot 10^{-119}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 3.8 \cdot 10^{-85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 0.54:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \phi_2\\ \end{array} \end{array} \end{array} \]

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 (* R_m (- lambda1))) (t_1 (* phi1 (- R_m))))
   (*
    R_s
    (if (<= phi2 -4.1e-212)
      t_1
      (if (<= phi2 1.6e-220)
        t_0
        (if (<= phi2 2.9e-140)
          (* R_m lambda2)
          (if (<= phi2 1.18e-119)
            t_0
            (if (<= phi2 3.8e-85)
              t_1
              (if (<= phi2 0.54) t_0 (* R_m 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 t_0 = R_m * -lambda1;
	double t_1 = phi1 * -R_m;
	double tmp;
	if (phi2 <= -4.1e-212) {
		tmp = t_1;
	} else if (phi2 <= 1.6e-220) {
		tmp = t_0;
	} else if (phi2 <= 2.9e-140) {
		tmp = R_m * lambda2;
	} else if (phi2 <= 1.18e-119) {
		tmp = t_0;
	} else if (phi2 <= 3.8e-85) {
		tmp = t_1;
	} else if (phi2 <= 0.54) {
		tmp = t_0;
	} else {
		tmp = R_m * phi2;
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = r_m * -lambda1
    t_1 = phi1 * -r_m
    if (phi2 <= (-4.1d-212)) then
        tmp = t_1
    else if (phi2 <= 1.6d-220) then
        tmp = t_0
    else if (phi2 <= 2.9d-140) then
        tmp = r_m * lambda2
    else if (phi2 <= 1.18d-119) then
        tmp = t_0
    else if (phi2 <= 3.8d-85) then
        tmp = t_1
    else if (phi2 <= 0.54d0) then
        tmp = t_0
    else
        tmp = r_m * phi2
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R_m * -lambda1;
	double t_1 = phi1 * -R_m;
	double tmp;
	if (phi2 <= -4.1e-212) {
		tmp = t_1;
	} else if (phi2 <= 1.6e-220) {
		tmp = t_0;
	} else if (phi2 <= 2.9e-140) {
		tmp = R_m * lambda2;
	} else if (phi2 <= 1.18e-119) {
		tmp = t_0;
	} else if (phi2 <= 3.8e-85) {
		tmp = t_1;
	} else if (phi2 <= 0.54) {
		tmp = t_0;
	} else {
		tmp = R_m * 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):
	t_0 = R_m * -lambda1
	t_1 = phi1 * -R_m
	tmp = 0
	if phi2 <= -4.1e-212:
		tmp = t_1
	elif phi2 <= 1.6e-220:
		tmp = t_0
	elif phi2 <= 2.9e-140:
		tmp = R_m * lambda2
	elif phi2 <= 1.18e-119:
		tmp = t_0
	elif phi2 <= 3.8e-85:
		tmp = t_1
	elif phi2 <= 0.54:
		tmp = t_0
	else:
		tmp = R_m * 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)
	t_0 = Float64(R_m * Float64(-lambda1))
	t_1 = Float64(phi1 * Float64(-R_m))
	tmp = 0.0
	if (phi2 <= -4.1e-212)
		tmp = t_1;
	elseif (phi2 <= 1.6e-220)
		tmp = t_0;
	elseif (phi2 <= 2.9e-140)
		tmp = Float64(R_m * lambda2);
	elseif (phi2 <= 1.18e-119)
		tmp = t_0;
	elseif (phi2 <= 3.8e-85)
		tmp = t_1;
	elseif (phi2 <= 0.54)
		tmp = t_0;
	else
		tmp = Float64(R_m * 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)
	t_0 = R_m * -lambda1;
	t_1 = phi1 * -R_m;
	tmp = 0.0;
	if (phi2 <= -4.1e-212)
		tmp = t_1;
	elseif (phi2 <= 1.6e-220)
		tmp = t_0;
	elseif (phi2 <= 2.9e-140)
		tmp = R_m * lambda2;
	elseif (phi2 <= 1.18e-119)
		tmp = t_0;
	elseif (phi2 <= 3.8e-85)
		tmp = t_1;
	elseif (phi2 <= 0.54)
		tmp = t_0;
	else
		tmp = R_m * 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_] := Block[{t$95$0 = N[(R$95$m * (-lambda1)), $MachinePrecision]}, Block[{t$95$1 = N[(phi1 * (-R$95$m)), $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi2, -4.1e-212], t$95$1, If[LessEqual[phi2, 1.6e-220], t$95$0, If[LessEqual[phi2, 2.9e-140], N[(R$95$m * lambda2), $MachinePrecision], If[LessEqual[phi2, 1.18e-119], t$95$0, If[LessEqual[phi2, 3.8e-85], t$95$1, If[LessEqual[phi2, 0.54], t$95$0, N[(R$95$m * phi2), $MachinePrecision]]]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if phi2 < -4.10000000000000014e-212 or 1.17999999999999996e-119 < phi2 < 3.7999999999999999e-85

   1. Initial program 56.7%

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

      1. hypot-define 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in phi1 around -inf 20.3%

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

      1. mul-1-neg 20.3%

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

      2. *-commutative 20.3%

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

      3. distribute-rgt-neg-in 20.3%

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

   7. Simplified 20.3%

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

   if -4.10000000000000014e-212 < phi2 < 1.60000000000000003e-220 or 2.89999999999999997e-140 < phi2 < 1.17999999999999996e-119 or 3.7999999999999999e-85 < phi2 < 0.54000000000000004

   1. Initial program 62.4%

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

      1. hypot-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in phi2 around 0 99.7%

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

   6. Taylor expanded in phi1 around 0 40.0%

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

      1. unpow2 40.0%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_2 \cdot \phi_2} + {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      2. unpow2 40.0%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}} \]

      3. hypot-define 58.0%

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

   8. Simplified 58.0%

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

   9. Taylor expanded in lambda1 around -inf 19.5%

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

       1. mul-1-neg 19.5%

          \[\leadsto \color{blue}{-R \cdot \lambda_1} \]

       2. *-commutative 19.5%

          \[\leadsto -\color{blue}{\lambda_1 \cdot R} \]

       3. distribute-rgt-neg-in 19.5%

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

   11. Simplified 19.5%

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

   if 1.60000000000000003e-220 < phi2 < 2.89999999999999997e-140

   1. Initial program 52.2%

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

      1. hypot-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi2 around 0 99.8%

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

   6. Taylor expanded in phi1 around 0 39.5%

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

      1. unpow2 39.5%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_2 \cdot \phi_2} + {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      2. unpow2 39.5%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}} \]

      3. hypot-define 69.7%

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

   8. Simplified 69.7%

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

   9. Taylor expanded in lambda2 around inf 20.0%

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

       1. *-commutative 20.0%

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

   11. Simplified 20.0%

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

   if 0.54000000000000004 < phi2

   1. Initial program 47.7%

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

      1. hypot-define 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in phi2 around inf 59.7%

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

      1. *-commutative 59.7%

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

   7. Simplified 59.7%

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

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -4.1 \cdot 10^{-212}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_2 \leq 1.6 \cdot 10^{-220}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 2.9 \cdot 10^{-140}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_2 \leq 1.18 \cdot 10^{-119}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 3.8 \cdot 10^{-85}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_2 \leq 0.54:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 32.0% accurate, 11.3× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2 \cdot 10^{-157}:\\ \;\;\;\;\phi_1 \cdot \left(-R\_m\right)\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{-253} \lor \neg \left(\phi_2 \leq 5.5 \cdot 10^{-181}\right) \land \phi_2 \leq 7.5 \cdot 10^{-27}:\\ \;\;\;\;R\_m \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \phi_2 - R\_m \cdot \phi_1\\ \end{array} \end{array} \]

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 -2e-157)
    (* phi1 (- R_m))
    (if (or (<= phi2 1.05e-253)
            (and (not (<= phi2 5.5e-181)) (<= phi2 7.5e-27)))
      (* R_m (* lambda2 (- 1.0 (/ lambda1 lambda2))))
      (- (* R_m phi2) (* R_m 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 (phi2 <= -2e-157) {
		tmp = phi1 * -R_m;
	} else if ((phi2 <= 1.05e-253) || (!(phi2 <= 5.5e-181) && (phi2 <= 7.5e-27))) {
		tmp = R_m * (lambda2 * (1.0 - (lambda1 / lambda2)));
	} else {
		tmp = (R_m * phi2) - (R_m * 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 (phi2 <= (-2d-157)) then
        tmp = phi1 * -r_m
    else if ((phi2 <= 1.05d-253) .or. (.not. (phi2 <= 5.5d-181)) .and. (phi2 <= 7.5d-27)) then
        tmp = r_m * (lambda2 * (1.0d0 - (lambda1 / lambda2)))
    else
        tmp = (r_m * phi2) - (r_m * 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 (phi2 <= -2e-157) {
		tmp = phi1 * -R_m;
	} else if ((phi2 <= 1.05e-253) || (!(phi2 <= 5.5e-181) && (phi2 <= 7.5e-27))) {
		tmp = R_m * (lambda2 * (1.0 - (lambda1 / lambda2)));
	} else {
		tmp = (R_m * phi2) - (R_m * 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 phi2 <= -2e-157:
		tmp = phi1 * -R_m
	elif (phi2 <= 1.05e-253) or (not (phi2 <= 5.5e-181) and (phi2 <= 7.5e-27)):
		tmp = R_m * (lambda2 * (1.0 - (lambda1 / lambda2)))
	else:
		tmp = (R_m * phi2) - (R_m * 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 (phi2 <= -2e-157)
		tmp = Float64(phi1 * Float64(-R_m));
	elseif ((phi2 <= 1.05e-253) || (!(phi2 <= 5.5e-181) && (phi2 <= 7.5e-27)))
		tmp = Float64(R_m * Float64(lambda2 * Float64(1.0 - Float64(lambda1 / lambda2))));
	else
		tmp = Float64(Float64(R_m * phi2) - Float64(R_m * 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 (phi2 <= -2e-157)
		tmp = phi1 * -R_m;
	elseif ((phi2 <= 1.05e-253) || (~((phi2 <= 5.5e-181)) && (phi2 <= 7.5e-27)))
		tmp = R_m * (lambda2 * (1.0 - (lambda1 / lambda2)));
	else
		tmp = (R_m * phi2) - (R_m * 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[phi2, -2e-157], N[(phi1 * (-R$95$m)), $MachinePrecision], If[Or[LessEqual[phi2, 1.05e-253], And[N[Not[LessEqual[phi2, 5.5e-181]], $MachinePrecision], LessEqual[phi2, 7.5e-27]]], N[(R$95$m * N[(lambda2 * N[(1.0 - N[(lambda1 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(R$95$m * phi2), $MachinePrecision] - N[(R$95$m * phi1), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.99999999999999989e-157

   1. Initial program 57.1%

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

      1. hypot-define 95.5%

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

   3. Simplified 95.5%

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

   5. Taylor expanded in phi1 around -inf 17.6%

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

      1. mul-1-neg 17.6%

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

      2. *-commutative 17.6%

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

      3. distribute-rgt-neg-in 17.6%

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

   7. Simplified 17.6%

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

   if -1.99999999999999989e-157 < phi2 < 1.0499999999999999e-253 or 5.50000000000000033e-181 < phi2 < 7.50000000000000029e-27

   1. Initial program 58.6%

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

      1. hypot-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi2 around 0 99.8%

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

   6. Taylor expanded in lambda2 around inf 41.4%

      \[\leadsto R \cdot \color{blue}{\left(\lambda_2 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) + -1 \cdot \frac{\lambda_1 \cdot \cos \left(0.5 \cdot \phi_1\right)}{\lambda_2}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 41.4%

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

      2. unsub-neg 41.4%

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

      3. associate-/l* 41.4%

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

   8. Simplified 41.4%

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

   9. Taylor expanded in phi1 around 0 33.1%

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

   if 1.0499999999999999e-253 < phi2 < 5.50000000000000033e-181 or 7.50000000000000029e-27 < phi2

   1. Initial program 50.3%

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

      1. hypot-define 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in phi2 around inf 60.8%

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

      1. associate-*r/ 60.8%

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

      2. mul-1-neg 60.8%

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

      3. *-commutative 60.8%

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

   7. Simplified 60.8%

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

   8. Taylor expanded in phi2 around 0 62.0%

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

      1. +-commutative 62.0%

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

      2. mul-1-neg 62.0%

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

      3. unsub-neg 62.0%

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

   10. Simplified 62.0%

       \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2 \cdot 10^{-157}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_2 \leq 1.05 \cdot 10^{-253} \lor \neg \left(\phi_2 \leq 5.5 \cdot 10^{-181}\right) \land \phi_2 \leq 7.5 \cdot 10^{-27}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 32.6% accurate, 11.3× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ \begin{array}{l} t_0 := R\_m \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq -6.2 \cdot 10^{-162}:\\ \;\;\;\;\phi_1 \cdot \left(-R\_m\right)\\ \mathbf{elif}\;\phi_2 \leq 1.15 \cdot 10^{-253}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 4.4 \cdot 10^{-181}:\\ \;\;\;\;R\_m \cdot \phi_2 - R\_m \cdot \phi_1\\ \mathbf{elif}\;\phi_2 \leq 4.4 \cdot 10^{-28}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (* R_m (* lambda2 (- 1.0 (/ lambda1 lambda2))))))
   (*
    R_s
    (if (<= phi2 -6.2e-162)
      (* phi1 (- R_m))
      (if (<= phi2 1.15e-253)
        t_0
        (if (<= phi2 4.4e-181)
          (- (* R_m phi2) (* R_m phi1))
          (if (<= phi2 4.4e-28)
            t_0
            (* R_m (* phi2 (- 1.0 (/ 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 t_0 = R_m * (lambda2 * (1.0 - (lambda1 / lambda2)));
	double tmp;
	if (phi2 <= -6.2e-162) {
		tmp = phi1 * -R_m;
	} else if (phi2 <= 1.15e-253) {
		tmp = t_0;
	} else if (phi2 <= 4.4e-181) {
		tmp = (R_m * phi2) - (R_m * phi1);
	} else if (phi2 <= 4.4e-28) {
		tmp = t_0;
	} else {
		tmp = R_m * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r_m * (lambda2 * (1.0d0 - (lambda1 / lambda2)))
    if (phi2 <= (-6.2d-162)) then
        tmp = phi1 * -r_m
    else if (phi2 <= 1.15d-253) then
        tmp = t_0
    else if (phi2 <= 4.4d-181) then
        tmp = (r_m * phi2) - (r_m * phi1)
    else if (phi2 <= 4.4d-28) then
        tmp = t_0
    else
        tmp = r_m * (phi2 * (1.0d0 - (phi1 / phi2)))
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R_m * (lambda2 * (1.0 - (lambda1 / lambda2)));
	double tmp;
	if (phi2 <= -6.2e-162) {
		tmp = phi1 * -R_m;
	} else if (phi2 <= 1.15e-253) {
		tmp = t_0;
	} else if (phi2 <= 4.4e-181) {
		tmp = (R_m * phi2) - (R_m * phi1);
	} else if (phi2 <= 4.4e-28) {
		tmp = t_0;
	} else {
		tmp = R_m * (phi2 * (1.0 - (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):
	t_0 = R_m * (lambda2 * (1.0 - (lambda1 / lambda2)))
	tmp = 0
	if phi2 <= -6.2e-162:
		tmp = phi1 * -R_m
	elif phi2 <= 1.15e-253:
		tmp = t_0
	elif phi2 <= 4.4e-181:
		tmp = (R_m * phi2) - (R_m * phi1)
	elif phi2 <= 4.4e-28:
		tmp = t_0
	else:
		tmp = R_m * (phi2 * (1.0 - (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)
	t_0 = Float64(R_m * Float64(lambda2 * Float64(1.0 - Float64(lambda1 / lambda2))))
	tmp = 0.0
	if (phi2 <= -6.2e-162)
		tmp = Float64(phi1 * Float64(-R_m));
	elseif (phi2 <= 1.15e-253)
		tmp = t_0;
	elseif (phi2 <= 4.4e-181)
		tmp = Float64(Float64(R_m * phi2) - Float64(R_m * phi1));
	elseif (phi2 <= 4.4e-28)
		tmp = t_0;
	else
		tmp = Float64(R_m * Float64(phi2 * Float64(1.0 - 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)
	t_0 = R_m * (lambda2 * (1.0 - (lambda1 / lambda2)));
	tmp = 0.0;
	if (phi2 <= -6.2e-162)
		tmp = phi1 * -R_m;
	elseif (phi2 <= 1.15e-253)
		tmp = t_0;
	elseif (phi2 <= 4.4e-181)
		tmp = (R_m * phi2) - (R_m * phi1);
	elseif (phi2 <= 4.4e-28)
		tmp = t_0;
	else
		tmp = R_m * (phi2 * (1.0 - (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_] := Block[{t$95$0 = N[(R$95$m * N[(lambda2 * N[(1.0 - N[(lambda1 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi2, -6.2e-162], N[(phi1 * (-R$95$m)), $MachinePrecision], If[LessEqual[phi2, 1.15e-253], t$95$0, If[LessEqual[phi2, 4.4e-181], N[(N[(R$95$m * phi2), $MachinePrecision] - N[(R$95$m * phi1), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 4.4e-28], t$95$0, N[(R$95$m * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if phi2 < -6.1999999999999997e-162

   1. Initial program 57.5%

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

      1. hypot-define 95.6%

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

   3. Simplified 95.6%

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

   5. Taylor expanded in phi1 around -inf 17.4%

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

      1. mul-1-neg 17.4%

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

      2. *-commutative 17.4%

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

      3. distribute-rgt-neg-in 17.4%

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

   7. Simplified 17.4%

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

   if -6.1999999999999997e-162 < phi2 < 1.15e-253 or 4.39999999999999994e-181 < phi2 < 4.39999999999999992e-28

   1. Initial program 58.1%

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

      1. hypot-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi2 around 0 99.8%

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

   6. Taylor expanded in lambda2 around inf 40.6%

      \[\leadsto R \cdot \color{blue}{\left(\lambda_2 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) + -1 \cdot \frac{\lambda_1 \cdot \cos \left(0.5 \cdot \phi_1\right)}{\lambda_2}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 40.6%

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

      2. unsub-neg 40.6%

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

      3. associate-/l* 40.6%

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

   8. Simplified 40.6%

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

   9. Taylor expanded in phi1 around 0 32.2%

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

   if 1.15e-253 < phi2 < 4.39999999999999994e-181

   1. Initial program 55.7%

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

      1. hypot-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi2 around inf 48.9%

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

      1. associate-*r/ 48.9%

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

      2. mul-1-neg 48.9%

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

      3. *-commutative 48.9%

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

   7. Simplified 48.9%

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

   8. Taylor expanded in phi2 around 0 56.0%

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

      1. +-commutative 56.0%

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

      2. mul-1-neg 56.0%

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

      3. unsub-neg 56.0%

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

   10. Simplified 56.0%

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

   if 4.39999999999999992e-28 < phi2

   1. Initial program 49.3%

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

   3. Taylor expanded in phi2 around inf 64.6%

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

      1. mul-1-neg 64.6%

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

      2. unsub-neg 64.6%

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

   5. Simplified 64.6%

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

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -6.2 \cdot 10^{-162}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_2 \leq 1.15 \cdot 10^{-253}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 4.4 \cdot 10^{-181}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{elif}\;\phi_2 \leq 4.4 \cdot 10^{-28}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 32.8% accurate, 11.3× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.02 \cdot 10^{-147}:\\ \;\;\;\;\phi_1 \cdot \left(-R\_m\right)\\ \mathbf{elif}\;\phi_2 \leq 1.15 \cdot 10^{-253}:\\ \;\;\;\;R\_m \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 6.8 \cdot 10^{-187}:\\ \;\;\;\;R\_m \cdot \phi_2 - R\_m \cdot \phi_1\\ \mathbf{elif}\;\phi_2 \leq 0.00046:\\ \;\;\;\;\lambda_2 \cdot \left(R\_m - \lambda_1 \cdot \frac{R\_m}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

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 (<= phi2 -1.02e-147)
    (* phi1 (- R_m))
    (if (<= phi2 1.15e-253)
      (* R_m (* lambda2 (- 1.0 (/ lambda1 lambda2))))
      (if (<= phi2 6.8e-187)
        (- (* R_m phi2) (* R_m phi1))
        (if (<= phi2 0.00046)
          (* lambda2 (- R_m (* lambda1 (/ R_m lambda2))))
          (* R_m (* phi2 (- 1.0 (/ 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 (phi2 <= -1.02e-147) {
		tmp = phi1 * -R_m;
	} else if (phi2 <= 1.15e-253) {
		tmp = R_m * (lambda2 * (1.0 - (lambda1 / lambda2)));
	} else if (phi2 <= 6.8e-187) {
		tmp = (R_m * phi2) - (R_m * phi1);
	} else if (phi2 <= 0.00046) {
		tmp = lambda2 * (R_m - (lambda1 * (R_m / lambda2)));
	} else {
		tmp = R_m * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= (-1.02d-147)) then
        tmp = phi1 * -r_m
    else if (phi2 <= 1.15d-253) then
        tmp = r_m * (lambda2 * (1.0d0 - (lambda1 / lambda2)))
    else if (phi2 <= 6.8d-187) then
        tmp = (r_m * phi2) - (r_m * phi1)
    else if (phi2 <= 0.00046d0) then
        tmp = lambda2 * (r_m - (lambda1 * (r_m / lambda2)))
    else
        tmp = r_m * (phi2 * (1.0d0 - (phi1 / phi2)))
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -1.02e-147) {
		tmp = phi1 * -R_m;
	} else if (phi2 <= 1.15e-253) {
		tmp = R_m * (lambda2 * (1.0 - (lambda1 / lambda2)));
	} else if (phi2 <= 6.8e-187) {
		tmp = (R_m * phi2) - (R_m * phi1);
	} else if (phi2 <= 0.00046) {
		tmp = lambda2 * (R_m - (lambda1 * (R_m / lambda2)));
	} else {
		tmp = R_m * (phi2 * (1.0 - (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 phi2 <= -1.02e-147:
		tmp = phi1 * -R_m
	elif phi2 <= 1.15e-253:
		tmp = R_m * (lambda2 * (1.0 - (lambda1 / lambda2)))
	elif phi2 <= 6.8e-187:
		tmp = (R_m * phi2) - (R_m * phi1)
	elif phi2 <= 0.00046:
		tmp = lambda2 * (R_m - (lambda1 * (R_m / lambda2)))
	else:
		tmp = R_m * (phi2 * (1.0 - (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 (phi2 <= -1.02e-147)
		tmp = Float64(phi1 * Float64(-R_m));
	elseif (phi2 <= 1.15e-253)
		tmp = Float64(R_m * Float64(lambda2 * Float64(1.0 - Float64(lambda1 / lambda2))));
	elseif (phi2 <= 6.8e-187)
		tmp = Float64(Float64(R_m * phi2) - Float64(R_m * phi1));
	elseif (phi2 <= 0.00046)
		tmp = Float64(lambda2 * Float64(R_m - Float64(lambda1 * Float64(R_m / lambda2))));
	else
		tmp = Float64(R_m * Float64(phi2 * Float64(1.0 - 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 (phi2 <= -1.02e-147)
		tmp = phi1 * -R_m;
	elseif (phi2 <= 1.15e-253)
		tmp = R_m * (lambda2 * (1.0 - (lambda1 / lambda2)));
	elseif (phi2 <= 6.8e-187)
		tmp = (R_m * phi2) - (R_m * phi1);
	elseif (phi2 <= 0.00046)
		tmp = lambda2 * (R_m - (lambda1 * (R_m / lambda2)));
	else
		tmp = R_m * (phi2 * (1.0 - (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[phi2, -1.02e-147], N[(phi1 * (-R$95$m)), $MachinePrecision], If[LessEqual[phi2, 1.15e-253], N[(R$95$m * N[(lambda2 * N[(1.0 - N[(lambda1 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 6.8e-187], N[(N[(R$95$m * phi2), $MachinePrecision] - N[(R$95$m * phi1), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 0.00046], N[(lambda2 * N[(R$95$m - N[(lambda1 * N[(R$95$m / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if phi2 < -1.02e-147

   1. Initial program 57.6%

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

      1. hypot-define 95.5%

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

   3. Simplified 95.5%

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

   5. Taylor expanded in phi1 around -inf 17.7%

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

      1. mul-1-neg 17.7%

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

      2. *-commutative 17.7%

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

      3. distribute-rgt-neg-in 17.7%

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

   7. Simplified 17.7%

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

   if -1.02e-147 < phi2 < 1.15e-253

   1. Initial program 62.8%

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

      1. hypot-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi2 around 0 99.8%

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

   6. Taylor expanded in lambda2 around inf 33.7%

      \[\leadsto R \cdot \color{blue}{\left(\lambda_2 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) + -1 \cdot \frac{\lambda_1 \cdot \cos \left(0.5 \cdot \phi_1\right)}{\lambda_2}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 33.7%

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

      2. unsub-neg 33.7%

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

      3. associate-/l* 33.8%

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

   8. Simplified 33.8%

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

   9. Taylor expanded in phi1 around 0 30.2%

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

   if 1.15e-253 < phi2 < 6.8000000000000003e-187

   1. Initial program 55.7%

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

      1. hypot-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi2 around inf 48.9%

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

      1. associate-*r/ 48.9%

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

      2. mul-1-neg 48.9%

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

      3. *-commutative 48.9%

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

   7. Simplified 48.9%

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

   8. Taylor expanded in phi2 around 0 56.0%

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

      1. +-commutative 56.0%

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

      2. mul-1-neg 56.0%

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

      3. unsub-neg 56.0%

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

   10. Simplified 56.0%

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

   if 6.8000000000000003e-187 < phi2 < 4.6000000000000001e-4

   1. Initial program 52.4%

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

      1. hypot-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi2 around 0 99.8%

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

   6. Taylor expanded in phi1 around 0 36.4%

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

      1. unpow2 36.4%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_2 \cdot \phi_2} + {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      2. unpow2 36.4%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}} \]

      3. hypot-define 56.6%

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

   8. Simplified 56.6%

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

   9. Taylor expanded in lambda2 around inf 38.2%

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

       1. mul-1-neg 38.2%

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

       2. unsub-neg 38.2%

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

       3. *-commutative 38.2%

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

       4. associate-*r/ 34.9%

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

   11. Simplified 34.9%

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

   if 4.6000000000000001e-4 < phi2

   1. Initial program 47.7%

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

   3. Taylor expanded in phi2 around inf 65.0%

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

      1. mul-1-neg 65.0%

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

      2. unsub-neg 65.0%

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

   5. Simplified 65.0%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.02 \cdot 10^{-147}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_2 \leq 1.15 \cdot 10^{-253}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 6.8 \cdot 10^{-187}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{elif}\;\phi_2 \leq 0.00046:\\ \;\;\;\;\lambda_2 \cdot \left(R - \lambda_1 \cdot \frac{R}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 31.8% accurate, 11.3× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.3 \cdot 10^{-156}:\\ \;\;\;\;\phi_1 \cdot \left(R\_m \cdot \frac{\phi_2}{\phi_1} - R\_m\right)\\ \mathbf{elif}\;\phi_2 \leq 1.15 \cdot 10^{-253}:\\ \;\;\;\;R\_m \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 7.2 \cdot 10^{-187}:\\ \;\;\;\;R\_m \cdot \phi_2 - R\_m \cdot \phi_1\\ \mathbf{elif}\;\phi_2 \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;\lambda_2 \cdot \left(R\_m - \lambda_1 \cdot \frac{R\_m}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

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 (<= phi2 -1.3e-156)
    (* phi1 (- (* R_m (/ phi2 phi1)) R_m))
    (if (<= phi2 1.15e-253)
      (* R_m (* lambda2 (- 1.0 (/ lambda1 lambda2))))
      (if (<= phi2 7.2e-187)
        (- (* R_m phi2) (* R_m phi1))
        (if (<= phi2 2.4e-5)
          (* lambda2 (- R_m (* lambda1 (/ R_m lambda2))))
          (* R_m (* phi2 (- 1.0 (/ 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 (phi2 <= -1.3e-156) {
		tmp = phi1 * ((R_m * (phi2 / phi1)) - R_m);
	} else if (phi2 <= 1.15e-253) {
		tmp = R_m * (lambda2 * (1.0 - (lambda1 / lambda2)));
	} else if (phi2 <= 7.2e-187) {
		tmp = (R_m * phi2) - (R_m * phi1);
	} else if (phi2 <= 2.4e-5) {
		tmp = lambda2 * (R_m - (lambda1 * (R_m / lambda2)));
	} else {
		tmp = R_m * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= (-1.3d-156)) then
        tmp = phi1 * ((r_m * (phi2 / phi1)) - r_m)
    else if (phi2 <= 1.15d-253) then
        tmp = r_m * (lambda2 * (1.0d0 - (lambda1 / lambda2)))
    else if (phi2 <= 7.2d-187) then
        tmp = (r_m * phi2) - (r_m * phi1)
    else if (phi2 <= 2.4d-5) then
        tmp = lambda2 * (r_m - (lambda1 * (r_m / lambda2)))
    else
        tmp = r_m * (phi2 * (1.0d0 - (phi1 / phi2)))
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -1.3e-156) {
		tmp = phi1 * ((R_m * (phi2 / phi1)) - R_m);
	} else if (phi2 <= 1.15e-253) {
		tmp = R_m * (lambda2 * (1.0 - (lambda1 / lambda2)));
	} else if (phi2 <= 7.2e-187) {
		tmp = (R_m * phi2) - (R_m * phi1);
	} else if (phi2 <= 2.4e-5) {
		tmp = lambda2 * (R_m - (lambda1 * (R_m / lambda2)));
	} else {
		tmp = R_m * (phi2 * (1.0 - (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 phi2 <= -1.3e-156:
		tmp = phi1 * ((R_m * (phi2 / phi1)) - R_m)
	elif phi2 <= 1.15e-253:
		tmp = R_m * (lambda2 * (1.0 - (lambda1 / lambda2)))
	elif phi2 <= 7.2e-187:
		tmp = (R_m * phi2) - (R_m * phi1)
	elif phi2 <= 2.4e-5:
		tmp = lambda2 * (R_m - (lambda1 * (R_m / lambda2)))
	else:
		tmp = R_m * (phi2 * (1.0 - (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 (phi2 <= -1.3e-156)
		tmp = Float64(phi1 * Float64(Float64(R_m * Float64(phi2 / phi1)) - R_m));
	elseif (phi2 <= 1.15e-253)
		tmp = Float64(R_m * Float64(lambda2 * Float64(1.0 - Float64(lambda1 / lambda2))));
	elseif (phi2 <= 7.2e-187)
		tmp = Float64(Float64(R_m * phi2) - Float64(R_m * phi1));
	elseif (phi2 <= 2.4e-5)
		tmp = Float64(lambda2 * Float64(R_m - Float64(lambda1 * Float64(R_m / lambda2))));
	else
		tmp = Float64(R_m * Float64(phi2 * Float64(1.0 - 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 (phi2 <= -1.3e-156)
		tmp = phi1 * ((R_m * (phi2 / phi1)) - R_m);
	elseif (phi2 <= 1.15e-253)
		tmp = R_m * (lambda2 * (1.0 - (lambda1 / lambda2)));
	elseif (phi2 <= 7.2e-187)
		tmp = (R_m * phi2) - (R_m * phi1);
	elseif (phi2 <= 2.4e-5)
		tmp = lambda2 * (R_m - (lambda1 * (R_m / lambda2)));
	else
		tmp = R_m * (phi2 * (1.0 - (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[phi2, -1.3e-156], N[(phi1 * N[(N[(R$95$m * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision] - R$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.15e-253], N[(R$95$m * N[(lambda2 * N[(1.0 - N[(lambda1 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 7.2e-187], N[(N[(R$95$m * phi2), $MachinePrecision] - N[(R$95$m * phi1), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 2.4e-5], N[(lambda2 * N[(R$95$m - N[(lambda1 * N[(R$95$m / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if phi2 < -1.3e-156

   1. Initial program 57.1%

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

      1. hypot-define 95.5%

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

   3. Simplified 95.5%

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

   5. Taylor expanded in phi2 around inf 16.7%

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

      1. associate-*r/ 16.7%

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

      2. mul-1-neg 16.7%

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

      3. *-commutative 16.7%

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

   7. Simplified 16.7%

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

   8. Taylor expanded in phi1 around inf 15.7%

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

      1. neg-mul-1 15.7%

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

      2. +-commutative 15.7%

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

      3. unsub-neg 15.7%

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

      4. associate-/l* 16.7%

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

   10. Simplified 16.7%

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

   if -1.3e-156 < phi2 < 1.15e-253

   1. Initial program 64.0%

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

      1. hypot-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi2 around 0 99.8%

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

   6. Taylor expanded in lambda2 around inf 34.3%

      \[\leadsto R \cdot \color{blue}{\left(\lambda_2 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) + -1 \cdot \frac{\lambda_1 \cdot \cos \left(0.5 \cdot \phi_1\right)}{\lambda_2}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 34.3%

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

      2. unsub-neg 34.3%

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

      3. associate-/l* 34.4%

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

   8. Simplified 34.4%

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

   9. Taylor expanded in phi1 around 0 30.8%

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

   if 1.15e-253 < phi2 < 7.19999999999999989e-187

   1. Initial program 55.7%

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

      1. hypot-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi2 around inf 48.9%

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

      1. associate-*r/ 48.9%

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

      2. mul-1-neg 48.9%

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

      3. *-commutative 48.9%

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

   7. Simplified 48.9%

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

   8. Taylor expanded in phi2 around 0 56.0%

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

      1. +-commutative 56.0%

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

      2. mul-1-neg 56.0%

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

      3. unsub-neg 56.0%

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

   10. Simplified 56.0%

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

   if 7.19999999999999989e-187 < phi2 < 2.4000000000000001e-5

   1. Initial program 52.4%

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

      1. hypot-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi2 around 0 99.8%

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

   6. Taylor expanded in phi1 around 0 36.4%

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

      1. unpow2 36.4%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_2 \cdot \phi_2} + {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      2. unpow2 36.4%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}} \]

      3. hypot-define 56.6%

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

   8. Simplified 56.6%

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

   9. Taylor expanded in lambda2 around inf 38.2%

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

       1. mul-1-neg 38.2%

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

       2. unsub-neg 38.2%

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

       3. *-commutative 38.2%

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

       4. associate-*r/ 34.9%

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

   11. Simplified 34.9%

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

   if 2.4000000000000001e-5 < phi2

   1. Initial program 47.7%

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

   3. Taylor expanded in phi2 around inf 65.0%

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

      1. mul-1-neg 65.0%

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

      2. unsub-neg 65.0%

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

   5. Simplified 65.0%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.3 \cdot 10^{-156}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_2 \leq 1.15 \cdot 10^{-253}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 7.2 \cdot 10^{-187}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{elif}\;\phi_2 \leq 2.4 \cdot 10^{-5}:\\ \;\;\;\;\lambda_2 \cdot \left(R - \lambda_1 \cdot \frac{R}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 31.6% accurate, 11.3× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.35 \cdot 10^{-211}:\\ \;\;\;\;\phi_2 \cdot \left(R\_m - \frac{R\_m \cdot \phi_1}{\phi_2}\right)\\ \mathbf{elif}\;\phi_2 \leq 1.15 \cdot 10^{-253}:\\ \;\;\;\;R\_m \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.15 \cdot 10^{-185}:\\ \;\;\;\;R\_m \cdot \phi_2 - R\_m \cdot \phi_1\\ \mathbf{elif}\;\phi_2 \leq 3.5 \cdot 10^{-5}:\\ \;\;\;\;\lambda_2 \cdot \left(R\_m - \lambda_1 \cdot \frac{R\_m}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

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 (<= phi2 -2.35e-211)
    (* phi2 (- R_m (/ (* R_m phi1) phi2)))
    (if (<= phi2 1.15e-253)
      (* R_m (* lambda2 (- 1.0 (/ lambda1 lambda2))))
      (if (<= phi2 2.15e-185)
        (- (* R_m phi2) (* R_m phi1))
        (if (<= phi2 3.5e-5)
          (* lambda2 (- R_m (* lambda1 (/ R_m lambda2))))
          (* R_m (* phi2 (- 1.0 (/ 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 (phi2 <= -2.35e-211) {
		tmp = phi2 * (R_m - ((R_m * phi1) / phi2));
	} else if (phi2 <= 1.15e-253) {
		tmp = R_m * (lambda2 * (1.0 - (lambda1 / lambda2)));
	} else if (phi2 <= 2.15e-185) {
		tmp = (R_m * phi2) - (R_m * phi1);
	} else if (phi2 <= 3.5e-5) {
		tmp = lambda2 * (R_m - (lambda1 * (R_m / lambda2)));
	} else {
		tmp = R_m * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= (-2.35d-211)) then
        tmp = phi2 * (r_m - ((r_m * phi1) / phi2))
    else if (phi2 <= 1.15d-253) then
        tmp = r_m * (lambda2 * (1.0d0 - (lambda1 / lambda2)))
    else if (phi2 <= 2.15d-185) then
        tmp = (r_m * phi2) - (r_m * phi1)
    else if (phi2 <= 3.5d-5) then
        tmp = lambda2 * (r_m - (lambda1 * (r_m / lambda2)))
    else
        tmp = r_m * (phi2 * (1.0d0 - (phi1 / phi2)))
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= -2.35e-211) {
		tmp = phi2 * (R_m - ((R_m * phi1) / phi2));
	} else if (phi2 <= 1.15e-253) {
		tmp = R_m * (lambda2 * (1.0 - (lambda1 / lambda2)));
	} else if (phi2 <= 2.15e-185) {
		tmp = (R_m * phi2) - (R_m * phi1);
	} else if (phi2 <= 3.5e-5) {
		tmp = lambda2 * (R_m - (lambda1 * (R_m / lambda2)));
	} else {
		tmp = R_m * (phi2 * (1.0 - (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 phi2 <= -2.35e-211:
		tmp = phi2 * (R_m - ((R_m * phi1) / phi2))
	elif phi2 <= 1.15e-253:
		tmp = R_m * (lambda2 * (1.0 - (lambda1 / lambda2)))
	elif phi2 <= 2.15e-185:
		tmp = (R_m * phi2) - (R_m * phi1)
	elif phi2 <= 3.5e-5:
		tmp = lambda2 * (R_m - (lambda1 * (R_m / lambda2)))
	else:
		tmp = R_m * (phi2 * (1.0 - (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 (phi2 <= -2.35e-211)
		tmp = Float64(phi2 * Float64(R_m - Float64(Float64(R_m * phi1) / phi2)));
	elseif (phi2 <= 1.15e-253)
		tmp = Float64(R_m * Float64(lambda2 * Float64(1.0 - Float64(lambda1 / lambda2))));
	elseif (phi2 <= 2.15e-185)
		tmp = Float64(Float64(R_m * phi2) - Float64(R_m * phi1));
	elseif (phi2 <= 3.5e-5)
		tmp = Float64(lambda2 * Float64(R_m - Float64(lambda1 * Float64(R_m / lambda2))));
	else
		tmp = Float64(R_m * Float64(phi2 * Float64(1.0 - 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 (phi2 <= -2.35e-211)
		tmp = phi2 * (R_m - ((R_m * phi1) / phi2));
	elseif (phi2 <= 1.15e-253)
		tmp = R_m * (lambda2 * (1.0 - (lambda1 / lambda2)));
	elseif (phi2 <= 2.15e-185)
		tmp = (R_m * phi2) - (R_m * phi1);
	elseif (phi2 <= 3.5e-5)
		tmp = lambda2 * (R_m - (lambda1 * (R_m / lambda2)));
	else
		tmp = R_m * (phi2 * (1.0 - (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[phi2, -2.35e-211], N[(phi2 * N[(R$95$m - N[(N[(R$95$m * phi1), $MachinePrecision] / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.15e-253], N[(R$95$m * N[(lambda2 * N[(1.0 - N[(lambda1 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 2.15e-185], N[(N[(R$95$m * phi2), $MachinePrecision] - N[(R$95$m * phi1), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 3.5e-5], N[(lambda2 * N[(R$95$m - N[(lambda1 * N[(R$95$m / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if phi2 < -2.3499999999999998e-211

   1. Initial program 57.5%

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

      1. hypot-define 95.9%

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

   3. Simplified 95.9%

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

      1. add-sqr-sqrt 55.7%

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

      2. pow2 55.7%

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

   6. Applied egg-rr 55.7%

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

   7. Taylor expanded in phi2 around inf 17.2%

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

      1. neg-mul-1 17.2%

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

      2. sub-neg 17.2%

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

   9. Simplified 17.2%

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

   if -2.3499999999999998e-211 < phi2 < 1.15e-253

   1. Initial program 64.7%

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

      1. hypot-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in phi2 around 0 99.7%

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

   6. Taylor expanded in lambda2 around inf 35.1%

      \[\leadsto R \cdot \color{blue}{\left(\lambda_2 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) + -1 \cdot \frac{\lambda_1 \cdot \cos \left(0.5 \cdot \phi_1\right)}{\lambda_2}\right)\right)} \]
   7. Step-by-step derivation

      1. mul-1-neg 35.1%

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

      2. unsub-neg 35.1%

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

      3. associate-/l* 35.1%

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

   8. Simplified 35.1%

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

   9. Taylor expanded in phi1 around 0 30.1%

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

   if 1.15e-253 < phi2 < 2.15e-185

   1. Initial program 55.7%

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

      1. hypot-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi2 around inf 48.9%

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

      1. associate-*r/ 48.9%

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

      2. mul-1-neg 48.9%

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

      3. *-commutative 48.9%

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

   7. Simplified 48.9%

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

   8. Taylor expanded in phi2 around 0 56.0%

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

      1. +-commutative 56.0%

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

      2. mul-1-neg 56.0%

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

      3. unsub-neg 56.0%

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

   10. Simplified 56.0%

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

   if 2.15e-185 < phi2 < 3.4999999999999997e-5

   1. Initial program 52.4%

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

      1. hypot-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi2 around 0 99.8%

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

   6. Taylor expanded in phi1 around 0 36.4%

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

      1. unpow2 36.4%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_2 \cdot \phi_2} + {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      2. unpow2 36.4%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}} \]

      3. hypot-define 56.6%

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

   8. Simplified 56.6%

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

   9. Taylor expanded in lambda2 around inf 38.2%

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

       1. mul-1-neg 38.2%

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

       2. unsub-neg 38.2%

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

       3. *-commutative 38.2%

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

       4. associate-*r/ 34.9%

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

   11. Simplified 34.9%

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

   if 3.4999999999999997e-5 < phi2

   1. Initial program 47.7%

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

   3. Taylor expanded in phi2 around inf 65.0%

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

      1. mul-1-neg 65.0%

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

      2. unsub-neg 65.0%

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

   5. Simplified 65.0%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.35 \cdot 10^{-211}:\\ \;\;\;\;\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)\\ \mathbf{elif}\;\phi_2 \leq 1.15 \cdot 10^{-253}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 2.15 \cdot 10^{-185}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{elif}\;\phi_2 \leq 3.5 \cdot 10^{-5}:\\ \;\;\;\;\lambda_2 \cdot \left(R - \lambda_1 \cdot \frac{R}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 31.4% accurate, 11.3× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ \begin{array}{l} t_0 := \lambda_1 \cdot \left(R\_m \cdot \frac{\lambda_2}{\lambda_1} - R\_m\right)\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.5 \cdot 10^{-211}:\\ \;\;\;\;\phi_2 \cdot \left(R\_m - \frac{R\_m \cdot \phi_1}{\phi_2}\right)\\ \mathbf{elif}\;\phi_2 \leq 1.15 \cdot 10^{-253}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 6.5 \cdot 10^{-180}:\\ \;\;\;\;R\_m \cdot \phi_2 - R\_m \cdot \phi_1\\ \mathbf{elif}\;\phi_2 \leq 2.4 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \end{array} \]

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 (* lambda1 (- (* R_m (/ lambda2 lambda1)) R_m))))
   (*
    R_s
    (if (<= phi2 -3.5e-211)
      (* phi2 (- R_m (/ (* R_m phi1) phi2)))
      (if (<= phi2 1.15e-253)
        t_0
        (if (<= phi2 6.5e-180)
          (- (* R_m phi2) (* R_m phi1))
          (if (<= phi2 2.4e-7)
            t_0
            (* R_m (* phi2 (- 1.0 (/ 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 t_0 = lambda1 * ((R_m * (lambda2 / lambda1)) - R_m);
	double tmp;
	if (phi2 <= -3.5e-211) {
		tmp = phi2 * (R_m - ((R_m * phi1) / phi2));
	} else if (phi2 <= 1.15e-253) {
		tmp = t_0;
	} else if (phi2 <= 6.5e-180) {
		tmp = (R_m * phi2) - (R_m * phi1);
	} else if (phi2 <= 2.4e-7) {
		tmp = t_0;
	} else {
		tmp = R_m * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = lambda1 * ((r_m * (lambda2 / lambda1)) - r_m)
    if (phi2 <= (-3.5d-211)) then
        tmp = phi2 * (r_m - ((r_m * phi1) / phi2))
    else if (phi2 <= 1.15d-253) then
        tmp = t_0
    else if (phi2 <= 6.5d-180) then
        tmp = (r_m * phi2) - (r_m * phi1)
    else if (phi2 <= 2.4d-7) then
        tmp = t_0
    else
        tmp = r_m * (phi2 * (1.0d0 - (phi1 / phi2)))
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = lambda1 * ((R_m * (lambda2 / lambda1)) - R_m);
	double tmp;
	if (phi2 <= -3.5e-211) {
		tmp = phi2 * (R_m - ((R_m * phi1) / phi2));
	} else if (phi2 <= 1.15e-253) {
		tmp = t_0;
	} else if (phi2 <= 6.5e-180) {
		tmp = (R_m * phi2) - (R_m * phi1);
	} else if (phi2 <= 2.4e-7) {
		tmp = t_0;
	} else {
		tmp = R_m * (phi2 * (1.0 - (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):
	t_0 = lambda1 * ((R_m * (lambda2 / lambda1)) - R_m)
	tmp = 0
	if phi2 <= -3.5e-211:
		tmp = phi2 * (R_m - ((R_m * phi1) / phi2))
	elif phi2 <= 1.15e-253:
		tmp = t_0
	elif phi2 <= 6.5e-180:
		tmp = (R_m * phi2) - (R_m * phi1)
	elif phi2 <= 2.4e-7:
		tmp = t_0
	else:
		tmp = R_m * (phi2 * (1.0 - (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)
	t_0 = Float64(lambda1 * Float64(Float64(R_m * Float64(lambda2 / lambda1)) - R_m))
	tmp = 0.0
	if (phi2 <= -3.5e-211)
		tmp = Float64(phi2 * Float64(R_m - Float64(Float64(R_m * phi1) / phi2)));
	elseif (phi2 <= 1.15e-253)
		tmp = t_0;
	elseif (phi2 <= 6.5e-180)
		tmp = Float64(Float64(R_m * phi2) - Float64(R_m * phi1));
	elseif (phi2 <= 2.4e-7)
		tmp = t_0;
	else
		tmp = Float64(R_m * Float64(phi2 * Float64(1.0 - 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)
	t_0 = lambda1 * ((R_m * (lambda2 / lambda1)) - R_m);
	tmp = 0.0;
	if (phi2 <= -3.5e-211)
		tmp = phi2 * (R_m - ((R_m * phi1) / phi2));
	elseif (phi2 <= 1.15e-253)
		tmp = t_0;
	elseif (phi2 <= 6.5e-180)
		tmp = (R_m * phi2) - (R_m * phi1);
	elseif (phi2 <= 2.4e-7)
		tmp = t_0;
	else
		tmp = R_m * (phi2 * (1.0 - (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_] := Block[{t$95$0 = N[(lambda1 * N[(N[(R$95$m * N[(lambda2 / lambda1), $MachinePrecision]), $MachinePrecision] - R$95$m), $MachinePrecision]), $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi2, -3.5e-211], N[(phi2 * N[(R$95$m - N[(N[(R$95$m * phi1), $MachinePrecision] / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.15e-253], t$95$0, If[LessEqual[phi2, 6.5e-180], N[(N[(R$95$m * phi2), $MachinePrecision] - N[(R$95$m * phi1), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 2.4e-7], t$95$0, N[(R$95$m * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if phi2 < -3.5e-211

   1. Initial program 57.5%

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

      1. hypot-define 95.9%

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

   3. Simplified 95.9%

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

      1. add-sqr-sqrt 55.7%

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

      2. pow2 55.7%

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

   6. Applied egg-rr 55.7%

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

   7. Taylor expanded in phi2 around inf 17.2%

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

      1. neg-mul-1 17.2%

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

      2. sub-neg 17.2%

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

   9. Simplified 17.2%

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

   if -3.5e-211 < phi2 < 1.15e-253 or 6.50000000000000013e-180 < phi2 < 2.39999999999999979e-7

   1. Initial program 59.4%

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

      1. hypot-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in phi2 around 0 99.7%

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

   6. Taylor expanded in phi1 around 0 40.6%

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

      1. unpow2 40.6%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_2 \cdot \phi_2} + {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      2. unpow2 40.6%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}} \]

      3. hypot-define 62.1%

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

   8. Simplified 62.1%

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

   9. Taylor expanded in lambda1 around -inf 31.9%

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

       1. mul-1-neg 31.9%

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

       2. distribute-rgt-neg-in 31.9%

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

       3. mul-1-neg 31.9%

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

       4. unsub-neg 31.9%

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

       5. associate-/l* 31.6%

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

   11. Simplified 31.6%

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

   if 1.15e-253 < phi2 < 6.50000000000000013e-180

   1. Initial program 55.7%

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

      1. hypot-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi2 around inf 48.9%

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

      1. associate-*r/ 48.9%

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

      2. mul-1-neg 48.9%

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

      3. *-commutative 48.9%

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

   7. Simplified 48.9%

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

   8. Taylor expanded in phi2 around 0 56.0%

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

      1. +-commutative 56.0%

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

      2. mul-1-neg 56.0%

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

      3. unsub-neg 56.0%

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

   10. Simplified 56.0%

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

   if 2.39999999999999979e-7 < phi2

   1. Initial program 47.7%

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

   3. Taylor expanded in phi2 around inf 65.0%

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

      1. mul-1-neg 65.0%

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

      2. unsub-neg 65.0%

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

   5. Simplified 65.0%

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

4. Final simplification 35.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.5 \cdot 10^{-211}:\\ \;\;\;\;\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)\\ \mathbf{elif}\;\phi_2 \leq 1.15 \cdot 10^{-253}:\\ \;\;\;\;\lambda_1 \cdot \left(R \cdot \frac{\lambda_2}{\lambda_1} - R\right)\\ \mathbf{elif}\;\phi_2 \leq 6.5 \cdot 10^{-180}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \mathbf{elif}\;\phi_2 \leq 2.4 \cdot 10^{-7}:\\ \;\;\;\;\lambda_1 \cdot \left(R \cdot \frac{\lambda_2}{\lambda_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 26.5% accurate, 17.3× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ \begin{array}{l} t_0 := R\_m \cdot \left(-\lambda_1\right)\\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 6.8 \cdot 10^{-218}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 3.5 \cdot 10^{-140}:\\ \;\;\;\;R\_m \cdot \lambda_2\\ \mathbf{elif}\;\phi_2 \leq 0.55:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \phi_2\\ \end{array} \end{array} \end{array} \]

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 (* R_m (- lambda1))))
   (*
    R_s
    (if (<= phi2 6.8e-218)
      t_0
      (if (<= phi2 3.5e-140)
        (* R_m lambda2)
        (if (<= phi2 0.55) t_0 (* R_m 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 t_0 = R_m * -lambda1;
	double tmp;
	if (phi2 <= 6.8e-218) {
		tmp = t_0;
	} else if (phi2 <= 3.5e-140) {
		tmp = R_m * lambda2;
	} else if (phi2 <= 0.55) {
		tmp = t_0;
	} else {
		tmp = R_m * phi2;
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = r_m * -lambda1
    if (phi2 <= 6.8d-218) then
        tmp = t_0
    else if (phi2 <= 3.5d-140) then
        tmp = r_m * lambda2
    else if (phi2 <= 0.55d0) then
        tmp = t_0
    else
        tmp = r_m * phi2
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R_m * -lambda1;
	double tmp;
	if (phi2 <= 6.8e-218) {
		tmp = t_0;
	} else if (phi2 <= 3.5e-140) {
		tmp = R_m * lambda2;
	} else if (phi2 <= 0.55) {
		tmp = t_0;
	} else {
		tmp = R_m * 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):
	t_0 = R_m * -lambda1
	tmp = 0
	if phi2 <= 6.8e-218:
		tmp = t_0
	elif phi2 <= 3.5e-140:
		tmp = R_m * lambda2
	elif phi2 <= 0.55:
		tmp = t_0
	else:
		tmp = R_m * 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)
	t_0 = Float64(R_m * Float64(-lambda1))
	tmp = 0.0
	if (phi2 <= 6.8e-218)
		tmp = t_0;
	elseif (phi2 <= 3.5e-140)
		tmp = Float64(R_m * lambda2);
	elseif (phi2 <= 0.55)
		tmp = t_0;
	else
		tmp = Float64(R_m * 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)
	t_0 = R_m * -lambda1;
	tmp = 0.0;
	if (phi2 <= 6.8e-218)
		tmp = t_0;
	elseif (phi2 <= 3.5e-140)
		tmp = R_m * lambda2;
	elseif (phi2 <= 0.55)
		tmp = t_0;
	else
		tmp = R_m * 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_] := Block[{t$95$0 = N[(R$95$m * (-lambda1)), $MachinePrecision]}, N[(R$95$s * If[LessEqual[phi2, 6.8e-218], t$95$0, If[LessEqual[phi2, 3.5e-140], N[(R$95$m * lambda2), $MachinePrecision], If[LessEqual[phi2, 0.55], t$95$0, N[(R$95$m * phi2), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 6.79999999999999971e-218 or 3.4999999999999998e-140 < phi2 < 0.55000000000000004

   1. Initial program 58.5%

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

      1. hypot-define 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in phi2 around 0 90.9%

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

   6. Taylor expanded in phi1 around 0 42.2%

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

      1. unpow2 42.2%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_2 \cdot \phi_2} + {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      2. unpow2 42.2%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}} \]

      3. hypot-define 59.0%

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

   8. Simplified 59.0%

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

   9. Taylor expanded in lambda1 around -inf 17.2%

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

       1. mul-1-neg 17.2%

          \[\leadsto \color{blue}{-R \cdot \lambda_1} \]

       2. *-commutative 17.2%

          \[\leadsto -\color{blue}{\lambda_1 \cdot R} \]

       3. distribute-rgt-neg-in 17.2%

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

   11. Simplified 17.2%

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

   if 6.79999999999999971e-218 < phi2 < 3.4999999999999998e-140

   1. Initial program 52.2%

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

      1. hypot-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi2 around 0 99.8%

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

   6. Taylor expanded in phi1 around 0 39.5%

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

      1. unpow2 39.5%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_2 \cdot \phi_2} + {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      2. unpow2 39.5%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}} \]

      3. hypot-define 69.7%

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

   8. Simplified 69.7%

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

   9. Taylor expanded in lambda2 around inf 20.0%

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

       1. *-commutative 20.0%

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

   11. Simplified 20.0%

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

   if 0.55000000000000004 < phi2

   1. Initial program 47.7%

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

      1. hypot-define 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in phi2 around inf 59.7%

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

      1. *-commutative 59.7%

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

   7. Simplified 59.7%

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

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 6.8 \cdot 10^{-218}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 3.5 \cdot 10^{-140}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_2 \leq 0.55:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 33.7% accurate, 27.4× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3 \cdot 10^{+175}:\\ \;\;\;\;R\_m \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \phi_2 - R\_m \cdot \phi_1\\ \end{array} \end{array} \]

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 -3e+175) (* R_m (- lambda1)) (- (* R_m phi2) (* R_m phi1)))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -3e+175) {
		tmp = R_m * -lambda1;
	} else {
		tmp = (R_m * phi2) - (R_m * phi1);
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-3d+175)) then
        tmp = r_m * -lambda1
    else
        tmp = (r_m * phi2) - (r_m * phi1)
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -3e+175) {
		tmp = R_m * -lambda1;
	} else {
		tmp = (R_m * phi2) - (R_m * phi1);
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -3e+175:
		tmp = R_m * -lambda1
	else:
		tmp = (R_m * phi2) - (R_m * phi1)
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -3e+175)
		tmp = Float64(R_m * Float64(-lambda1));
	else
		tmp = Float64(Float64(R_m * phi2) - Float64(R_m * phi1));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -3e+175)
		tmp = R_m * -lambda1;
	else
		tmp = (R_m * phi2) - (R_m * phi1);
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda1, -3e+175], N[(R$95$m * (-lambda1)), $MachinePrecision], N[(N[(R$95$m * phi2), $MachinePrecision] - N[(R$95$m * phi1), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -3.0000000000000002e175

   1. Initial program 34.3%

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

      1. hypot-define 92.6%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in phi2 around 0 78.5%

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

   6. Taylor expanded in phi1 around 0 34.3%

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

      1. unpow2 34.3%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_2 \cdot \phi_2} + {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      2. unpow2 34.3%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}} \]

      3. hypot-define 62.8%

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

   8. Simplified 62.8%

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

   9. Taylor expanded in lambda1 around -inf 57.0%

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

       1. mul-1-neg 57.0%

          \[\leadsto \color{blue}{-R \cdot \lambda_1} \]

       2. *-commutative 57.0%

          \[\leadsto -\color{blue}{\lambda_1 \cdot R} \]

       3. distribute-rgt-neg-in 57.0%

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

   11. Simplified 57.0%

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

   if -3.0000000000000002e175 < lambda1

   1. Initial program 58.1%

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

      1. hypot-define 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in phi2 around inf 32.3%

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

      1. associate-*r/ 32.3%

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

      2. mul-1-neg 32.3%

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

      3. *-commutative 32.3%

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

   7. Simplified 32.3%

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

   8. Taylor expanded in phi2 around 0 33.1%

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

      1. +-commutative 33.1%

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

      2. mul-1-neg 33.1%

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

      3. unsub-neg 33.1%

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

   10. Simplified 33.1%

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

4. Final simplification 35.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3 \cdot 10^{+175}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2 - R \cdot \phi_1\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 26.7% accurate, 41.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1:\\ \;\;\;\;R\_m \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \phi_2\\ \end{array} \end{array} \]

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 1.0) (* R_m lambda2) (* R_m phi2))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if phi2 < 1

   1. Initial program 58.1%

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

      1. hypot-define 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in phi2 around 0 91.6%

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

   6. Taylor expanded in phi1 around 0 42.0%

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

      1. unpow2 42.0%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_2 \cdot \phi_2} + {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      2. unpow2 42.0%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}} \]

      3. hypot-define 59.8%

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

   8. Simplified 59.8%

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

   9. Taylor expanded in lambda2 around inf 14.7%

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

       1. *-commutative 14.7%

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

   11. Simplified 14.7%

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

   if 1 < phi2

   1. Initial program 47.7%

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

      1. hypot-define 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in phi2 around inf 59.7%

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

      1. *-commutative 59.7%

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

   7. Simplified 59.7%

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

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 14.6% accurate, 109.7× speedup

Math

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

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 lambda2)))

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 * lambda2);
}

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 * (r_m * lambda2)
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 * (R_m * lambda2);
}

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 * lambda2)

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 * lambda2))
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 * lambda2);
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 * lambda2), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.4%

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

   1. hypot-define 96.5%

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

3. Simplified 96.5%

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

5. Taylor expanded in phi2 around 0 86.8%

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

6. Taylor expanded in phi1 around 0 41.2%

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

   1. unpow2 41.2%

      \[\leadsto R \cdot \sqrt{\color{blue}{\phi_2 \cdot \phi_2} + {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

   2. unpow2 41.2%

      \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}} \]

   3. hypot-define 61.0%

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

8. Simplified 61.0%

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

9. Taylor expanded in lambda2 around inf 12.7%

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

    1. *-commutative 12.7%

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

11. Simplified 12.7%

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

12. Final simplification 12.7%

    \[\leadsto R \cdot \lambda_2 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024059 
(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))))))