Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3

Percentage Accurate: 50.9% → 81.7%

Time: 9.8s

Alternatives: 7

Speedup: 6.2×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \frac{x \cdot x - t_0}{x \cdot x + t_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y 4.0) y))) (/ (- (* x x) t_0) (+ (* x x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = (y * 4.0d0) * y
    code = ((x * x) - t_0) / ((x * x) + t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Python

def code(x, y):
	t_0 = (y * 4.0) * y
	return ((x * x) - t_0) / ((x * x) + t_0)

Julia

function code(x, y)
	t_0 = Float64(Float64(y * 4.0) * y)
	return Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = (y * 4.0) * y;
	tmp = ((x * x) - t_0) / ((x * x) + t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 50.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \frac{x \cdot x - t_0}{x \cdot x + t_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y 4.0) y))) (/ (- (* x x) t_0) (+ (* x x) t_0))))

C

double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    t_0 = (y * 4.0d0) * y
    code = ((x * x) - t_0) / ((x * x) + t_0)
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * 4.0) * y;
	return ((x * x) - t_0) / ((x * x) + t_0);
}

Python

def code(x, y):
	t_0 = (y * 4.0) * y
	return ((x * x) - t_0) / ((x * x) + t_0)

Julia

function code(x, y)
	t_0 = Float64(Float64(y * 4.0) * y)
	return Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
end

MATLAB

function tmp = code(x, y)
	t_0 = (y * 4.0) * y;
	tmp = ((x * x) - t_0) / ((x * x) + t_0);
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 81.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-305}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+223}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot -4\right)}{\mathsf{fma}\left(4, y \cdot y, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \frac{\frac{y}{x}}{\frac{x}{y}}, 1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 2e-305)
   -1.0
   (if (<= (* x x) 5e+223)
     (/ (fma x x (* (* y y) -4.0)) (fma 4.0 (* y y) (* x x)))
     (+ 1.0 (* -8.0 (* 2.0 (log (fma 0.5 (/ (/ y x) (/ x y)) 1.0))))))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2e-305) {
		tmp = -1.0;
	} else if ((x * x) <= 5e+223) {
		tmp = fma(x, x, ((y * y) * -4.0)) / fma(4.0, (y * y), (x * x));
	} else {
		tmp = 1.0 + (-8.0 * (2.0 * log(fma(0.5, ((y / x) / (x / y)), 1.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 2e-305)
		tmp = -1.0;
	elseif (Float64(x * x) <= 5e+223)
		tmp = Float64(fma(x, x, Float64(Float64(y * y) * -4.0)) / fma(4.0, Float64(y * y), Float64(x * x)));
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(2.0 * log(fma(0.5, Float64(Float64(y / x) / Float64(x / y)), 1.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-305], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 5e+223], N[(N[(x * x + N[(N[(y * y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] / N[(4.0 * N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-8.0 * N[(2.0 * N[Log[N[(0.5 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1.99999999999999999e-305

   1. Initial program 42.9%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around 0 87.2%

      \[\leadsto \color{blue}{-1} \]

   if 1.99999999999999999e-305 < (*.f64 x x) < 4.99999999999999985e223

   1. Initial program 80.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Step-by-step derivation

      1. fma-neg 80.5%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, x, -\left(y \cdot 4\right) \cdot y\right)}}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      2. *-commutative 80.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, -\color{blue}{y \cdot \left(y \cdot 4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      3. associate-*r* 80.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, -\color{blue}{\left(y \cdot y\right) \cdot 4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      4. distribute-rgt-neg-in 80.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \color{blue}{\left(y \cdot y\right) \cdot \left(-4\right)}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      5. metadata-eval 80.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot \color{blue}{-4}\right)}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

      6. +-commutative 80.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot -4\right)}{\color{blue}{\left(y \cdot 4\right) \cdot y + x \cdot x}} \]

      7. *-commutative 80.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot -4\right)}{\color{blue}{\left(4 \cdot y\right)} \cdot y + x \cdot x} \]

      8. associate-*l* 80.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot -4\right)}{\color{blue}{4 \cdot \left(y \cdot y\right)} + x \cdot x} \]

      9. fma-def 80.5%

         \[\leadsto \frac{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot -4\right)}{\color{blue}{\mathsf{fma}\left(4, y \cdot y, x \cdot x\right)}} \]

   3. Simplified 80.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot -4\right)}{\mathsf{fma}\left(4, y \cdot y, x \cdot x\right)}} \]

   if 4.99999999999999985e223 < (*.f64 x x)

   1. Initial program 16.3%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in y around 0 70.2%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. add-log-exp 70.2%

         \[\leadsto 1 + -8 \cdot \color{blue}{\log \left(e^{\frac{{y}^{2}}{{x}^{2}}}\right)} \]

      2. add-sqr-sqrt 70.2%

         \[\leadsto 1 + -8 \cdot \log \color{blue}{\left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}} \cdot \sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)} \]

      3. log-prod 70.2%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(\log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right)} \]

      4. add-sqr-sqrt 70.2%

         \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{\color{blue}{\sqrt{\frac{{y}^{2}}{{x}^{2}}} \cdot \sqrt{\frac{{y}^{2}}{{x}^{2}}}}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      5. pow2 70.2%

         \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{\color{blue}{{\left(\sqrt{\frac{{y}^{2}}{{x}^{2}}}\right)}^{2}}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      6. sqrt-div 70.2%

         \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\color{blue}{\left(\frac{\sqrt{{y}^{2}}}{\sqrt{{x}^{2}}}\right)}}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      7. unpow2 70.2%

         \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\left(\frac{\sqrt{\color{blue}{y \cdot y}}}{\sqrt{{x}^{2}}}\right)}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      8. sqrt-prod 40.0%

         \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\left(\frac{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}{\sqrt{{x}^{2}}}\right)}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      9. add-sqr-sqrt 70.2%

         \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\left(\frac{\color{blue}{y}}{\sqrt{{x}^{2}}}\right)}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      10. unpow2 70.2%

          \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\left(\frac{y}{\sqrt{\color{blue}{x \cdot x}}}\right)}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      11. sqrt-prod 41.3%

          \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\left(\frac{y}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      12. add-sqr-sqrt 70.2%

          \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\left(\frac{y}{\color{blue}{x}}\right)}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

   4. Applied egg-rr 86.0%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\log \left(\sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}}\right) + \log \left(\sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}}\right)\right)} \]
   5. Step-by-step derivation

      1. count-2 86.0%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}}\right)\right)} \]

   6. Simplified 86.0%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}}\right)\right)} \]

   7. Taylor expanded in y around 0 70.2%

      \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \color{blue}{\left(1 + 0.5 \cdot \frac{{y}^{2}}{{x}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative 70.2%

         \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \color{blue}{\left(0.5 \cdot \frac{{y}^{2}}{{x}^{2}} + 1\right)}\right) \]

      2. fma-def 70.2%

         \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \color{blue}{\left(\mathsf{fma}\left(0.5, \frac{{y}^{2}}{{x}^{2}}, 1\right)\right)}\right) \]

      3. unpow2 70.2%

         \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \frac{\color{blue}{y \cdot y}}{{x}^{2}}, 1\right)\right)\right) \]

      4. unpow2 70.2%

         \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \frac{y \cdot y}{\color{blue}{x \cdot x}}, 1\right)\right)\right) \]

      5. times-frac 87.4%

         \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \color{blue}{\frac{y}{x} \cdot \frac{y}{x}}, 1\right)\right)\right) \]

      6. unpow2 87.4%

         \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \color{blue}{{\left(\frac{y}{x}\right)}^{2}}, 1\right)\right)\right) \]

   9. Simplified 87.4%

      \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \color{blue}{\left(\mathsf{fma}\left(0.5, {\left(\frac{y}{x}\right)}^{2}, 1\right)\right)}\right) \]
   10. Step-by-step derivation

       1. pow2 87.4%

          \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \color{blue}{\frac{y}{x} \cdot \frac{y}{x}}, 1\right)\right)\right) \]

       2. clear-num 87.4%

          \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \frac{y}{x} \cdot \color{blue}{\frac{1}{\frac{x}{y}}}, 1\right)\right)\right) \]

       3. un-div-inv 87.4%

          \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}}, 1\right)\right)\right) \]

   11. Applied egg-rr 87.4%

       \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}}, 1\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-305}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+223}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot -4\right)}{\mathsf{fma}\left(4, y \cdot y, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \frac{\frac{y}{x}}{\frac{x}{y}}, 1\right)\right)\right)\\ \end{array} \]

Alternative 2: 81.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-305}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+223}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \frac{\frac{y}{x}}{\frac{x}{y}}, 1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 2e-305)
     -1.0
     (if (<= (* x x) 5e+223)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (+ 1.0 (* -8.0 (* 2.0 (log (fma 0.5 (/ (/ y x) (/ x y)) 1.0)))))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 2e-305) {
		tmp = -1.0;
	} else if ((x * x) <= 5e+223) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (-8.0 * (2.0 * log(fma(0.5, ((y / x) / (x / y)), 1.0))));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 2e-305)
		tmp = -1.0;
	elseif (Float64(x * x) <= 5e+223)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(2.0 * log(fma(0.5, Float64(Float64(y / x) / Float64(x / y)), 1.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 2e-305], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 5e+223], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-8.0 * N[(2.0 * N[Log[N[(0.5 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1.99999999999999999e-305

   1. Initial program 42.9%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around 0 87.2%

      \[\leadsto \color{blue}{-1} \]

   if 1.99999999999999999e-305 < (*.f64 x x) < 4.99999999999999985e223

   1. Initial program 80.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   if 4.99999999999999985e223 < (*.f64 x x)

   1. Initial program 16.3%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in y around 0 70.2%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. add-log-exp 70.2%

         \[\leadsto 1 + -8 \cdot \color{blue}{\log \left(e^{\frac{{y}^{2}}{{x}^{2}}}\right)} \]

      2. add-sqr-sqrt 70.2%

         \[\leadsto 1 + -8 \cdot \log \color{blue}{\left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}} \cdot \sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)} \]

      3. log-prod 70.2%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(\log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right)} \]

      4. add-sqr-sqrt 70.2%

         \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{\color{blue}{\sqrt{\frac{{y}^{2}}{{x}^{2}}} \cdot \sqrt{\frac{{y}^{2}}{{x}^{2}}}}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      5. pow2 70.2%

         \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{\color{blue}{{\left(\sqrt{\frac{{y}^{2}}{{x}^{2}}}\right)}^{2}}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      6. sqrt-div 70.2%

         \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\color{blue}{\left(\frac{\sqrt{{y}^{2}}}{\sqrt{{x}^{2}}}\right)}}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      7. unpow2 70.2%

         \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\left(\frac{\sqrt{\color{blue}{y \cdot y}}}{\sqrt{{x}^{2}}}\right)}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      8. sqrt-prod 40.0%

         \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\left(\frac{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}{\sqrt{{x}^{2}}}\right)}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      9. add-sqr-sqrt 70.2%

         \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\left(\frac{\color{blue}{y}}{\sqrt{{x}^{2}}}\right)}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      10. unpow2 70.2%

          \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\left(\frac{y}{\sqrt{\color{blue}{x \cdot x}}}\right)}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      11. sqrt-prod 41.3%

          \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\left(\frac{y}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      12. add-sqr-sqrt 70.2%

          \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\left(\frac{y}{\color{blue}{x}}\right)}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

   4. Applied egg-rr 86.0%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\log \left(\sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}}\right) + \log \left(\sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}}\right)\right)} \]
   5. Step-by-step derivation

      1. count-2 86.0%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}}\right)\right)} \]

   6. Simplified 86.0%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}}\right)\right)} \]

   7. Taylor expanded in y around 0 70.2%

      \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \color{blue}{\left(1 + 0.5 \cdot \frac{{y}^{2}}{{x}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative 70.2%

         \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \color{blue}{\left(0.5 \cdot \frac{{y}^{2}}{{x}^{2}} + 1\right)}\right) \]

      2. fma-def 70.2%

         \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \color{blue}{\left(\mathsf{fma}\left(0.5, \frac{{y}^{2}}{{x}^{2}}, 1\right)\right)}\right) \]

      3. unpow2 70.2%

         \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \frac{\color{blue}{y \cdot y}}{{x}^{2}}, 1\right)\right)\right) \]

      4. unpow2 70.2%

         \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \frac{y \cdot y}{\color{blue}{x \cdot x}}, 1\right)\right)\right) \]

      5. times-frac 87.4%

         \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \color{blue}{\frac{y}{x} \cdot \frac{y}{x}}, 1\right)\right)\right) \]

      6. unpow2 87.4%

         \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \color{blue}{{\left(\frac{y}{x}\right)}^{2}}, 1\right)\right)\right) \]

   9. Simplified 87.4%

      \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \color{blue}{\left(\mathsf{fma}\left(0.5, {\left(\frac{y}{x}\right)}^{2}, 1\right)\right)}\right) \]
   10. Step-by-step derivation

       1. pow2 87.4%

          \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \color{blue}{\frac{y}{x} \cdot \frac{y}{x}}, 1\right)\right)\right) \]

       2. clear-num 87.4%

          \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \frac{y}{x} \cdot \color{blue}{\frac{1}{\frac{x}{y}}}, 1\right)\right)\right) \]

       3. un-div-inv 87.4%

          \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}}, 1\right)\right)\right) \]

   11. Applied egg-rr 87.4%

       \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}}, 1\right)\right)\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-305}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+223}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \frac{\frac{y}{x}}{\frac{x}{y}}, 1\right)\right)\right)\\ \end{array} \]

Alternative 3: 81.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-305}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+223}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;1 + -16 \cdot \mathsf{log1p}\left(0.5 \cdot {\left(\frac{y}{x}\right)}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 2e-305)
     -1.0
     (if (<= (* x x) 5e+223)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (+ 1.0 (* -16.0 (log1p (* 0.5 (pow (/ y x) 2.0)))))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 2e-305) {
		tmp = -1.0;
	} else if ((x * x) <= 5e+223) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (-16.0 * log1p((0.5 * pow((y / x), 2.0))));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 2e-305) {
		tmp = -1.0;
	} else if ((x * x) <= 5e+223) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (-16.0 * Math.log1p((0.5 * Math.pow((y / x), 2.0))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if (x * x) <= 2e-305:
		tmp = -1.0
	elif (x * x) <= 5e+223:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	else:
		tmp = 1.0 + (-16.0 * math.log1p((0.5 * math.pow((y / x), 2.0))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 2e-305)
		tmp = -1.0;
	elseif (Float64(x * x) <= 5e+223)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = Float64(1.0 + Float64(-16.0 * log1p(Float64(0.5 * (Float64(y / x) ^ 2.0)))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 2e-305], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 5e+223], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-16.0 * N[Log[1 + N[(0.5 * N[Power[N[(y / x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1.99999999999999999e-305

   1. Initial program 42.9%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around 0 87.2%

      \[\leadsto \color{blue}{-1} \]

   if 1.99999999999999999e-305 < (*.f64 x x) < 4.99999999999999985e223

   1. Initial program 80.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   if 4.99999999999999985e223 < (*.f64 x x)

   1. Initial program 16.3%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in y around 0 70.2%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. add-log-exp 70.2%

         \[\leadsto 1 + -8 \cdot \color{blue}{\log \left(e^{\frac{{y}^{2}}{{x}^{2}}}\right)} \]

      2. add-sqr-sqrt 70.2%

         \[\leadsto 1 + -8 \cdot \log \color{blue}{\left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}} \cdot \sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)} \]

      3. log-prod 70.2%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(\log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right)} \]

      4. add-sqr-sqrt 70.2%

         \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{\color{blue}{\sqrt{\frac{{y}^{2}}{{x}^{2}}} \cdot \sqrt{\frac{{y}^{2}}{{x}^{2}}}}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      5. pow2 70.2%

         \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{\color{blue}{{\left(\sqrt{\frac{{y}^{2}}{{x}^{2}}}\right)}^{2}}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      6. sqrt-div 70.2%

         \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\color{blue}{\left(\frac{\sqrt{{y}^{2}}}{\sqrt{{x}^{2}}}\right)}}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      7. unpow2 70.2%

         \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\left(\frac{\sqrt{\color{blue}{y \cdot y}}}{\sqrt{{x}^{2}}}\right)}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      8. sqrt-prod 40.0%

         \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\left(\frac{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}{\sqrt{{x}^{2}}}\right)}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      9. add-sqr-sqrt 70.2%

         \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\left(\frac{\color{blue}{y}}{\sqrt{{x}^{2}}}\right)}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      10. unpow2 70.2%

          \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\left(\frac{y}{\sqrt{\color{blue}{x \cdot x}}}\right)}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      11. sqrt-prod 41.3%

          \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\left(\frac{y}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      12. add-sqr-sqrt 70.2%

          \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\left(\frac{y}{\color{blue}{x}}\right)}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

   4. Applied egg-rr 86.0%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\log \left(\sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}}\right) + \log \left(\sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}}\right)\right)} \]
   5. Step-by-step derivation

      1. count-2 86.0%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}}\right)\right)} \]

   6. Simplified 86.0%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}}\right)\right)} \]

   7. Taylor expanded in y around 0 70.2%

      \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \color{blue}{\left(1 + 0.5 \cdot \frac{{y}^{2}}{{x}^{2}}\right)}\right) \]
   8. Step-by-step derivation

      1. +-commutative 70.2%

         \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \color{blue}{\left(0.5 \cdot \frac{{y}^{2}}{{x}^{2}} + 1\right)}\right) \]

      2. fma-def 70.2%

         \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \color{blue}{\left(\mathsf{fma}\left(0.5, \frac{{y}^{2}}{{x}^{2}}, 1\right)\right)}\right) \]

      3. unpow2 70.2%

         \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \frac{\color{blue}{y \cdot y}}{{x}^{2}}, 1\right)\right)\right) \]

      4. unpow2 70.2%

         \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \frac{y \cdot y}{\color{blue}{x \cdot x}}, 1\right)\right)\right) \]

      5. times-frac 87.4%

         \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \color{blue}{\frac{y}{x} \cdot \frac{y}{x}}, 1\right)\right)\right) \]

      6. unpow2 87.4%

         \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \color{blue}{{\left(\frac{y}{x}\right)}^{2}}, 1\right)\right)\right) \]

   9. Simplified 87.4%

      \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \color{blue}{\left(\mathsf{fma}\left(0.5, {\left(\frac{y}{x}\right)}^{2}, 1\right)\right)}\right) \]
   10. Step-by-step derivation

       1. expm1-log1p-u 85.6%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, {\left(\frac{y}{x}\right)}^{2}, 1\right)\right)\right)\right)\right)} \]

       2. expm1-udef 85.6%

          \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(-8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, {\left(\frac{y}{x}\right)}^{2}, 1\right)\right)\right)\right)} - 1\right)} \]

       3. associate-*r* 85.6%

          \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(-8 \cdot 2\right) \cdot \log \left(\mathsf{fma}\left(0.5, {\left(\frac{y}{x}\right)}^{2}, 1\right)\right)}\right)} - 1\right) \]

       4. *-commutative 85.6%

          \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{\log \left(\mathsf{fma}\left(0.5, {\left(\frac{y}{x}\right)}^{2}, 1\right)\right) \cdot \left(-8 \cdot 2\right)}\right)} - 1\right) \]

       5. fma-udef 85.6%

          \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\log \color{blue}{\left(0.5 \cdot {\left(\frac{y}{x}\right)}^{2} + 1\right)} \cdot \left(-8 \cdot 2\right)\right)} - 1\right) \]

       6. +-commutative 85.6%

          \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\log \color{blue}{\left(1 + 0.5 \cdot {\left(\frac{y}{x}\right)}^{2}\right)} \cdot \left(-8 \cdot 2\right)\right)} - 1\right) \]

       7. log1p-udef 85.6%

          \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{log1p}\left(0.5 \cdot {\left(\frac{y}{x}\right)}^{2}\right)} \cdot \left(-8 \cdot 2\right)\right)} - 1\right) \]

       8. metadata-eval 85.6%

          \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\mathsf{log1p}\left(0.5 \cdot {\left(\frac{y}{x}\right)}^{2}\right) \cdot \color{blue}{-16}\right)} - 1\right) \]

   11. Applied egg-rr 85.6%

       \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{log1p}\left(0.5 \cdot {\left(\frac{y}{x}\right)}^{2}\right) \cdot -16\right)} - 1\right)} \]
   12. Step-by-step derivation

       1. expm1-def 85.6%

          \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(0.5 \cdot {\left(\frac{y}{x}\right)}^{2}\right) \cdot -16\right)\right)} \]

       2. expm1-log1p 87.4%

          \[\leadsto 1 + \color{blue}{\mathsf{log1p}\left(0.5 \cdot {\left(\frac{y}{x}\right)}^{2}\right) \cdot -16} \]

       3. *-commutative 87.4%

          \[\leadsto 1 + \color{blue}{-16 \cdot \mathsf{log1p}\left(0.5 \cdot {\left(\frac{y}{x}\right)}^{2}\right)} \]

   13. Simplified 87.4%

       \[\leadsto 1 + \color{blue}{-16 \cdot \mathsf{log1p}\left(0.5 \cdot {\left(\frac{y}{x}\right)}^{2}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-305}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+223}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -16 \cdot \mathsf{log1p}\left(0.5 \cdot {\left(\frac{y}{x}\right)}^{2}\right)\\ \end{array} \]

Alternative 4: 81.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-305}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+223}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-8 \cdot \left(1 + \frac{\frac{y}{x}}{\frac{x}{y}}\right) + 8\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 2e-305)
     -1.0
     (if (<= (* x x) 5e+223)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (+ 1.0 (+ (* -8.0 (+ 1.0 (/ (/ y x) (/ x y)))) 8.0))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 2e-305) {
		tmp = -1.0;
	} else if ((x * x) <= 5e+223) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + ((-8.0 * (1.0 + ((y / x) / (x / y)))) + 8.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    if ((x * x) <= 2d-305) then
        tmp = -1.0d0
    else if ((x * x) <= 5d+223) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else
        tmp = 1.0d0 + (((-8.0d0) * (1.0d0 + ((y / x) / (x / y)))) + 8.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 2e-305) {
		tmp = -1.0;
	} else if ((x * x) <= 5e+223) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + ((-8.0 * (1.0 + ((y / x) / (x / y)))) + 8.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if (x * x) <= 2e-305:
		tmp = -1.0
	elif (x * x) <= 5e+223:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	else:
		tmp = 1.0 + ((-8.0 * (1.0 + ((y / x) / (x / y)))) + 8.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 2e-305)
		tmp = -1.0;
	elseif (Float64(x * x) <= 5e+223)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = Float64(1.0 + Float64(Float64(-8.0 * Float64(1.0 + Float64(Float64(y / x) / Float64(x / y)))) + 8.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if ((x * x) <= 2e-305)
		tmp = -1.0;
	elseif ((x * x) <= 5e+223)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	else
		tmp = 1.0 + ((-8.0 * (1.0 + ((y / x) / (x / y)))) + 8.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 2e-305], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 5e+223], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(-8.0 * N[(1.0 + N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 8.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1.99999999999999999e-305

   1. Initial program 42.9%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around 0 87.2%

      \[\leadsto \color{blue}{-1} \]

   if 1.99999999999999999e-305 < (*.f64 x x) < 4.99999999999999985e223

   1. Initial program 80.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   if 4.99999999999999985e223 < (*.f64 x x)

   1. Initial program 16.3%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in y around 0 70.2%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. add-log-exp 70.2%

         \[\leadsto 1 + -8 \cdot \color{blue}{\log \left(e^{\frac{{y}^{2}}{{x}^{2}}}\right)} \]

      2. add-sqr-sqrt 70.2%

         \[\leadsto 1 + -8 \cdot \log \color{blue}{\left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}} \cdot \sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)} \]

      3. log-prod 70.2%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(\log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right)} \]

      4. add-sqr-sqrt 70.2%

         \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{\color{blue}{\sqrt{\frac{{y}^{2}}{{x}^{2}}} \cdot \sqrt{\frac{{y}^{2}}{{x}^{2}}}}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      5. pow2 70.2%

         \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{\color{blue}{{\left(\sqrt{\frac{{y}^{2}}{{x}^{2}}}\right)}^{2}}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      6. sqrt-div 70.2%

         \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\color{blue}{\left(\frac{\sqrt{{y}^{2}}}{\sqrt{{x}^{2}}}\right)}}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      7. unpow2 70.2%

         \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\left(\frac{\sqrt{\color{blue}{y \cdot y}}}{\sqrt{{x}^{2}}}\right)}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      8. sqrt-prod 40.0%

         \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\left(\frac{\color{blue}{\sqrt{y} \cdot \sqrt{y}}}{\sqrt{{x}^{2}}}\right)}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      9. add-sqr-sqrt 70.2%

         \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\left(\frac{\color{blue}{y}}{\sqrt{{x}^{2}}}\right)}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      10. unpow2 70.2%

          \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\left(\frac{y}{\sqrt{\color{blue}{x \cdot x}}}\right)}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      11. sqrt-prod 41.3%

          \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\left(\frac{y}{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right)}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

      12. add-sqr-sqrt 70.2%

          \[\leadsto 1 + -8 \cdot \left(\log \left(\sqrt{e^{{\left(\frac{y}{\color{blue}{x}}\right)}^{2}}}\right) + \log \left(\sqrt{e^{\frac{{y}^{2}}{{x}^{2}}}}\right)\right) \]

   4. Applied egg-rr 86.0%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\log \left(\sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}}\right) + \log \left(\sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}}\right)\right)} \]
   5. Step-by-step derivation

      1. count-2 86.0%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}}\right)\right)} \]

   6. Simplified 86.0%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(2 \cdot \log \left(\sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}}\right)\right)} \]
   7. Step-by-step derivation

      1. add-log-exp 86.0%

         \[\leadsto 1 + -8 \cdot \color{blue}{\log \left(e^{2 \cdot \log \left(\sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}}\right)}\right)} \]

      2. *-commutative 86.0%

         \[\leadsto 1 + -8 \cdot \log \left(e^{\color{blue}{\log \left(\sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}}\right) \cdot 2}}\right) \]

      3. exp-to-pow 86.0%

         \[\leadsto 1 + -8 \cdot \log \color{blue}{\left({\left(\sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}}\right)}^{2}\right)} \]

      4. pow2 86.0%

         \[\leadsto 1 + -8 \cdot \log \color{blue}{\left(\sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}} \cdot \sqrt{e^{{\left(\frac{y}{x}\right)}^{2}}}\right)} \]

      5. add-sqr-sqrt 86.0%

         \[\leadsto 1 + -8 \cdot \log \color{blue}{\left(e^{{\left(\frac{y}{x}\right)}^{2}}\right)} \]

      6. expm1-log1p-u 86.0%

         \[\leadsto 1 + -8 \cdot \log \left(e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)\right)}}\right) \]

      7. expm1-def 86.0%

         \[\leadsto 1 + -8 \cdot \log \left(e^{\color{blue}{e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)} - 1}}\right) \]

      8. add-log-exp 86.4%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)} - 1\right)} \]

      9. sub-neg 86.4%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)} + \left(-1\right)\right)} \]

      10. distribute-rgt-in 86.4%

          \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)} \cdot -8 + \left(-1\right) \cdot -8\right)} \]

      11. log1p-udef 86.4%

          \[\leadsto 1 + \left(e^{\color{blue}{\log \left(1 + {\left(\frac{y}{x}\right)}^{2}\right)}} \cdot -8 + \left(-1\right) \cdot -8\right) \]

      12. rem-exp-log 86.4%

          \[\leadsto 1 + \left(\color{blue}{\left(1 + {\left(\frac{y}{x}\right)}^{2}\right)} \cdot -8 + \left(-1\right) \cdot -8\right) \]

      13. +-commutative 86.4%

          \[\leadsto 1 + \left(\color{blue}{\left({\left(\frac{y}{x}\right)}^{2} + 1\right)} \cdot -8 + \left(-1\right) \cdot -8\right) \]

      14. metadata-eval 86.4%

          \[\leadsto 1 + \left(\left({\left(\frac{y}{x}\right)}^{2} + 1\right) \cdot -8 + \color{blue}{-1} \cdot -8\right) \]

   8. Applied egg-rr 86.4%

      \[\leadsto 1 + \color{blue}{\left(\left({\left(\frac{y}{x}\right)}^{2} + 1\right) \cdot -8 + 8\right)} \]
   9. Step-by-step derivation

      1. pow2 87.4%

         \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \color{blue}{\frac{y}{x} \cdot \frac{y}{x}}, 1\right)\right)\right) \]

      2. clear-num 87.4%

         \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \frac{y}{x} \cdot \color{blue}{\frac{1}{\frac{x}{y}}}, 1\right)\right)\right) \]

      3. un-div-inv 87.4%

         \[\leadsto 1 + -8 \cdot \left(2 \cdot \log \left(\mathsf{fma}\left(0.5, \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}}, 1\right)\right)\right) \]

   10. Applied egg-rr 86.4%

       \[\leadsto 1 + \left(\left(\color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} + 1\right) \cdot -8 + 8\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-305}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+223}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(-8 \cdot \left(1 + \frac{\frac{y}{x}}{\frac{x}{y}}\right) + 8\right)\\ \end{array} \]

Alternative 5: 62.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{-67}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 3.2e-67) -1.0 (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 3.2e-67) {
		tmp = -1.0;
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 3.2d-67) then
        tmp = -1.0d0
    else
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 3.2e-67) {
		tmp = -1.0;
	} else {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 3.2e-67:
		tmp = -1.0
	else:
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 3.2e-67)
		tmp = -1.0;
	else
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 3.2e-67)
		tmp = -1.0;
	else
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 3.2e-67], -1.0, N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.20000000000000021e-67

   1. Initial program 51.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around 0 56.4%

      \[\leadsto \color{blue}{-1} \]

   if 3.20000000000000021e-67 < x

   1. Initial program 50.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in y around 0 69.3%

      \[\leadsto \color{blue}{1 + -8 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. unpow2 69.3%

         \[\leadsto 1 + -8 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. unpow2 69.3%

         \[\leadsto 1 + -8 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 76.5%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

   4. Applied egg-rr 76.5%

      \[\leadsto 1 + -8 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{-67}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \end{array} \]

Alternative 6: 61.8% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-65}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 5.5e-65) -1.0 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5.5e-65) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 5.5d-65) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 5.5e-65) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 5.5e-65:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5.5e-65)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 5.5e-65)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5.5e-65], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 5.4999999999999999e-65

   1. Initial program 51.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around 0 56.4%

      \[\leadsto \color{blue}{-1} \]

   if 5.4999999999999999e-65 < x

   1. Initial program 50.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

   2. Taylor expanded in x around inf 75.2%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-65}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 50.5% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -1.0)

C

double code(double x, double y) {
	return -1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

function tmp = code(x, y)
	tmp = -1.0;
end

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 51.1%

   \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]

2. Taylor expanded in x around 0 45.7%

   \[\leadsto \color{blue}{-1} \]

3. Final simplification 45.7%

   \[\leadsto -1 \]

Developer target: 51.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot 4\\ t_1 := x \cdot x + t_0\\ t_2 := \frac{t_0}{t_1}\\ t_3 := \left(y \cdot 4\right) \cdot y\\ \mathbf{if}\;\frac{x \cdot x - t_3}{x \cdot x + t_3} < 0.9743233849626781:\\ \;\;\;\;\frac{x \cdot x}{t_1} - t_2\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{t_1}}\right)}^{2} - t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y y) 4.0))
        (t_1 (+ (* x x) t_0))
        (t_2 (/ t_0 t_1))
        (t_3 (* (* y 4.0) y)))
   (if (< (/ (- (* x x) t_3) (+ (* x x) t_3)) 0.9743233849626781)
     (- (/ (* x x) t_1) t_2)
     (- (pow (/ x (sqrt t_1)) 2.0) t_2))))

C

double code(double x, double y) {
	double t_0 = (y * y) * 4.0;
	double t_1 = (x * x) + t_0;
	double t_2 = t_0 / t_1;
	double t_3 = (y * 4.0) * y;
	double tmp;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781) {
		tmp = ((x * x) / t_1) - t_2;
	} else {
		tmp = pow((x / sqrt(t_1)), 2.0) - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (y * y) * 4.0d0
    t_1 = (x * x) + t_0
    t_2 = t_0 / t_1
    t_3 = (y * 4.0d0) * y
    if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781d0) then
        tmp = ((x * x) / t_1) - t_2
    else
        tmp = ((x / sqrt(t_1)) ** 2.0d0) - t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * y) * 4.0;
	double t_1 = (x * x) + t_0;
	double t_2 = t_0 / t_1;
	double t_3 = (y * 4.0) * y;
	double tmp;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781) {
		tmp = ((x * x) / t_1) - t_2;
	} else {
		tmp = Math.pow((x / Math.sqrt(t_1)), 2.0) - t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) * 4.0
	t_1 = (x * x) + t_0
	t_2 = t_0 / t_1
	t_3 = (y * 4.0) * y
	tmp = 0
	if (((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781:
		tmp = ((x * x) / t_1) - t_2
	else:
		tmp = math.pow((x / math.sqrt(t_1)), 2.0) - t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * 4.0)
	t_1 = Float64(Float64(x * x) + t_0)
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(Float64(y * 4.0) * y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * x) - t_3) / Float64(Float64(x * x) + t_3)) < 0.9743233849626781)
		tmp = Float64(Float64(Float64(x * x) / t_1) - t_2);
	else
		tmp = Float64((Float64(x / sqrt(t_1)) ^ 2.0) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) * 4.0;
	t_1 = (x * x) + t_0;
	t_2 = t_0 / t_1;
	t_3 = (y * 4.0) * y;
	tmp = 0.0;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781)
		tmp = ((x * x) / t_1) - t_2;
	else
		tmp = ((x / sqrt(t_1)) ^ 2.0) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[Less[N[(N[(N[(x * x), $MachinePrecision] - t$95$3), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], 0.9743233849626781], N[(N[(N[(x * x), $MachinePrecision] / t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[Power[N[(x / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]

Reproduce

herbie shell --seed 2023309 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))) 0.9743233849626781) (- (/ (* x x) (+ (* x x) (* (* y y) 4.0))) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4.0)))) 2.0) (/ (* (* y y) 4.0) (+ (* x x) (* (* y y) 4.0)))))

  (/ (- (* x x) (* (* y 4.0) y)) (+ (* x x) (* (* y 4.0) y))))