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

Percentage Accurate: 50.9% → 79.0%

Time: 10.2s

Alternatives: 7

Speedup: 1.2×

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: 79.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-108}:\\ \;\;\;\;\mathsf{log1p}\left(0.5 \cdot \left({\left(\frac{x}{y}\right)}^{2} \cdot e^{-1}\right) + \mathsf{expm1}\left(-1\right)\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+69}:\\ \;\;\;\;{\left(\frac{{x}^{2} + {y}^{2} \cdot 4}{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-8}{\frac{x}{y} \cdot \frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 4e-108)
   (log1p (+ (* 0.5 (* (pow (/ x y) 2.0) (exp -1.0))) (expm1 -1.0)))
   (if (<= (* x x) 5e+69)
     (pow
      (/ (+ (pow x 2.0) (* (pow y 2.0) 4.0)) (fma y (* y -4.0) (pow x 2.0)))
      -1.0)
     (+ 1.0 (/ -8.0 (* (/ x y) (/ x y)))))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 4e-108) {
		tmp = log1p(((0.5 * (pow((x / y), 2.0) * exp(-1.0))) + expm1(-1.0)));
	} else if ((x * x) <= 5e+69) {
		tmp = pow(((pow(x, 2.0) + (pow(y, 2.0) * 4.0)) / fma(y, (y * -4.0), pow(x, 2.0))), -1.0);
	} else {
		tmp = 1.0 + (-8.0 / ((x / y) * (x / y)));
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 4e-108)
		tmp = log1p(Float64(Float64(0.5 * Float64((Float64(x / y) ^ 2.0) * exp(-1.0))) + expm1(-1.0)));
	elseif (Float64(x * x) <= 5e+69)
		tmp = Float64(Float64((x ^ 2.0) + Float64((y ^ 2.0) * 4.0)) / fma(y, Float64(y * -4.0), (x ^ 2.0))) ^ -1.0;
	else
		tmp = Float64(1.0 + Float64(-8.0 / Float64(Float64(x / y) * Float64(x / y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 4e-108], N[Log[1 + N[(N[(0.5 * N[(N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision] * N[Exp[-1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Exp[-1.0] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e+69], N[Power[N[(N[(N[Power[x, 2.0], $MachinePrecision] + N[(N[Power[y, 2.0], $MachinePrecision] * 4.0), $MachinePrecision]), $MachinePrecision] / N[(y * N[(y * -4.0), $MachinePrecision] + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision], N[(1.0 + N[(-8.0 / N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 4.00000000000000016e-108

   1. Initial program 60.4%

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

      1. clear-num 60.4%

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

      2. inv-pow 60.4%

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

      3. +-commutative 60.4%

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

      4. *-commutative 60.4%

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

      5. associate-*l* 60.4%

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

      6. fma-def 60.4%

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

      7. pow2 60.4%

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

      8. pow2 60.4%

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

      9. sub-neg 60.4%

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

      10. +-commutative 60.4%

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

      11. *-commutative 60.4%

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

      12. distribute-rgt-neg-in 60.4%

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

      13. fma-def 60.4%

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

      14. distribute-rgt-neg-in 60.4%

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

      15. metadata-eval 60.4%

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

      16. pow2 60.4%

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

   4. Applied egg-rr 60.4%

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

   5. Taylor expanded in x around 0 62.5%

      \[\leadsto {\color{blue}{\left(\left(-0.5 \cdot \frac{{x}^{2}}{{y}^{2}} + -0.125 \cdot \frac{{x}^{4}}{{y}^{4}}\right) - 1\right)}}^{-1} \]
   6. Step-by-step derivation

      1. associate--l+ 62.5%

         \[\leadsto {\color{blue}{\left(-0.5 \cdot \frac{{x}^{2}}{{y}^{2}} + \left(-0.125 \cdot \frac{{x}^{4}}{{y}^{4}} - 1\right)\right)}}^{-1} \]

      2. *-commutative 62.5%

         \[\leadsto {\left(\color{blue}{\frac{{x}^{2}}{{y}^{2}} \cdot -0.5} + \left(-0.125 \cdot \frac{{x}^{4}}{{y}^{4}} - 1\right)\right)}^{-1} \]

      3. fma-def 62.5%

         \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(\frac{{x}^{2}}{{y}^{2}}, -0.5, -0.125 \cdot \frac{{x}^{4}}{{y}^{4}} - 1\right)\right)}}^{-1} \]

      4. unpow2 62.5%

         \[\leadsto {\left(\mathsf{fma}\left(\frac{\color{blue}{x \cdot x}}{{y}^{2}}, -0.5, -0.125 \cdot \frac{{x}^{4}}{{y}^{4}} - 1\right)\right)}^{-1} \]

      5. unpow2 62.5%

         \[\leadsto {\left(\mathsf{fma}\left(\frac{x \cdot x}{\color{blue}{y \cdot y}}, -0.5, -0.125 \cdot \frac{{x}^{4}}{{y}^{4}} - 1\right)\right)}^{-1} \]

      6. times-frac 62.5%

         \[\leadsto {\left(\mathsf{fma}\left(\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}, -0.5, -0.125 \cdot \frac{{x}^{4}}{{y}^{4}} - 1\right)\right)}^{-1} \]

      7. unpow2 62.5%

         \[\leadsto {\left(\mathsf{fma}\left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}}, -0.5, -0.125 \cdot \frac{{x}^{4}}{{y}^{4}} - 1\right)\right)}^{-1} \]

      8. fma-neg 62.5%

         \[\leadsto {\left(\mathsf{fma}\left({\left(\frac{x}{y}\right)}^{2}, -0.5, \color{blue}{\mathsf{fma}\left(-0.125, \frac{{x}^{4}}{{y}^{4}}, -1\right)}\right)\right)}^{-1} \]

   7. Simplified 87.5%

      \[\leadsto {\color{blue}{\left(\mathsf{fma}\left({\left(\frac{x}{y}\right)}^{2}, -0.5, \mathsf{fma}\left(-0.125, {\left(\frac{x}{y}\right)}^{4}, -1\right)\right)\right)}}^{-1} \]
   8. Step-by-step derivation

      1. log1p-expm1-u 87.5%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\mathsf{fma}\left({\left(\frac{x}{y}\right)}^{2}, -0.5, \mathsf{fma}\left(-0.125, {\left(\frac{x}{y}\right)}^{4}, -1\right)\right)\right)}^{-1}\right)\right)} \]

      2. unpow-1 87.5%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{1}{\mathsf{fma}\left({\left(\frac{x}{y}\right)}^{2}, -0.5, \mathsf{fma}\left(-0.125, {\left(\frac{x}{y}\right)}^{4}, -1\right)\right)}}\right)\right) \]

   9. Applied egg-rr 87.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\mathsf{fma}\left({\left(\frac{x}{y}\right)}^{2}, -0.5, \mathsf{fma}\left(-0.125, {\left(\frac{x}{y}\right)}^{4}, -1\right)\right)}\right)\right)} \]

   10. Taylor expanded in x around 0 80.0%

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

       1. +-commutative 80.0%

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

       2. associate--l+ 80.0%

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

       3. associate-/l* 80.0%

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

       4. associate-/r/ 80.0%

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

       5. unpow2 80.0%

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

       6. unpow2 80.0%

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

       7. times-frac 88.9%

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

       8. unpow2 88.9%

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

       9. expm1-def 88.9%

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

   12. Simplified 88.9%

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

   if 4.00000000000000016e-108 < (*.f64 x x) < 5.00000000000000036e69

   1. Initial program 86.0%

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

      1. clear-num 86.0%

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

      2. inv-pow 86.0%

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

      3. +-commutative 86.0%

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

      4. *-commutative 86.0%

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

      5. associate-*l* 86.0%

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

      6. fma-def 86.0%

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

      7. pow2 86.0%

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

      8. pow2 86.0%

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

      9. sub-neg 86.0%

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

      10. +-commutative 86.0%

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

      11. *-commutative 86.0%

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

      12. distribute-rgt-neg-in 86.0%

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

      13. fma-def 86.0%

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

      14. distribute-rgt-neg-in 86.0%

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

      15. metadata-eval 86.0%

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

      16. pow2 86.0%

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

   4. Applied egg-rr 86.0%

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

      1. fma-udef 86.0%

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

      2. +-commutative 86.0%

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

      3. *-commutative 86.0%

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

   6. Applied egg-rr 86.0%

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

   if 5.00000000000000036e69 < (*.f64 x x)

   1. Initial program 26.8%

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

      1. clear-num 26.8%

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

      2. inv-pow 26.8%

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

      3. +-commutative 26.8%

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

      4. *-commutative 26.8%

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

      5. associate-*l* 26.8%

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

      6. fma-def 26.8%

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

      7. pow2 26.8%

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

      8. pow2 26.8%

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

      9. sub-neg 26.8%

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

      10. +-commutative 26.8%

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

      11. *-commutative 26.8%

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

      12. distribute-rgt-neg-in 26.8%

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

      13. fma-def 26.8%

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

      14. distribute-rgt-neg-in 26.8%

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

      15. metadata-eval 26.8%

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

      16. pow2 26.8%

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

   4. Applied egg-rr 26.8%

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

   5. Taylor expanded in y around 0 77.2%

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

      1. associate-*r/ 77.2%

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

      2. associate-/l* 77.2%

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

      3. unpow2 77.2%

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

      4. unpow2 77.2%

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

      5. times-frac 82.6%

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

      6. unpow2 82.6%

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

   7. Simplified 82.6%

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

   8. Taylor expanded in x around inf 77.2%

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

      1. associate-*r/ 77.2%

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

      2. associate-/l* 77.2%

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

      3. unpow2 77.2%

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

      4. unpow2 77.2%

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

      5. times-frac 83.0%

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

      6. unpow2 83.0%

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

   10. Simplified 83.0%

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

       1. pow2 83.0%

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

   12. Applied egg-rr 83.0%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-108}:\\ \;\;\;\;\mathsf{log1p}\left(0.5 \cdot \left({\left(\frac{x}{y}\right)}^{2} \cdot e^{-1}\right) + \mathsf{expm1}\left(-1\right)\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+69}:\\ \;\;\;\;{\left(\frac{{x}^{2} + {y}^{2} \cdot 4}{\mathsf{fma}\left(y, y \cdot -4, {x}^{2}\right)}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-8}{\frac{x}{y} \cdot \frac{x}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-108}:\\ \;\;\;\;\mathsf{log1p}\left(0.5 \cdot \left({\left(\frac{x}{y}\right)}^{2} \cdot e^{-1}\right) + \mathsf{expm1}\left(-1\right)\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+69}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-8}{\frac{x}{y} \cdot \frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 4e-108)
     (log1p (+ (* 0.5 (* (pow (/ x y) 2.0) (exp -1.0))) (expm1 -1.0)))
     (if (<= (* x x) 5e+69)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (+ 1.0 (/ -8.0 (* (/ x y) (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 4e-108) {
		tmp = log1p(((0.5 * (pow((x / y), 2.0) * exp(-1.0))) + expm1(-1.0)));
	} else if ((x * x) <= 5e+69) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (-8.0 / ((x / y) * (x / y)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 4e-108) {
		tmp = Math.log1p(((0.5 * (Math.pow((x / y), 2.0) * Math.exp(-1.0))) + Math.expm1(-1.0)));
	} else if ((x * x) <= 5e+69) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (-8.0 / ((x / y) * (x / y)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if (x * x) <= 4e-108:
		tmp = math.log1p(((0.5 * (math.pow((x / y), 2.0) * math.exp(-1.0))) + math.expm1(-1.0)))
	elif (x * x) <= 5e+69:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	else:
		tmp = 1.0 + (-8.0 / ((x / y) * (x / y)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 4e-108)
		tmp = log1p(Float64(Float64(0.5 * Float64((Float64(x / y) ^ 2.0) * exp(-1.0))) + expm1(-1.0)));
	elseif (Float64(x * x) <= 5e+69)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = Float64(1.0 + Float64(-8.0 / Float64(Float64(x / y) * Float64(x / y))));
	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], 4e-108], N[Log[1 + N[(N[(0.5 * N[(N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision] * N[Exp[-1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(Exp[-1.0] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e+69], 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[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 4.00000000000000016e-108

   1. Initial program 60.4%

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

      1. clear-num 60.4%

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

      2. inv-pow 60.4%

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

      3. +-commutative 60.4%

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

      4. *-commutative 60.4%

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

      5. associate-*l* 60.4%

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

      6. fma-def 60.4%

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

      7. pow2 60.4%

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

      8. pow2 60.4%

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

      9. sub-neg 60.4%

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

      10. +-commutative 60.4%

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

      11. *-commutative 60.4%

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

      12. distribute-rgt-neg-in 60.4%

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

      13. fma-def 60.4%

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

      14. distribute-rgt-neg-in 60.4%

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

      15. metadata-eval 60.4%

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

      16. pow2 60.4%

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

   4. Applied egg-rr 60.4%

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

   5. Taylor expanded in x around 0 62.5%

      \[\leadsto {\color{blue}{\left(\left(-0.5 \cdot \frac{{x}^{2}}{{y}^{2}} + -0.125 \cdot \frac{{x}^{4}}{{y}^{4}}\right) - 1\right)}}^{-1} \]
   6. Step-by-step derivation

      1. associate--l+ 62.5%

         \[\leadsto {\color{blue}{\left(-0.5 \cdot \frac{{x}^{2}}{{y}^{2}} + \left(-0.125 \cdot \frac{{x}^{4}}{{y}^{4}} - 1\right)\right)}}^{-1} \]

      2. *-commutative 62.5%

         \[\leadsto {\left(\color{blue}{\frac{{x}^{2}}{{y}^{2}} \cdot -0.5} + \left(-0.125 \cdot \frac{{x}^{4}}{{y}^{4}} - 1\right)\right)}^{-1} \]

      3. fma-def 62.5%

         \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(\frac{{x}^{2}}{{y}^{2}}, -0.5, -0.125 \cdot \frac{{x}^{4}}{{y}^{4}} - 1\right)\right)}}^{-1} \]

      4. unpow2 62.5%

         \[\leadsto {\left(\mathsf{fma}\left(\frac{\color{blue}{x \cdot x}}{{y}^{2}}, -0.5, -0.125 \cdot \frac{{x}^{4}}{{y}^{4}} - 1\right)\right)}^{-1} \]

      5. unpow2 62.5%

         \[\leadsto {\left(\mathsf{fma}\left(\frac{x \cdot x}{\color{blue}{y \cdot y}}, -0.5, -0.125 \cdot \frac{{x}^{4}}{{y}^{4}} - 1\right)\right)}^{-1} \]

      6. times-frac 62.5%

         \[\leadsto {\left(\mathsf{fma}\left(\color{blue}{\frac{x}{y} \cdot \frac{x}{y}}, -0.5, -0.125 \cdot \frac{{x}^{4}}{{y}^{4}} - 1\right)\right)}^{-1} \]

      7. unpow2 62.5%

         \[\leadsto {\left(\mathsf{fma}\left(\color{blue}{{\left(\frac{x}{y}\right)}^{2}}, -0.5, -0.125 \cdot \frac{{x}^{4}}{{y}^{4}} - 1\right)\right)}^{-1} \]

      8. fma-neg 62.5%

         \[\leadsto {\left(\mathsf{fma}\left({\left(\frac{x}{y}\right)}^{2}, -0.5, \color{blue}{\mathsf{fma}\left(-0.125, \frac{{x}^{4}}{{y}^{4}}, -1\right)}\right)\right)}^{-1} \]

   7. Simplified 87.5%

      \[\leadsto {\color{blue}{\left(\mathsf{fma}\left({\left(\frac{x}{y}\right)}^{2}, -0.5, \mathsf{fma}\left(-0.125, {\left(\frac{x}{y}\right)}^{4}, -1\right)\right)\right)}}^{-1} \]
   8. Step-by-step derivation

      1. log1p-expm1-u 87.5%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\left(\mathsf{fma}\left({\left(\frac{x}{y}\right)}^{2}, -0.5, \mathsf{fma}\left(-0.125, {\left(\frac{x}{y}\right)}^{4}, -1\right)\right)\right)}^{-1}\right)\right)} \]

      2. unpow-1 87.5%

         \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{1}{\mathsf{fma}\left({\left(\frac{x}{y}\right)}^{2}, -0.5, \mathsf{fma}\left(-0.125, {\left(\frac{x}{y}\right)}^{4}, -1\right)\right)}}\right)\right) \]

   9. Applied egg-rr 87.5%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{1}{\mathsf{fma}\left({\left(\frac{x}{y}\right)}^{2}, -0.5, \mathsf{fma}\left(-0.125, {\left(\frac{x}{y}\right)}^{4}, -1\right)\right)}\right)\right)} \]

   10. Taylor expanded in x around 0 80.0%

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

       1. +-commutative 80.0%

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

       2. associate--l+ 80.0%

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

       3. associate-/l* 80.0%

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

       4. associate-/r/ 80.0%

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

       5. unpow2 80.0%

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

       6. unpow2 80.0%

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

       7. times-frac 88.9%

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

       8. unpow2 88.9%

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

       9. expm1-def 88.9%

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

   12. Simplified 88.9%

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

   if 4.00000000000000016e-108 < (*.f64 x x) < 5.00000000000000036e69

   1. Initial program 86.0%

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

   if 5.00000000000000036e69 < (*.f64 x x)

   1. Initial program 26.8%

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

      1. clear-num 26.8%

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

      2. inv-pow 26.8%

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

      3. +-commutative 26.8%

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

      4. *-commutative 26.8%

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

      5. associate-*l* 26.8%

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

      6. fma-def 26.8%

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

      7. pow2 26.8%

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

      8. pow2 26.8%

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

      9. sub-neg 26.8%

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

      10. +-commutative 26.8%

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

      11. *-commutative 26.8%

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

      12. distribute-rgt-neg-in 26.8%

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

      13. fma-def 26.8%

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

      14. distribute-rgt-neg-in 26.8%

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

      15. metadata-eval 26.8%

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

      16. pow2 26.8%

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

   4. Applied egg-rr 26.8%

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

   5. Taylor expanded in y around 0 77.2%

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

      1. associate-*r/ 77.2%

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

      2. associate-/l* 77.2%

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

      3. unpow2 77.2%

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

      4. unpow2 77.2%

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

      5. times-frac 82.6%

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

      6. unpow2 82.6%

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

   7. Simplified 82.6%

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

   8. Taylor expanded in x around inf 77.2%

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

      1. associate-*r/ 77.2%

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

      2. associate-/l* 77.2%

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

      3. unpow2 77.2%

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

      4. unpow2 77.2%

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

      5. times-frac 83.0%

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

      6. unpow2 83.0%

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

   10. Simplified 83.0%

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

       1. pow2 83.0%

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

   12. Applied egg-rr 83.0%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-108}:\\ \;\;\;\;\mathsf{log1p}\left(0.5 \cdot \left({\left(\frac{x}{y}\right)}^{2} \cdot e^{-1}\right) + \mathsf{expm1}\left(-1\right)\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+69}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-8}{\frac{x}{y} \cdot \frac{x}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-276}:\\ \;\;\;\;-1 + 0.5 \cdot t_1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+69}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-8}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))) (t_1 (* (/ x y) (/ x y))))
   (if (<= (* x x) 2e-276)
     (+ -1.0 (* 0.5 t_1))
     (if (<= (* x x) 5e+69)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (+ 1.0 (/ -8.0 t_1))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = (x / y) * (x / y);
	double tmp;
	if ((x * x) <= 2e-276) {
		tmp = -1.0 + (0.5 * t_1);
	} else if ((x * x) <= 5e+69) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (-8.0 / 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 = (x / y) * (x / y)
    if ((x * x) <= 2d-276) then
        tmp = (-1.0d0) + (0.5d0 * t_1)
    else if ((x * x) <= 5d+69) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else
        tmp = 1.0d0 + ((-8.0d0) / 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 = (x / y) * (x / y);
	double tmp;
	if ((x * x) <= 2e-276) {
		tmp = -1.0 + (0.5 * t_1);
	} else if ((x * x) <= 5e+69) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (-8.0 / t_1);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = (x / y) * (x / y)
	tmp = 0
	if (x * x) <= 2e-276:
		tmp = -1.0 + (0.5 * t_1)
	elif (x * x) <= 5e+69:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	else:
		tmp = 1.0 + (-8.0 / t_1)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(Float64(x / y) * Float64(x / y))
	tmp = 0.0
	if (Float64(x * x) <= 2e-276)
		tmp = Float64(-1.0 + Float64(0.5 * t_1));
	elseif (Float64(x * x) <= 5e+69)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = Float64(1.0 + Float64(-8.0 / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	t_1 = (x / y) * (x / y);
	tmp = 0.0;
	if ((x * x) <= 2e-276)
		tmp = -1.0 + (0.5 * t_1);
	elseif ((x * x) <= 5e+69)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	else
		tmp = 1.0 + (-8.0 / 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[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 2e-276], N[(-1.0 + N[(0.5 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e+69], 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 / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 2e-276

   1. Initial program 54.3%

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

   3. Taylor expanded in x around 0 82.9%

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

      1. unpow2 82.9%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{y}^{2}} - 1 \]

      2. unpow2 82.9%

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

      3. times-frac 94.8%

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

   5. Applied egg-rr 94.8%

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

   if 2e-276 < (*.f64 x x) < 5.00000000000000036e69

   1. Initial program 81.0%

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

   if 5.00000000000000036e69 < (*.f64 x x)

   1. Initial program 26.8%

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

      1. clear-num 26.8%

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

      2. inv-pow 26.8%

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

      3. +-commutative 26.8%

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

      4. *-commutative 26.8%

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

      5. associate-*l* 26.8%

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

      6. fma-def 26.8%

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

      7. pow2 26.8%

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

      8. pow2 26.8%

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

      9. sub-neg 26.8%

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

      10. +-commutative 26.8%

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

      11. *-commutative 26.8%

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

      12. distribute-rgt-neg-in 26.8%

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

      13. fma-def 26.8%

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

      14. distribute-rgt-neg-in 26.8%

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

      15. metadata-eval 26.8%

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

      16. pow2 26.8%

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

   4. Applied egg-rr 26.8%

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

   5. Taylor expanded in y around 0 77.2%

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

      1. associate-*r/ 77.2%

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

      2. associate-/l* 77.2%

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

      3. unpow2 77.2%

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

      4. unpow2 77.2%

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

      5. times-frac 82.6%

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

      6. unpow2 82.6%

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

   7. Simplified 82.6%

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

   8. Taylor expanded in x around inf 77.2%

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

      1. associate-*r/ 77.2%

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

      2. associate-/l* 77.2%

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

      3. unpow2 77.2%

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

      4. unpow2 77.2%

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

      5. times-frac 83.0%

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

      6. unpow2 83.0%

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

   10. Simplified 83.0%

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

       1. pow2 83.0%

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

   12. Applied egg-rr 83.0%

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

4. Final simplification 85.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-276}:\\ \;\;\;\;-1 + 0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+69}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-8}{\frac{x}{y} \cdot \frac{x}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-39}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 3.3 \lor \neg \left(x \leq 2.15 \cdot 10^{+20}\right):\\ \;\;\;\;1 + \frac{-8}{\frac{x}{y} \cdot \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.1e-39)
   -1.0
   (if (or (<= x 3.3) (not (<= x 2.15e+20)))
     (+ 1.0 (/ -8.0 (* (/ x y) (/ x y))))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.1e-39) {
		tmp = -1.0;
	} else if ((x <= 3.3) || !(x <= 2.15e+20)) {
		tmp = 1.0 + (-8.0 / ((x / y) * (x / y)));
	} 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 <= 1.1d-39) then
        tmp = -1.0d0
    else if ((x <= 3.3d0) .or. (.not. (x <= 2.15d+20))) then
        tmp = 1.0d0 + ((-8.0d0) / ((x / y) * (x / y)))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.1e-39) {
		tmp = -1.0;
	} else if ((x <= 3.3) || !(x <= 2.15e+20)) {
		tmp = 1.0 + (-8.0 / ((x / y) * (x / y)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.1e-39:
		tmp = -1.0
	elif (x <= 3.3) or not (x <= 2.15e+20):
		tmp = 1.0 + (-8.0 / ((x / y) * (x / y)))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.1e-39)
		tmp = -1.0;
	elseif ((x <= 3.3) || !(x <= 2.15e+20))
		tmp = Float64(1.0 + Float64(-8.0 / Float64(Float64(x / y) * Float64(x / y))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.1e-39)
		tmp = -1.0;
	elseif ((x <= 3.3) || ~((x <= 2.15e+20)))
		tmp = 1.0 + (-8.0 / ((x / y) * (x / y)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.1e-39], -1.0, If[Or[LessEqual[x, 3.3], N[Not[LessEqual[x, 2.15e+20]], $MachinePrecision]], N[(1.0 + N[(-8.0 / N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1e-39 or 3.2999999999999998 < x < 2.15e20

   1. Initial program 55.4%

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

   3. Taylor expanded in x around 0 62.4%

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

   if 1.1e-39 < x < 3.2999999999999998 or 2.15e20 < x

   1. Initial program 36.1%

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

      1. clear-num 36.1%

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

      2. inv-pow 36.1%

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

      3. +-commutative 36.1%

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

      4. *-commutative 36.1%

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

      5. associate-*l* 36.1%

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

      6. fma-def 36.1%

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

      7. pow2 36.1%

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

      8. pow2 36.1%

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

      9. sub-neg 36.1%

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

      10. +-commutative 36.1%

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

      11. *-commutative 36.1%

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

      12. distribute-rgt-neg-in 36.1%

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

      13. fma-def 36.1%

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

      14. distribute-rgt-neg-in 36.1%

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

      15. metadata-eval 36.1%

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

      16. pow2 36.1%

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

   4. Applied egg-rr 36.1%

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

   5. Taylor expanded in y around 0 74.1%

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

      1. associate-*r/ 74.1%

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

      2. associate-/l* 74.1%

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

      3. unpow2 74.1%

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

      4. unpow2 74.1%

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

      5. times-frac 79.7%

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

      6. unpow2 79.7%

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

   7. Simplified 79.7%

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

   8. Taylor expanded in x around inf 74.2%

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

      1. associate-*r/ 74.2%

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

      2. associate-/l* 74.2%

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

      3. unpow2 74.2%

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

      4. unpow2 74.2%

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

      5. times-frac 80.2%

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

      6. unpow2 80.2%

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

   10. Simplified 80.2%

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

       1. pow2 80.2%

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

   12. Applied egg-rr 80.2%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.1 \cdot 10^{-39}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 3.3 \lor \neg \left(x \leq 2.15 \cdot 10^{+20}\right):\\ \;\;\;\;1 + \frac{-8}{\frac{x}{y} \cdot \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq 8 \cdot 10^{-39} \lor \neg \left(x \leq 0.92\right) \land x \leq 2.2 \cdot 10^{+23}:\\ \;\;\;\;-1 + 0.5 \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-8}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ x y) (/ x y))))
   (if (or (<= x 8e-39) (and (not (<= x 0.92)) (<= x 2.2e+23)))
     (+ -1.0 (* 0.5 t_0))
     (+ 1.0 (/ -8.0 t_0)))))

C

double code(double x, double y) {
	double t_0 = (x / y) * (x / y);
	double tmp;
	if ((x <= 8e-39) || (!(x <= 0.92) && (x <= 2.2e+23))) {
		tmp = -1.0 + (0.5 * t_0);
	} else {
		tmp = 1.0 + (-8.0 / t_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 = (x / y) * (x / y)
    if ((x <= 8d-39) .or. (.not. (x <= 0.92d0)) .and. (x <= 2.2d+23)) then
        tmp = (-1.0d0) + (0.5d0 * t_0)
    else
        tmp = 1.0d0 + ((-8.0d0) / t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x / y) * (x / y);
	double tmp;
	if ((x <= 8e-39) || (!(x <= 0.92) && (x <= 2.2e+23))) {
		tmp = -1.0 + (0.5 * t_0);
	} else {
		tmp = 1.0 + (-8.0 / t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) * (x / y)
	tmp = 0
	if (x <= 8e-39) or (not (x <= 0.92) and (x <= 2.2e+23)):
		tmp = -1.0 + (0.5 * t_0)
	else:
		tmp = 1.0 + (-8.0 / t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) * Float64(x / y))
	tmp = 0.0
	if ((x <= 8e-39) || (!(x <= 0.92) && (x <= 2.2e+23)))
		tmp = Float64(-1.0 + Float64(0.5 * t_0));
	else
		tmp = Float64(1.0 + Float64(-8.0 / t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / y) * (x / y);
	tmp = 0.0;
	if ((x <= 8e-39) || (~((x <= 0.92)) && (x <= 2.2e+23)))
		tmp = -1.0 + (0.5 * t_0);
	else
		tmp = 1.0 + (-8.0 / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[x, 8e-39], And[N[Not[LessEqual[x, 0.92]], $MachinePrecision], LessEqual[x, 2.2e+23]]], N[(-1.0 + N[(0.5 * t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-8.0 / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 7.99999999999999943e-39 or 0.92000000000000004 < x < 2.20000000000000008e23

   1. Initial program 55.4%

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

   3. Taylor expanded in x around 0 57.9%

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

      1. unpow2 57.9%

         \[\leadsto 0.5 \cdot \frac{\color{blue}{x \cdot x}}{{y}^{2}} - 1 \]

      2. unpow2 57.9%

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

      3. times-frac 63.7%

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

   5. Applied egg-rr 63.7%

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

   if 7.99999999999999943e-39 < x < 0.92000000000000004 or 2.20000000000000008e23 < x

   1. Initial program 36.1%

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

      1. clear-num 36.1%

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

      2. inv-pow 36.1%

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

      3. +-commutative 36.1%

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

      4. *-commutative 36.1%

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

      5. associate-*l* 36.1%

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

      6. fma-def 36.1%

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

      7. pow2 36.1%

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

      8. pow2 36.1%

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

      9. sub-neg 36.1%

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

      10. +-commutative 36.1%

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

      11. *-commutative 36.1%

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

      12. distribute-rgt-neg-in 36.1%

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

      13. fma-def 36.1%

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

      14. distribute-rgt-neg-in 36.1%

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

      15. metadata-eval 36.1%

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

      16. pow2 36.1%

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

   4. Applied egg-rr 36.1%

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

   5. Taylor expanded in y around 0 74.1%

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

      1. associate-*r/ 74.1%

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

      2. associate-/l* 74.1%

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

      3. unpow2 74.1%

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

      4. unpow2 74.1%

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

      5. times-frac 79.7%

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

      6. unpow2 79.7%

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

   7. Simplified 79.7%

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

   8. Taylor expanded in x around inf 74.2%

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

      1. associate-*r/ 74.2%

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

      2. associate-/l* 74.2%

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

      3. unpow2 74.2%

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

      4. unpow2 74.2%

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

      5. times-frac 80.2%

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

      6. unpow2 80.2%

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

   10. Simplified 80.2%

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

       1. pow2 80.2%

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

   12. Applied egg-rr 80.2%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 8 \cdot 10^{-39} \lor \neg \left(x \leq 0.92\right) \land x \leq 2.2 \cdot 10^{+23}:\\ \;\;\;\;-1 + 0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-8}{\frac{x}{y} \cdot \frac{x}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-42}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 0.05:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+20}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4e-42) -1.0 (if (<= x 0.05) 1.0 (if (<= x 2e+20) -1.0 1.0))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4e-42) {
		tmp = -1.0;
	} else if (x <= 0.05) {
		tmp = 1.0;
	} else if (x <= 2e+20) {
		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 <= 4d-42) then
        tmp = -1.0d0
    else if (x <= 0.05d0) then
        tmp = 1.0d0
    else if (x <= 2d+20) 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 <= 4e-42) {
		tmp = -1.0;
	} else if (x <= 0.05) {
		tmp = 1.0;
	} else if (x <= 2e+20) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 4e-42:
		tmp = -1.0
	elif x <= 0.05:
		tmp = 1.0
	elif x <= 2e+20:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4e-42)
		tmp = -1.0;
	elseif (x <= 0.05)
		tmp = 1.0;
	elseif (x <= 2e+20)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 4e-42)
		tmp = -1.0;
	elseif (x <= 0.05)
		tmp = 1.0;
	elseif (x <= 2e+20)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4e-42], -1.0, If[LessEqual[x, 0.05], 1.0, If[LessEqual[x, 2e+20], -1.0, 1.0]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.00000000000000015e-42 or 0.050000000000000003 < x < 2e20

   1. Initial program 55.4%

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

   3. Taylor expanded in x around 0 62.4%

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

   if 4.00000000000000015e-42 < x < 0.050000000000000003 or 2e20 < x

   1. Initial program 36.1%

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

   3. Taylor expanded in x around inf 79.5%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{-42}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 0.05:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2 \cdot 10^{+20}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.8% 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 50.0%

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

3. Taylor expanded in x around 0 50.8%

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

4. Final simplification 50.8%

   \[\leadsto -1 \]
5. Add Preprocessing

Developer target: 51.3% 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 2024013 
(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))))