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

Percentage Accurate: 52.1% → 81.8%

Time: 7.2s

Alternatives: 8

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: 52.1% 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: 81.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 10^{-296}:\\ \;\;\;\;0.5 \cdot \frac{\frac{x}{y}}{\frac{y}{x}} + -1\\ \mathbf{elif}\;x \cdot x \leq 10^{+269}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(x, x, t_0\right)}{x \cdot x - t_0}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot \frac{y}{x}, -8, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 1e-296)
     (+ (* 0.5 (/ (/ x y) (/ y x))) -1.0)
     (if (<= (* x x) 1e+269)
       (pow (/ (fma x x t_0) (- (* x x) t_0)) -1.0)
       (fma (* (/ y x) (/ y x)) -8.0 1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 1e-296) {
		tmp = (0.5 * ((x / y) / (y / x))) + -1.0;
	} else if ((x * x) <= 1e+269) {
		tmp = pow((fma(x, x, t_0) / ((x * x) - t_0)), -1.0);
	} else {
		tmp = fma(((y / x) * (y / x)), -8.0, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 1e-296)
		tmp = Float64(Float64(0.5 * Float64(Float64(x / y) / Float64(y / x))) + -1.0);
	elseif (Float64(x * x) <= 1e+269)
		tmp = Float64(fma(x, x, t_0) / Float64(Float64(x * x) - t_0)) ^ -1.0;
	else
		tmp = fma(Float64(Float64(y / x) * Float64(y / x)), -8.0, 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], 1e-296], N[(N[(0.5 * N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1e+269], N[Power[N[(N[(x * x + t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] * -8.0 + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1e-296

   1. Initial program 51.2%

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

   2. Taylor expanded in x around 0 72.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   3. Step-by-step derivation

      1. fma-neg 72.8%

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

      2. unpow2 72.8%

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

      3. unpow2 72.8%

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

      4. times-frac 87.5%

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

      5. metadata-eval 87.5%

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

   4. Simplified 87.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)} \]
   5. Step-by-step derivation

      1. fma-udef 87.5%

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

      2. pow2 87.5%

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

   6. Applied egg-rr 87.5%

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

      1. unpow2 87.5%

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

      2. clear-num 87.5%

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

      3. un-div-inv 87.5%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + -1 \]

   8. Applied egg-rr 87.5%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + -1 \]

   if 1e-296 < (*.f64 x x) < 1e269

   1. Initial program 83.9%

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

      1. clear-num 83.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x \cdot x - \left(y \cdot 4\right) \cdot y}}} \]

      2. inv-pow 83.9%

         \[\leadsto \color{blue}{{\left(\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x \cdot x - \left(y \cdot 4\right) \cdot y}\right)}^{-1}} \]

      3. fma-def 83.9%

         \[\leadsto {\left(\frac{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}{x \cdot x - \left(y \cdot 4\right) \cdot y}\right)}^{-1} \]

      4. *-commutative 83.9%

         \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y \cdot 4\right)}\right)}{x \cdot x - \left(y \cdot 4\right) \cdot y}\right)}^{-1} \]

      5. sub-neg 83.9%

         \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\color{blue}{x \cdot x + \left(-\left(y \cdot 4\right) \cdot y\right)}}\right)}^{-1} \]

      6. +-commutative 83.9%

         \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\color{blue}{\left(-\left(y \cdot 4\right) \cdot y\right) + x \cdot x}}\right)}^{-1} \]

      7. *-commutative 83.9%

         \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\left(-\color{blue}{y \cdot \left(y \cdot 4\right)}\right) + x \cdot x}\right)}^{-1} \]

      8. distribute-rgt-neg-in 83.9%

         \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\color{blue}{y \cdot \left(-y \cdot 4\right)} + x \cdot x}\right)}^{-1} \]

      9. fma-def 83.9%

         \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\color{blue}{\mathsf{fma}\left(y, -y \cdot 4, x \cdot x\right)}}\right)}^{-1} \]

      10. distribute-rgt-neg-in 83.9%

          \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(-4\right)}, x \cdot x\right)}\right)}^{-1} \]

      11. metadata-eval 83.9%

          \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\mathsf{fma}\left(y, y \cdot \color{blue}{-4}, x \cdot x\right)}\right)}^{-1} \]

   3. Applied egg-rr 83.9%

      \[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\mathsf{fma}\left(y, y \cdot -4, x \cdot x\right)}\right)}^{-1}} \]
   4. Step-by-step derivation

      1. add-sqr-sqrt 41.9%

         \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\mathsf{fma}\left(y, \color{blue}{\sqrt{y \cdot -4} \cdot \sqrt{y \cdot -4}}, x \cdot x\right)}\right)}^{-1} \]

      2. sqrt-unprod 69.0%

         \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\mathsf{fma}\left(y, \color{blue}{\sqrt{\left(y \cdot -4\right) \cdot \left(y \cdot -4\right)}}, x \cdot x\right)}\right)}^{-1} \]

      3. swap-sqr 69.0%

         \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\mathsf{fma}\left(y, \sqrt{\color{blue}{\left(y \cdot y\right) \cdot \left(-4 \cdot -4\right)}}, x \cdot x\right)}\right)}^{-1} \]

      4. metadata-eval 69.0%

         \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\mathsf{fma}\left(y, \sqrt{\left(y \cdot y\right) \cdot \color{blue}{16}}, x \cdot x\right)}\right)}^{-1} \]

      5. metadata-eval 69.0%

         \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\mathsf{fma}\left(y, \sqrt{\left(y \cdot y\right) \cdot \color{blue}{\left(4 \cdot 4\right)}}, x \cdot x\right)}\right)}^{-1} \]

      6. swap-sqr 69.0%

         \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\mathsf{fma}\left(y, \sqrt{\color{blue}{\left(y \cdot 4\right) \cdot \left(y \cdot 4\right)}}, x \cdot x\right)}\right)}^{-1} \]

      7. sqrt-unprod 27.0%

         \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\mathsf{fma}\left(y, \color{blue}{\sqrt{y \cdot 4} \cdot \sqrt{y \cdot 4}}, x \cdot x\right)}\right)}^{-1} \]

      8. add-sqr-sqrt 53.8%

         \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\mathsf{fma}\left(y, \color{blue}{y \cdot 4}, x \cdot x\right)}\right)}^{-1} \]

      9. fma-def 53.8%

         \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\color{blue}{y \cdot \left(y \cdot 4\right) + x \cdot x}}\right)}^{-1} \]

      10. *-commutative 53.8%

          \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\color{blue}{\left(y \cdot 4\right) \cdot y} + x \cdot x}\right)}^{-1} \]

      11. +-commutative 53.8%

          \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\color{blue}{x \cdot x + \left(y \cdot 4\right) \cdot y}}\right)}^{-1} \]

      12. *-commutative 53.8%

          \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{x \cdot x + \color{blue}{y \cdot \left(y \cdot 4\right)}}\right)}^{-1} \]

      13. add-sqr-sqrt 27.0%

          \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{x \cdot x + y \cdot \color{blue}{\left(\sqrt{y \cdot 4} \cdot \sqrt{y \cdot 4}\right)}}\right)}^{-1} \]

      14. sqrt-unprod 69.0%

          \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{x \cdot x + y \cdot \color{blue}{\sqrt{\left(y \cdot 4\right) \cdot \left(y \cdot 4\right)}}}\right)}^{-1} \]

      15. swap-sqr 69.0%

          \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{x \cdot x + y \cdot \sqrt{\color{blue}{\left(y \cdot y\right) \cdot \left(4 \cdot 4\right)}}}\right)}^{-1} \]

      16. metadata-eval 69.0%

          \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{x \cdot x + y \cdot \sqrt{\left(y \cdot y\right) \cdot \color{blue}{16}}}\right)}^{-1} \]

      17. metadata-eval 69.0%

          \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{x \cdot x + y \cdot \sqrt{\left(y \cdot y\right) \cdot \color{blue}{\left(-4 \cdot -4\right)}}}\right)}^{-1} \]

      18. swap-sqr 69.0%

          \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{x \cdot x + y \cdot \sqrt{\color{blue}{\left(y \cdot -4\right) \cdot \left(y \cdot -4\right)}}}\right)}^{-1} \]

      19. sqrt-unprod 41.9%

          \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{x \cdot x + y \cdot \color{blue}{\left(\sqrt{y \cdot -4} \cdot \sqrt{y \cdot -4}\right)}}\right)}^{-1} \]

      20. add-sqr-sqrt 83.9%

          \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{x \cdot x + y \cdot \color{blue}{\left(y \cdot -4\right)}}\right)}^{-1} \]

      21. associate-*r* 83.9%

          \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{x \cdot x + \color{blue}{\left(y \cdot y\right) \cdot -4}}\right)}^{-1} \]

      22. metadata-eval 83.9%

          \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{x \cdot x + \left(y \cdot y\right) \cdot \color{blue}{\left(-4\right)}}\right)}^{-1} \]

      23. distribute-rgt-neg-in 83.9%

          \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{x \cdot x + \color{blue}{\left(-\left(y \cdot y\right) \cdot 4\right)}}\right)}^{-1} \]

      24. associate-*r* 83.9%

          \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{x \cdot x + \left(-\color{blue}{y \cdot \left(y \cdot 4\right)}\right)}\right)}^{-1} \]

   5. Applied egg-rr 83.9%

      \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\color{blue}{x \cdot x - y \cdot \left(y \cdot 4\right)}}\right)}^{-1} \]

   if 1e269 < (*.f64 x x)

   1. Initial program 11.5%

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

   2. Taylor expanded in x around inf 82.1%

      \[\leadsto \color{blue}{\left(1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) - 4 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. associate--l+ 82.1%

         \[\leadsto \color{blue}{1 + \left(-4 \cdot \frac{{y}^{2}}{{x}^{2}} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right)} \]

      2. distribute-rgt-out-- 82.1%

         \[\leadsto 1 + \color{blue}{\frac{{y}^{2}}{{x}^{2}} \cdot \left(-4 - 4\right)} \]

      3. metadata-eval 82.1%

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

      4. *-commutative 82.1%

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

      5. +-commutative 82.1%

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

      6. *-commutative 82.1%

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

      7. fma-def 82.1%

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

      8. unpow2 82.1%

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

      9. unpow2 82.1%

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

      10. times-frac 88.4%

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

   4. Simplified 88.4%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-296}:\\ \;\;\;\;0.5 \cdot \frac{\frac{x}{y}}{\frac{y}{x}} + -1\\ \mathbf{elif}\;x \cdot x \leq 10^{+269}:\\ \;\;\;\;{\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{x \cdot x - y \cdot \left(y \cdot 4\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot \frac{y}{x}, -8, 1\right)\\ \end{array} \]

Alternative 2: 81.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 10^{-296}:\\ \;\;\;\;0.5 \cdot \frac{\frac{x}{y}}{\frac{y}{x}} + -1\\ \mathbf{elif}\;x \cdot x \leq 10^{+269}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot \frac{y}{x}, -8, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 1e-296)
     (+ (* 0.5 (/ (/ x y) (/ y x))) -1.0)
     (if (<= (* x x) 1e+269)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (fma (* (/ y x) (/ y x)) -8.0 1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 1e-296) {
		tmp = (0.5 * ((x / y) / (y / x))) + -1.0;
	} else if ((x * x) <= 1e+269) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = fma(((y / x) * (y / x)), -8.0, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 1e-296)
		tmp = Float64(Float64(0.5 * Float64(Float64(x / y) / Float64(y / x))) + -1.0);
	elseif (Float64(x * x) <= 1e+269)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = fma(Float64(Float64(y / x) * Float64(y / x)), -8.0, 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], 1e-296], N[(N[(0.5 * N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1e+269], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] * -8.0 + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1e-296

   1. Initial program 51.2%

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

   2. Taylor expanded in x around 0 72.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   3. Step-by-step derivation

      1. fma-neg 72.8%

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

      2. unpow2 72.8%

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

      3. unpow2 72.8%

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

      4. times-frac 87.5%

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

      5. metadata-eval 87.5%

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

   4. Simplified 87.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)} \]
   5. Step-by-step derivation

      1. fma-udef 87.5%

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

      2. pow2 87.5%

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

   6. Applied egg-rr 87.5%

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

      1. unpow2 87.5%

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

      2. clear-num 87.5%

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

      3. un-div-inv 87.5%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + -1 \]

   8. Applied egg-rr 87.5%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + -1 \]

   if 1e-296 < (*.f64 x x) < 1e269

   1. Initial program 83.9%

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

   if 1e269 < (*.f64 x x)

   1. Initial program 11.5%

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

   2. Taylor expanded in x around inf 82.1%

      \[\leadsto \color{blue}{\left(1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) - 4 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   3. Step-by-step derivation

      1. associate--l+ 82.1%

         \[\leadsto \color{blue}{1 + \left(-4 \cdot \frac{{y}^{2}}{{x}^{2}} - 4 \cdot \frac{{y}^{2}}{{x}^{2}}\right)} \]

      2. distribute-rgt-out-- 82.1%

         \[\leadsto 1 + \color{blue}{\frac{{y}^{2}}{{x}^{2}} \cdot \left(-4 - 4\right)} \]

      3. metadata-eval 82.1%

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

      4. *-commutative 82.1%

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

      5. +-commutative 82.1%

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

      6. *-commutative 82.1%

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

      7. fma-def 82.1%

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

      8. unpow2 82.1%

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

      9. unpow2 82.1%

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

      10. times-frac 88.4%

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

   4. Simplified 88.4%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-296}:\\ \;\;\;\;0.5 \cdot \frac{\frac{x}{y}}{\frac{y}{x}} + -1\\ \mathbf{elif}\;x \cdot x \leq 10^{+269}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot \frac{y}{x}, -8, 1\right)\\ \end{array} \]

Alternative 3: 81.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{if}\;x \cdot x \leq 10^{-296}:\\ \;\;\;\;0.5 \cdot \frac{\frac{x}{y}}{\frac{y}{x}} + -1\\ \mathbf{elif}\;x \cdot x \leq 10^{+269}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \left(1 + t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))) (t_1 (* -4.0 (/ (/ y x) (/ x y)))))
   (if (<= (* x x) 1e-296)
     (+ (* 0.5 (/ (/ x y) (/ y x))) -1.0)
     (if (<= (* x x) 1e+269)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (+ t_1 (+ 1.0 t_1))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = -4.0 * ((y / x) / (x / y));
	double tmp;
	if ((x * x) <= 1e-296) {
		tmp = (0.5 * ((x / y) / (y / x))) + -1.0;
	} else if ((x * x) <= 1e+269) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = t_1 + (1.0 + t_1);
	}
	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) :: tmp
    t_0 = y * (y * 4.0d0)
    t_1 = (-4.0d0) * ((y / x) / (x / y))
    if ((x * x) <= 1d-296) then
        tmp = (0.5d0 * ((x / y) / (y / x))) + (-1.0d0)
    else if ((x * x) <= 1d+269) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else
        tmp = t_1 + (1.0d0 + t_1)
    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 = -4.0 * ((y / x) / (x / y));
	double tmp;
	if ((x * x) <= 1e-296) {
		tmp = (0.5 * ((x / y) / (y / x))) + -1.0;
	} else if ((x * x) <= 1e+269) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = t_1 + (1.0 + t_1);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = -4.0 * ((y / x) / (x / y))
	tmp = 0
	if (x * x) <= 1e-296:
		tmp = (0.5 * ((x / y) / (y / x))) + -1.0
	elif (x * x) <= 1e+269:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	else:
		tmp = t_1 + (1.0 + t_1)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(-4.0 * Float64(Float64(y / x) / Float64(x / y)))
	tmp = 0.0
	if (Float64(x * x) <= 1e-296)
		tmp = Float64(Float64(0.5 * Float64(Float64(x / y) / Float64(y / x))) + -1.0);
	elseif (Float64(x * x) <= 1e+269)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = Float64(t_1 + Float64(1.0 + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	t_1 = -4.0 * ((y / x) / (x / y));
	tmp = 0.0;
	if ((x * x) <= 1e-296)
		tmp = (0.5 * ((x / y) / (y / x))) + -1.0;
	elseif ((x * x) <= 1e+269)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	else
		tmp = t_1 + (1.0 + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 1e-296], N[(N[(0.5 * N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1e+269], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1e-296

   1. Initial program 51.2%

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

   2. Taylor expanded in x around 0 72.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   3. Step-by-step derivation

      1. fma-neg 72.8%

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

      2. unpow2 72.8%

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

      3. unpow2 72.8%

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

      4. times-frac 87.5%

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

      5. metadata-eval 87.5%

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

   4. Simplified 87.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)} \]
   5. Step-by-step derivation

      1. fma-udef 87.5%

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

      2. pow2 87.5%

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

   6. Applied egg-rr 87.5%

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

      1. unpow2 87.5%

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

      2. clear-num 87.5%

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

      3. un-div-inv 87.5%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + -1 \]

   8. Applied egg-rr 87.5%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + -1 \]

   if 1e-296 < (*.f64 x x) < 1e269

   1. Initial program 83.9%

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

   if 1e269 < (*.f64 x x)

   1. Initial program 11.5%

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

      1. clear-num 11.5%

         \[\leadsto \color{blue}{\frac{1}{\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x \cdot x - \left(y \cdot 4\right) \cdot y}}} \]

      2. inv-pow 11.5%

         \[\leadsto \color{blue}{{\left(\frac{x \cdot x + \left(y \cdot 4\right) \cdot y}{x \cdot x - \left(y \cdot 4\right) \cdot y}\right)}^{-1}} \]

      3. fma-def 11.5%

         \[\leadsto {\left(\frac{\color{blue}{\mathsf{fma}\left(x, x, \left(y \cdot 4\right) \cdot y\right)}}{x \cdot x - \left(y \cdot 4\right) \cdot y}\right)}^{-1} \]

      4. *-commutative 11.5%

         \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, \color{blue}{y \cdot \left(y \cdot 4\right)}\right)}{x \cdot x - \left(y \cdot 4\right) \cdot y}\right)}^{-1} \]

      5. sub-neg 11.5%

         \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\color{blue}{x \cdot x + \left(-\left(y \cdot 4\right) \cdot y\right)}}\right)}^{-1} \]

      6. +-commutative 11.5%

         \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\color{blue}{\left(-\left(y \cdot 4\right) \cdot y\right) + x \cdot x}}\right)}^{-1} \]

      7. *-commutative 11.5%

         \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\left(-\color{blue}{y \cdot \left(y \cdot 4\right)}\right) + x \cdot x}\right)}^{-1} \]

      8. distribute-rgt-neg-in 11.5%

         \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\color{blue}{y \cdot \left(-y \cdot 4\right)} + x \cdot x}\right)}^{-1} \]

      9. fma-def 11.5%

         \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\color{blue}{\mathsf{fma}\left(y, -y \cdot 4, x \cdot x\right)}}\right)}^{-1} \]

      10. distribute-rgt-neg-in 11.5%

          \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(-4\right)}, x \cdot x\right)}\right)}^{-1} \]

      11. metadata-eval 11.5%

          \[\leadsto {\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\mathsf{fma}\left(y, y \cdot \color{blue}{-4}, x \cdot x\right)}\right)}^{-1} \]

   3. Applied egg-rr 11.5%

      \[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}{\mathsf{fma}\left(y, y \cdot -4, x \cdot x\right)}\right)}^{-1}} \]

   4. Taylor expanded in x around inf 82.1%

      \[\leadsto \color{blue}{\left(1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) - 4 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   5. Step-by-step derivation

      1. cancel-sign-sub-inv 82.1%

         \[\leadsto \color{blue}{\left(1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) + \left(-4\right) \cdot \frac{{y}^{2}}{{x}^{2}}} \]

      2. metadata-eval 82.1%

         \[\leadsto \left(1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}\right) + \color{blue}{-4} \cdot \frac{{y}^{2}}{{x}^{2}} \]

      3. unpow2 82.1%

         \[\leadsto \left(1 + -4 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}}\right) + -4 \cdot \frac{{y}^{2}}{{x}^{2}} \]

      4. unpow2 82.1%

         \[\leadsto \left(1 + -4 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}}\right) + -4 \cdot \frac{{y}^{2}}{{x}^{2}} \]

      5. times-frac 82.1%

         \[\leadsto \left(1 + -4 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)}\right) + -4 \cdot \frac{{y}^{2}}{{x}^{2}} \]

      6. unpow2 82.1%

         \[\leadsto \left(1 + -4 \cdot \color{blue}{{\left(\frac{y}{x}\right)}^{2}}\right) + -4 \cdot \frac{{y}^{2}}{{x}^{2}} \]

      7. unpow2 82.1%

         \[\leadsto \left(1 + -4 \cdot {\left(\frac{y}{x}\right)}^{2}\right) + -4 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      8. unpow2 82.1%

         \[\leadsto \left(1 + -4 \cdot {\left(\frac{y}{x}\right)}^{2}\right) + -4 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      9. times-frac 88.4%

         \[\leadsto \left(1 + -4 \cdot {\left(\frac{y}{x}\right)}^{2}\right) + -4 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

      10. unpow2 88.4%

          \[\leadsto \left(1 + -4 \cdot {\left(\frac{y}{x}\right)}^{2}\right) + -4 \cdot \color{blue}{{\left(\frac{y}{x}\right)}^{2}} \]

   6. Simplified 88.4%

      \[\leadsto \color{blue}{\left(1 + -4 \cdot {\left(\frac{y}{x}\right)}^{2}\right) + -4 \cdot {\left(\frac{y}{x}\right)}^{2}} \]
   7. Step-by-step derivation

      1. unpow2 88.3%

         \[\leadsto 1 + -4 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

      2. clear-num 88.3%

         \[\leadsto 1 + -4 \cdot \left(\frac{y}{x} \cdot \color{blue}{\frac{1}{\frac{x}{y}}}\right) \]

      3. un-div-inv 88.3%

         \[\leadsto 1 + -4 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} \]

   8. Applied egg-rr 88.4%

      \[\leadsto \left(1 + -4 \cdot {\left(\frac{y}{x}\right)}^{2}\right) + -4 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} \]
   9. Step-by-step derivation

      1. unpow2 88.3%

         \[\leadsto 1 + -4 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

      2. clear-num 88.3%

         \[\leadsto 1 + -4 \cdot \left(\frac{y}{x} \cdot \color{blue}{\frac{1}{\frac{x}{y}}}\right) \]

      3. un-div-inv 88.3%

         \[\leadsto 1 + -4 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} \]

   10. Applied egg-rr 88.4%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-296}:\\ \;\;\;\;0.5 \cdot \frac{\frac{x}{y}}{\frac{y}{x}} + -1\\ \mathbf{elif}\;x \cdot x \leq 10^{+269}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}} + \left(1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\right)\\ \end{array} \]

Alternative 4: 81.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 10^{-296}:\\ \;\;\;\;0.5 \cdot \frac{\frac{x}{y}}{\frac{y}{x}} + -1\\ \mathbf{elif}\;x \cdot x \leq 10^{+269}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 1e-296)
     (+ (* 0.5 (/ (/ x y) (/ y x))) -1.0)
     (if (<= (* x x) 1e+269)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (+ 1.0 (* -4.0 (/ (/ y x) (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 1e-296) {
		tmp = (0.5 * ((x / y) / (y / x))) + -1.0;
	} else if ((x * x) <= 1e+269) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	}
	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) <= 1d-296) then
        tmp = (0.5d0 * ((x / y) / (y / x))) + (-1.0d0)
    else if ((x * x) <= 1d+269) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else
        tmp = 1.0d0 + ((-4.0d0) * ((y / x) / (x / y)))
    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) <= 1e-296) {
		tmp = (0.5 * ((x / y) / (y / x))) + -1.0;
	} else if ((x * x) <= 1e+269) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if (x * x) <= 1e-296:
		tmp = (0.5 * ((x / y) / (y / x))) + -1.0
	elif (x * x) <= 1e+269:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	else:
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 1e-296)
		tmp = Float64(Float64(0.5 * Float64(Float64(x / y) / Float64(y / x))) + -1.0);
	elseif (Float64(x * x) <= 1e+269)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = Float64(1.0 + Float64(-4.0 * Float64(Float64(y / x) / Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if ((x * x) <= 1e-296)
		tmp = (0.5 * ((x / y) / (y / x))) + -1.0;
	elseif ((x * x) <= 1e+269)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	else
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	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], 1e-296], N[(N[(0.5 * N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 1e+269], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-4.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 1e-296

   1. Initial program 51.2%

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

   2. Taylor expanded in x around 0 72.8%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   3. Step-by-step derivation

      1. fma-neg 72.8%

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

      2. unpow2 72.8%

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

      3. unpow2 72.8%

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

      4. times-frac 87.5%

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

      5. metadata-eval 87.5%

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

   4. Simplified 87.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)} \]
   5. Step-by-step derivation

      1. fma-udef 87.5%

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

      2. pow2 87.5%

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

   6. Applied egg-rr 87.5%

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

      1. unpow2 87.5%

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

      2. clear-num 87.5%

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

      3. un-div-inv 87.5%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + -1 \]

   8. Applied egg-rr 87.5%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + -1 \]

   if 1e-296 < (*.f64 x x) < 1e269

   1. Initial program 83.9%

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

   if 1e269 < (*.f64 x x)

   1. Initial program 11.5%

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

   2. Taylor expanded in x around inf 11.6%

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

      1. unpow2 11.6%

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

   4. Simplified 11.6%

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

   5. Taylor expanded in x around inf 82.1%

      \[\leadsto \color{blue}{1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 82.1%

         \[\leadsto 1 + -4 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. unpow2 82.1%

         \[\leadsto 1 + -4 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 88.3%

         \[\leadsto 1 + -4 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

      4. unpow2 88.3%

         \[\leadsto 1 + -4 \cdot \color{blue}{{\left(\frac{y}{x}\right)}^{2}} \]

   7. Simplified 88.3%

      \[\leadsto \color{blue}{1 + -4 \cdot {\left(\frac{y}{x}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 88.3%

         \[\leadsto 1 + -4 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

      2. clear-num 88.3%

         \[\leadsto 1 + -4 \cdot \left(\frac{y}{x} \cdot \color{blue}{\frac{1}{\frac{x}{y}}}\right) \]

      3. un-div-inv 88.3%

         \[\leadsto 1 + -4 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} \]

   9. Applied egg-rr 88.3%

      \[\leadsto 1 + -4 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-296}:\\ \;\;\;\;0.5 \cdot \frac{\frac{x}{y}}{\frac{y}{x}} + -1\\ \mathbf{elif}\;x \cdot x \leq 10^{+269}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \end{array} \]

Alternative 5: 74.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-126}:\\ \;\;\;\;0.5 \cdot \frac{\frac{x}{y}}{\frac{y}{x}} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 5e-126)
   (+ (* 0.5 (/ (/ x y) (/ y x))) -1.0)
   (+ 1.0 (* -4.0 (/ (/ y x) (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 5e-126) {
		tmp = (0.5 * ((x / y) / (y / x))) + -1.0;
	} else {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x * x) <= 5d-126) then
        tmp = (0.5d0 * ((x / y) / (y / x))) + (-1.0d0)
    else
        tmp = 1.0d0 + ((-4.0d0) * ((y / x) / (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * x) <= 5e-126) {
		tmp = (0.5 * ((x / y) / (y / x))) + -1.0;
	} else {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 5e-126:
		tmp = (0.5 * ((x / y) / (y / x))) + -1.0
	else:
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 5e-126)
		tmp = Float64(Float64(0.5 * Float64(Float64(x / y) / Float64(y / x))) + -1.0);
	else
		tmp = Float64(1.0 + Float64(-4.0 * Float64(Float64(y / x) / Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 5e-126)
		tmp = (0.5 * ((x / y) / (y / x))) + -1.0;
	else
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-126], N[(N[(0.5 * N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(1.0 + N[(-4.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 5.00000000000000006e-126

   1. Initial program 61.0%

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

   2. Taylor expanded in x around 0 71.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   3. Step-by-step derivation

      1. fma-neg 71.4%

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

      2. unpow2 71.4%

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

      3. unpow2 71.4%

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

      4. times-frac 80.9%

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

      5. metadata-eval 80.9%

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

   4. Simplified 80.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)} \]
   5. Step-by-step derivation

      1. fma-udef 80.9%

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

      2. pow2 80.9%

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

   6. Applied egg-rr 80.9%

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

      1. unpow2 80.9%

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

      2. clear-num 80.9%

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

      3. un-div-inv 80.9%

         \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + -1 \]

   8. Applied egg-rr 80.9%

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}} + -1 \]

   if 5.00000000000000006e-126 < (*.f64 x x)

   1. Initial program 48.3%

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

   2. Taylor expanded in x around inf 37.6%

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

      1. unpow2 37.6%

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

   4. Simplified 37.6%

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

   5. Taylor expanded in x around inf 74.0%

      \[\leadsto \color{blue}{1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 74.0%

         \[\leadsto 1 + -4 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. unpow2 74.0%

         \[\leadsto 1 + -4 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 77.2%

         \[\leadsto 1 + -4 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

      4. unpow2 77.2%

         \[\leadsto 1 + -4 \cdot \color{blue}{{\left(\frac{y}{x}\right)}^{2}} \]

   7. Simplified 77.2%

      \[\leadsto \color{blue}{1 + -4 \cdot {\left(\frac{y}{x}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 77.2%

         \[\leadsto 1 + -4 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

      2. clear-num 77.2%

         \[\leadsto 1 + -4 \cdot \left(\frac{y}{x} \cdot \color{blue}{\frac{1}{\frac{x}{y}}}\right) \]

      3. un-div-inv 77.2%

         \[\leadsto 1 + -4 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} \]

   9. Applied egg-rr 77.2%

      \[\leadsto 1 + -4 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-126}:\\ \;\;\;\;0.5 \cdot \frac{\frac{x}{y}}{\frac{y}{x}} + -1\\ \mathbf{else}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \end{array} \]

Alternative 6: 63.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-63}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 5.5e-63) -1.0 (+ 1.0 (* -4.0 (/ (/ y x) (/ x y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5.5e-63) {
		tmp = -1.0;
	} else {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	}
	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-63) then
        tmp = -1.0d0
    else
        tmp = 1.0d0 + ((-4.0d0) * ((y / x) / (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 5.5e-63) {
		tmp = -1.0;
	} else {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 5.5e-63:
		tmp = -1.0
	else:
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5.5e-63)
		tmp = -1.0;
	else
		tmp = Float64(1.0 + Float64(-4.0 * Float64(Float64(y / x) / Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 5.5e-63)
		tmp = -1.0;
	else
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5.5e-63], -1.0, N[(1.0 + N[(-4.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.50000000000000043e-63

   1. Initial program 56.1%

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

   2. Taylor expanded in x around 0 52.6%

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

   if 5.50000000000000043e-63 < x

   1. Initial program 46.4%

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

   2. Taylor expanded in x around inf 32.9%

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

      1. unpow2 32.9%

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

   4. Simplified 32.9%

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

   5. Taylor expanded in x around inf 67.3%

      \[\leadsto \color{blue}{1 + -4 \cdot \frac{{y}^{2}}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 67.3%

         \[\leadsto 1 + -4 \cdot \frac{\color{blue}{y \cdot y}}{{x}^{2}} \]

      2. unpow2 67.3%

         \[\leadsto 1 + -4 \cdot \frac{y \cdot y}{\color{blue}{x \cdot x}} \]

      3. times-frac 71.0%

         \[\leadsto 1 + -4 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

      4. unpow2 71.0%

         \[\leadsto 1 + -4 \cdot \color{blue}{{\left(\frac{y}{x}\right)}^{2}} \]

   7. Simplified 71.0%

      \[\leadsto \color{blue}{1 + -4 \cdot {\left(\frac{y}{x}\right)}^{2}} \]
   8. Step-by-step derivation

      1. unpow2 71.0%

         \[\leadsto 1 + -4 \cdot \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \]

      2. clear-num 71.0%

         \[\leadsto 1 + -4 \cdot \left(\frac{y}{x} \cdot \color{blue}{\frac{1}{\frac{x}{y}}}\right) \]

      3. un-div-inv 71.0%

         \[\leadsto 1 + -4 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} \]

   9. Applied egg-rr 71.0%

      \[\leadsto 1 + -4 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.5 \cdot 10^{-63}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \end{array} \]

Alternative 7: 63.1% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-27}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 5e-27) -1.0 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5e-27) {
		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 <= 5d-27) 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 <= 5e-27) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 5e-27:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5e-27)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 5e-27)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5e-27], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 5.0000000000000002e-27

   1. Initial program 56.7%

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

   2. Taylor expanded in x around 0 53.3%

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

   if 5.0000000000000002e-27 < x

   1. Initial program 43.9%

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

   2. Taylor expanded in x around inf 73.5%

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

4. Final simplification 58.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-27}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 50.4% 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 53.4%

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

2. Taylor expanded in x around 0 46.2%

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

3. Final simplification 46.2%

   \[\leadsto -1 \]

Developer target: 52.5% 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 2023200 
(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))))