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

Percentage Accurate: 50.2% → 99.9%

Time: 7.8s

Alternatives: 9

Speedup: 19.0×

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: 50.2% 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: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(x, y \cdot 2\right)\\ \frac{\mathsf{fma}\left(y, 2, x\right)}{t\_0} \cdot \frac{x + y \cdot -2}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (hypot x (* y 2.0))))
   (* (/ (fma y 2.0 x) t_0) (/ (+ x (* y -2.0)) t_0))))

C

double code(double x, double y) {
	double t_0 = hypot(x, (y * 2.0));
	return (fma(y, 2.0, x) / t_0) * ((x + (y * -2.0)) / t_0);
}

Julia

function code(x, y)
	t_0 = hypot(x, Float64(y * 2.0))
	return Float64(Float64(fma(y, 2.0, x) / t_0) * Float64(Float64(x + Float64(y * -2.0)) / t_0))
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Sqrt[x ^ 2 + N[(y * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]}, N[(N[(N[(y * 2.0 + x), $MachinePrecision] / t$95$0), $MachinePrecision] * N[(N[(x + N[(y * -2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 51.5%

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

   1. add-sqr-sqrt 51.5%

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

   2. difference-of-squares 51.6%

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

   3. *-commutative 51.6%

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

   4. associate-*r* 51.5%

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

   5. sqrt-prod 51.5%

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

   6. sqrt-unprod 24.9%

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

   7. add-sqr-sqrt 37.2%

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

   8. metadata-eval 37.2%

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

   9. *-commutative 37.2%

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

   10. associate-*r* 37.2%

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

   11. sqrt-prod 37.2%

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

   12. sqrt-unprod 24.9%

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

   13. add-sqr-sqrt 51.6%

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

   14. metadata-eval 51.6%

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

4. Applied egg-rr 51.6%

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

   1. add-sqr-sqrt 51.6%

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

   2. times-frac 52.1%

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

   3. +-commutative 52.1%

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

   4. fma-define 52.1%

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

   5. add-sqr-sqrt 52.1%

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

   6. hypot-define 52.2%

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

   7. *-commutative 52.2%

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

   8. associate-*r* 52.2%

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

   9. metadata-eval 52.2%

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

   10. swap-sqr 52.2%

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

   11. sqrt-unprod 25.6%

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

   12. add-sqr-sqrt 52.2%

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

6. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, y \cdot 2\right)} \cdot \frac{x + y \cdot -2}{\mathsf{hypot}\left(x, y \cdot 2\right)}} \]
7. Add Preprocessing

Alternative 2: 81.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;1 + \frac{y}{x} \cdot \left(\frac{y}{x} \cdot -8\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+245}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.375, {\left(\frac{x}{y}\right)}^{2}, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 0.0)
     (+ 1.0 (* (/ y x) (* (/ y x) -8.0)))
     (if (<= t_0 1e+245)
       (/ (fma x x (* y (* y -4.0))) (fma x x t_0))
       (fma 0.375 (pow (/ x y) 2.0) -1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 + ((y / x) * ((y / x) * -8.0));
	} else if (t_0 <= 1e+245) {
		tmp = fma(x, x, (y * (y * -4.0))) / fma(x, x, t_0);
	} else {
		tmp = fma(0.375, pow((x / y), 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(1.0 + Float64(Float64(y / x) * Float64(Float64(y / x) * -8.0)));
	elseif (t_0 <= 1e+245)
		tmp = Float64(fma(x, x, Float64(y * Float64(y * -4.0))) / fma(x, x, t_0));
	else
		tmp = fma(0.375, (Float64(x / y) ^ 2.0), -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(1.0 + N[(N[(y / x), $MachinePrecision] * N[(N[(y / x), $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+245], N[(N[(x * x + N[(y * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x + t$95$0), $MachinePrecision]), $MachinePrecision], N[(0.375 * N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 0.0

   1. Initial program 50.9%

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

   3. Taylor expanded in y around 0 80.0%

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

      1. *-commutative 80.0%

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

      2. pow2 80.0%

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

      3. add-sqr-sqrt 80.0%

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

      4. associate-*l* 80.0%

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

      5. sqrt-div 80.0%

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

      6. sqrt-pow1 80.0%

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

      7. metadata-eval 80.0%

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

      8. pow1 80.0%

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

      9. sqrt-prod 41.8%

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

      10. add-sqr-sqrt 80.0%

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

      11. sqrt-div 80.0%

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

      12. sqrt-pow1 80.5%

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

      13. metadata-eval 80.5%

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

      14. pow1 80.5%

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

      15. sqrt-prod 47.3%

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

      16. add-sqr-sqrt 96.3%

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

   5. Applied egg-rr 96.3%

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

   if 0.0 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.00000000000000004e245

   1. Initial program 80.3%

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

      1. fma-neg 80.3%

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

      2. *-commutative 80.3%

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

      3. distribute-rgt-neg-in 80.3%

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

      4. distribute-rgt-neg-in 80.3%

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

      5. metadata-eval 80.3%

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

      6. fma-define 80.3%

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

      7. *-commutative 80.3%

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

   3. Simplified 80.3%

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

   if 1.00000000000000004e245 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 11.9%

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

      1. add-sqr-sqrt 11.9%

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

      2. difference-of-squares 11.9%

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

      3. *-commutative 11.9%

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

      4. associate-*r* 11.9%

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

      5. sqrt-prod 11.9%

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

      6. sqrt-unprod 3.5%

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

      7. add-sqr-sqrt 3.7%

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

      8. metadata-eval 3.7%

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

      9. *-commutative 3.7%

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

      10. associate-*r* 3.7%

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

      11. sqrt-prod 3.7%

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

      12. sqrt-unprod 3.5%

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

      13. add-sqr-sqrt 11.9%

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

      14. metadata-eval 11.9%

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

   4. Applied egg-rr 11.9%

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

      1. add-sqr-sqrt 11.9%

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

      2. times-frac 14.6%

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

      3. +-commutative 14.6%

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

      4. fma-define 14.6%

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

      5. add-sqr-sqrt 14.6%

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

      6. hypot-define 14.6%

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

      7. *-commutative 14.6%

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

      8. associate-*r* 14.6%

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

      9. metadata-eval 14.6%

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

      10. swap-sqr 14.6%

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

      11. sqrt-unprod 5.0%

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

      12. add-sqr-sqrt 14.6%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in y around inf 45.8%

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

   8. Taylor expanded in x around 0 76.4%

      \[\leadsto \color{blue}{0.375 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   9. Step-by-step derivation

      1. fma-neg 76.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.375, \frac{{x}^{2}}{{y}^{2}}, -1\right)} \]

      2. unpow2 76.4%

         \[\leadsto \mathsf{fma}\left(0.375, \frac{\color{blue}{x \cdot x}}{{y}^{2}}, -1\right) \]

      3. unpow2 76.4%

         \[\leadsto \mathsf{fma}\left(0.375, \frac{x \cdot x}{\color{blue}{y \cdot y}}, -1\right) \]

      4. times-frac 88.1%

         \[\leadsto \mathsf{fma}\left(0.375, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}, -1\right) \]

      5. unpow2 88.1%

         \[\leadsto \mathsf{fma}\left(0.375, \color{blue}{{\left(\frac{x}{y}\right)}^{2}}, -1\right) \]

      6. metadata-eval 88.1%

         \[\leadsto \mathsf{fma}\left(0.375, {\left(\frac{x}{y}\right)}^{2}, \color{blue}{-1}\right) \]

   10. Simplified 88.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.375, {\left(\frac{x}{y}\right)}^{2}, -1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 0:\\ \;\;\;\;1 + \frac{y}{x} \cdot \left(\frac{y}{x} \cdot -8\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{+245}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.375, {\left(\frac{x}{y}\right)}^{2}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;1 + \frac{y}{x} \cdot \left(\frac{y}{x} \cdot -8\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+245}:\\ \;\;\;\;\frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{t\_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.375, {\left(\frac{x}{y}\right)}^{2}, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 0.0)
     (+ 1.0 (* (/ y x) (* (/ y x) -8.0)))
     (if (<= t_0 1e+245)
       (/ (* (+ x (* y 2.0)) (- x (* y 2.0))) (+ t_0 (* x x)))
       (fma 0.375 (pow (/ x y) 2.0) -1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 + ((y / x) * ((y / x) * -8.0));
	} else if (t_0 <= 1e+245) {
		tmp = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_0 + (x * x));
	} else {
		tmp = fma(0.375, pow((x / y), 2.0), -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(1.0 + Float64(Float64(y / x) * Float64(Float64(y / x) * -8.0)));
	elseif (t_0 <= 1e+245)
		tmp = Float64(Float64(Float64(x + Float64(y * 2.0)) * Float64(x - Float64(y * 2.0))) / Float64(t_0 + Float64(x * x)));
	else
		tmp = fma(0.375, (Float64(x / y) ^ 2.0), -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(1.0 + N[(N[(y / x), $MachinePrecision] * N[(N[(y / x), $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+245], N[(N[(N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision] * N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.375 * N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 0.0

   1. Initial program 50.9%

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

   3. Taylor expanded in y around 0 80.0%

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

      1. *-commutative 80.0%

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

      2. pow2 80.0%

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

      3. add-sqr-sqrt 80.0%

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

      4. associate-*l* 80.0%

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

      5. sqrt-div 80.0%

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

      6. sqrt-pow1 80.0%

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

      7. metadata-eval 80.0%

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

      8. pow1 80.0%

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

      9. sqrt-prod 41.8%

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

      10. add-sqr-sqrt 80.0%

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

      11. sqrt-div 80.0%

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

      12. sqrt-pow1 80.5%

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

      13. metadata-eval 80.5%

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

      14. pow1 80.5%

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

      15. sqrt-prod 47.3%

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

      16. add-sqr-sqrt 96.3%

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

   5. Applied egg-rr 96.3%

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

   if 0.0 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.00000000000000004e245

   1. Initial program 80.3%

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

      1. add-sqr-sqrt 80.3%

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

      2. difference-of-squares 80.3%

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

      3. *-commutative 80.3%

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

      4. associate-*r* 80.1%

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

      5. sqrt-prod 80.1%

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

      6. sqrt-unprod 39.1%

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

      7. add-sqr-sqrt 54.7%

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

      8. metadata-eval 54.7%

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

      9. *-commutative 54.7%

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

      10. associate-*r* 54.7%

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

      11. sqrt-prod 54.7%

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

      12. sqrt-unprod 39.1%

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

      13. add-sqr-sqrt 80.3%

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

      14. metadata-eval 80.3%

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

   4. Applied egg-rr 80.3%

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

   if 1.00000000000000004e245 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 11.9%

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

      1. add-sqr-sqrt 11.9%

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

      2. difference-of-squares 11.9%

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

      3. *-commutative 11.9%

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

      4. associate-*r* 11.9%

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

      5. sqrt-prod 11.9%

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

      6. sqrt-unprod 3.5%

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

      7. add-sqr-sqrt 3.7%

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

      8. metadata-eval 3.7%

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

      9. *-commutative 3.7%

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

      10. associate-*r* 3.7%

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

      11. sqrt-prod 3.7%

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

      12. sqrt-unprod 3.5%

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

      13. add-sqr-sqrt 11.9%

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

      14. metadata-eval 11.9%

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

   4. Applied egg-rr 11.9%

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

      1. add-sqr-sqrt 11.9%

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

      2. times-frac 14.6%

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

      3. +-commutative 14.6%

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

      4. fma-define 14.6%

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

      5. add-sqr-sqrt 14.6%

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

      6. hypot-define 14.6%

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

      7. *-commutative 14.6%

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

      8. associate-*r* 14.6%

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

      9. metadata-eval 14.6%

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

      10. swap-sqr 14.6%

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

      11. sqrt-unprod 5.0%

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

      12. add-sqr-sqrt 14.6%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in y around inf 45.8%

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

   8. Taylor expanded in x around 0 76.4%

      \[\leadsto \color{blue}{0.375 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   9. Step-by-step derivation

      1. fma-neg 76.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.375, \frac{{x}^{2}}{{y}^{2}}, -1\right)} \]

      2. unpow2 76.4%

         \[\leadsto \mathsf{fma}\left(0.375, \frac{\color{blue}{x \cdot x}}{{y}^{2}}, -1\right) \]

      3. unpow2 76.4%

         \[\leadsto \mathsf{fma}\left(0.375, \frac{x \cdot x}{\color{blue}{y \cdot y}}, -1\right) \]

      4. times-frac 88.1%

         \[\leadsto \mathsf{fma}\left(0.375, \color{blue}{\frac{x}{y} \cdot \frac{x}{y}}, -1\right) \]

      5. unpow2 88.1%

         \[\leadsto \mathsf{fma}\left(0.375, \color{blue}{{\left(\frac{x}{y}\right)}^{2}}, -1\right) \]

      6. metadata-eval 88.1%

         \[\leadsto \mathsf{fma}\left(0.375, {\left(\frac{x}{y}\right)}^{2}, \color{blue}{-1}\right) \]

   10. Simplified 88.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.375, {\left(\frac{x}{y}\right)}^{2}, -1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 0:\\ \;\;\;\;1 + \frac{y}{x} \cdot \left(\frac{y}{x} \cdot -8\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{+245}:\\ \;\;\;\;\frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.375, {\left(\frac{x}{y}\right)}^{2}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{x}{y} \cdot 0.5\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;1 + \frac{y}{x} \cdot \left(\frac{y}{x} \cdot -8\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+245}:\\ \;\;\;\;\frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{t\_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t\_1\right) \cdot \left(-1 + t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))) (t_1 (* (/ x y) 0.5)))
   (if (<= t_0 0.0)
     (+ 1.0 (* (/ y x) (* (/ y x) -8.0)))
     (if (<= t_0 1e+245)
       (/ (* (+ x (* y 2.0)) (- x (* y 2.0))) (+ t_0 (* x x)))
       (* (+ 1.0 t_1) (+ -1.0 t_1))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = (x / y) * 0.5;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 + ((y / x) * ((y / x) * -8.0));
	} else if (t_0 <= 1e+245) {
		tmp = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_0 + (x * x));
	} else {
		tmp = (1.0 + 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 = (x / y) * 0.5d0
    if (t_0 <= 0.0d0) then
        tmp = 1.0d0 + ((y / x) * ((y / x) * (-8.0d0)))
    else if (t_0 <= 1d+245) then
        tmp = ((x + (y * 2.0d0)) * (x - (y * 2.0d0))) / (t_0 + (x * x))
    else
        tmp = (1.0d0 + 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 = (x / y) * 0.5;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 + ((y / x) * ((y / x) * -8.0));
	} else if (t_0 <= 1e+245) {
		tmp = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_0 + (x * x));
	} else {
		tmp = (1.0 + t_1) * (-1.0 + t_1);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = (x / y) * 0.5
	tmp = 0
	if t_0 <= 0.0:
		tmp = 1.0 + ((y / x) * ((y / x) * -8.0))
	elif t_0 <= 1e+245:
		tmp = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_0 + (x * x))
	else:
		tmp = (1.0 + t_1) * (-1.0 + t_1)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(Float64(x / y) * 0.5)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(1.0 + Float64(Float64(y / x) * Float64(Float64(y / x) * -8.0)));
	elseif (t_0 <= 1e+245)
		tmp = Float64(Float64(Float64(x + Float64(y * 2.0)) * Float64(x - Float64(y * 2.0))) / Float64(t_0 + Float64(x * x)));
	else
		tmp = Float64(Float64(1.0 + 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 = (x / y) * 0.5;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = 1.0 + ((y / x) * ((y / x) * -8.0));
	elseif (t_0 <= 1e+245)
		tmp = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_0 + (x * x));
	else
		tmp = (1.0 + 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[(N[(x / y), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(1.0 + N[(N[(y / x), $MachinePrecision] * N[(N[(y / x), $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+245], N[(N[(N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision] * N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$1), $MachinePrecision] * N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 0.0

   1. Initial program 50.9%

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

   3. Taylor expanded in y around 0 80.0%

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

      1. *-commutative 80.0%

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

      2. pow2 80.0%

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

      3. add-sqr-sqrt 80.0%

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

      4. associate-*l* 80.0%

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

      5. sqrt-div 80.0%

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

      6. sqrt-pow1 80.0%

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

      7. metadata-eval 80.0%

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

      8. pow1 80.0%

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

      9. sqrt-prod 41.8%

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

      10. add-sqr-sqrt 80.0%

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

      11. sqrt-div 80.0%

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

      12. sqrt-pow1 80.5%

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

      13. metadata-eval 80.5%

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

      14. pow1 80.5%

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

      15. sqrt-prod 47.3%

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

      16. add-sqr-sqrt 96.3%

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

   5. Applied egg-rr 96.3%

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

   if 0.0 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.00000000000000004e245

   1. Initial program 80.3%

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

      1. add-sqr-sqrt 80.3%

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

      2. difference-of-squares 80.3%

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

      3. *-commutative 80.3%

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

      4. associate-*r* 80.1%

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

      5. sqrt-prod 80.1%

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

      6. sqrt-unprod 39.1%

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

      7. add-sqr-sqrt 54.7%

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

      8. metadata-eval 54.7%

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

      9. *-commutative 54.7%

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

      10. associate-*r* 54.7%

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

      11. sqrt-prod 54.7%

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

      12. sqrt-unprod 39.1%

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

      13. add-sqr-sqrt 80.3%

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

      14. metadata-eval 80.3%

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

   4. Applied egg-rr 80.3%

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

   if 1.00000000000000004e245 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 11.9%

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

      1. add-sqr-sqrt 11.9%

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

      2. difference-of-squares 11.9%

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

      3. *-commutative 11.9%

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

      4. associate-*r* 11.9%

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

      5. sqrt-prod 11.9%

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

      6. sqrt-unprod 3.5%

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

      7. add-sqr-sqrt 3.7%

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

      8. metadata-eval 3.7%

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

      9. *-commutative 3.7%

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

      10. associate-*r* 3.7%

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

      11. sqrt-prod 3.7%

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

      12. sqrt-unprod 3.5%

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

      13. add-sqr-sqrt 11.9%

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

      14. metadata-eval 11.9%

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

   4. Applied egg-rr 11.9%

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

      1. add-sqr-sqrt 11.9%

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

      2. times-frac 14.6%

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

      3. +-commutative 14.6%

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

      4. fma-define 14.6%

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

      5. add-sqr-sqrt 14.6%

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

      6. hypot-define 14.6%

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

      7. *-commutative 14.6%

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

      8. associate-*r* 14.6%

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

      9. metadata-eval 14.6%

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

      10. swap-sqr 14.6%

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

      11. sqrt-unprod 5.0%

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

      12. add-sqr-sqrt 14.6%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in y around inf 45.8%

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

   8. Taylor expanded in x around 0 88.0%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 0:\\ \;\;\;\;1 + \frac{y}{x} \cdot \left(\frac{y}{x} \cdot -8\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{+245}:\\ \;\;\;\;\frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y} \cdot 0.5\right) \cdot \left(-1 + \frac{x}{y} \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{x}{y} \cdot 0.5\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;1 + \frac{y}{x} \cdot \left(\frac{y}{x} \cdot -8\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+245}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{t\_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t\_1\right) \cdot \left(-1 + t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))) (t_1 (* (/ x y) 0.5)))
   (if (<= t_0 0.0)
     (+ 1.0 (* (/ y x) (* (/ y x) -8.0)))
     (if (<= t_0 1e+245)
       (/ (- (* x x) t_0) (+ t_0 (* x x)))
       (* (+ 1.0 t_1) (+ -1.0 t_1))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = (x / y) * 0.5;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 + ((y / x) * ((y / x) * -8.0));
	} else if (t_0 <= 1e+245) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = (1.0 + 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 = (x / y) * 0.5d0
    if (t_0 <= 0.0d0) then
        tmp = 1.0d0 + ((y / x) * ((y / x) * (-8.0d0)))
    else if (t_0 <= 1d+245) then
        tmp = ((x * x) - t_0) / (t_0 + (x * x))
    else
        tmp = (1.0d0 + 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 = (x / y) * 0.5;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 + ((y / x) * ((y / x) * -8.0));
	} else if (t_0 <= 1e+245) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = (1.0 + t_1) * (-1.0 + t_1);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = (x / y) * 0.5
	tmp = 0
	if t_0 <= 0.0:
		tmp = 1.0 + ((y / x) * ((y / x) * -8.0))
	elif t_0 <= 1e+245:
		tmp = ((x * x) - t_0) / (t_0 + (x * x))
	else:
		tmp = (1.0 + t_1) * (-1.0 + t_1)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(Float64(x / y) * 0.5)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(1.0 + Float64(Float64(y / x) * Float64(Float64(y / x) * -8.0)));
	elseif (t_0 <= 1e+245)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(t_0 + Float64(x * x)));
	else
		tmp = Float64(Float64(1.0 + 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 = (x / y) * 0.5;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = 1.0 + ((y / x) * ((y / x) * -8.0));
	elseif (t_0 <= 1e+245)
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	else
		tmp = (1.0 + 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[(N[(x / y), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(1.0 + N[(N[(y / x), $MachinePrecision] * N[(N[(y / x), $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+245], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$1), $MachinePrecision] * N[(-1.0 + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 0.0

   1. Initial program 50.9%

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

   3. Taylor expanded in y around 0 80.0%

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

      1. *-commutative 80.0%

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

      2. pow2 80.0%

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

      3. add-sqr-sqrt 80.0%

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

      4. associate-*l* 80.0%

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

      5. sqrt-div 80.0%

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

      6. sqrt-pow1 80.0%

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

      7. metadata-eval 80.0%

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

      8. pow1 80.0%

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

      9. sqrt-prod 41.8%

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

      10. add-sqr-sqrt 80.0%

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

      11. sqrt-div 80.0%

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

      12. sqrt-pow1 80.5%

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

      13. metadata-eval 80.5%

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

      14. pow1 80.5%

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

      15. sqrt-prod 47.3%

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

      16. add-sqr-sqrt 96.3%

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

   5. Applied egg-rr 96.3%

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

   if 0.0 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.00000000000000004e245

   1. Initial program 80.3%

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

   if 1.00000000000000004e245 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 11.9%

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

      1. add-sqr-sqrt 11.9%

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

      2. difference-of-squares 11.9%

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

      3. *-commutative 11.9%

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

      4. associate-*r* 11.9%

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

      5. sqrt-prod 11.9%

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

      6. sqrt-unprod 3.5%

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

      7. add-sqr-sqrt 3.7%

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

      8. metadata-eval 3.7%

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

      9. *-commutative 3.7%

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

      10. associate-*r* 3.7%

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

      11. sqrt-prod 3.7%

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

      12. sqrt-unprod 3.5%

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

      13. add-sqr-sqrt 11.9%

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

      14. metadata-eval 11.9%

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

   4. Applied egg-rr 11.9%

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

      1. add-sqr-sqrt 11.9%

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

      2. times-frac 14.6%

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

      3. +-commutative 14.6%

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

      4. fma-define 14.6%

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

      5. add-sqr-sqrt 14.6%

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

      6. hypot-define 14.6%

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

      7. *-commutative 14.6%

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

      8. associate-*r* 14.6%

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

      9. metadata-eval 14.6%

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

      10. swap-sqr 14.6%

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

      11. sqrt-unprod 5.0%

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

      12. add-sqr-sqrt 14.6%

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

   6. Applied egg-rr 98.8%

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

   7. Taylor expanded in y around inf 45.8%

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

   8. Taylor expanded in x around 0 88.0%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 0:\\ \;\;\;\;1 + \frac{y}{x} \cdot \left(\frac{y}{x} \cdot -8\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{+245}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y} \cdot 0.5\right) \cdot \left(-1 + \frac{x}{y} \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{y} \cdot 0.5\\ \mathbf{if}\;y \leq 8.5 \cdot 10^{-22}:\\ \;\;\;\;1 + \frac{y}{x} \cdot \left(\frac{y}{x} \cdot -8\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + t\_0\right) \cdot \left(-1 + t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (/ x y) 0.5)))
   (if (<= y 8.5e-22)
     (+ 1.0 (* (/ y x) (* (/ y x) -8.0)))
     (* (+ 1.0 t_0) (+ -1.0 t_0)))))

C

double code(double x, double y) {
	double t_0 = (x / y) * 0.5;
	double tmp;
	if (y <= 8.5e-22) {
		tmp = 1.0 + ((y / x) * ((y / x) * -8.0));
	} else {
		tmp = (1.0 + t_0) * (-1.0 + 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) * 0.5d0
    if (y <= 8.5d-22) then
        tmp = 1.0d0 + ((y / x) * ((y / x) * (-8.0d0)))
    else
        tmp = (1.0d0 + t_0) * ((-1.0d0) + t_0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x / y) * 0.5;
	double tmp;
	if (y <= 8.5e-22) {
		tmp = 1.0 + ((y / x) * ((y / x) * -8.0));
	} else {
		tmp = (1.0 + t_0) * (-1.0 + t_0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x / y) * 0.5
	tmp = 0
	if y <= 8.5e-22:
		tmp = 1.0 + ((y / x) * ((y / x) * -8.0))
	else:
		tmp = (1.0 + t_0) * (-1.0 + t_0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x / y) * 0.5)
	tmp = 0.0
	if (y <= 8.5e-22)
		tmp = Float64(1.0 + Float64(Float64(y / x) * Float64(Float64(y / x) * -8.0)));
	else
		tmp = Float64(Float64(1.0 + t_0) * Float64(-1.0 + t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x / y) * 0.5;
	tmp = 0.0;
	if (y <= 8.5e-22)
		tmp = 1.0 + ((y / x) * ((y / x) * -8.0));
	else
		tmp = (1.0 + t_0) * (-1.0 + t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[y, 8.5e-22], N[(1.0 + N[(N[(y / x), $MachinePrecision] * N[(N[(y / x), $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + t$95$0), $MachinePrecision] * N[(-1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 8.5000000000000001e-22

   1. Initial program 55.2%

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

   3. Taylor expanded in y around 0 54.2%

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

      1. *-commutative 54.2%

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

      2. pow2 54.2%

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

      3. add-sqr-sqrt 54.2%

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

      4. associate-*l* 54.2%

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

      5. sqrt-div 54.2%

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

      6. sqrt-pow1 52.8%

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

      7. metadata-eval 52.8%

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

      8. pow1 52.8%

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

      9. sqrt-prod 32.1%

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

      10. add-sqr-sqrt 53.4%

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

      11. sqrt-div 53.4%

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

      12. sqrt-pow1 54.6%

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

      13. metadata-eval 54.6%

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

      14. pow1 54.6%

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

      15. sqrt-prod 34.4%

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

      16. add-sqr-sqrt 60.5%

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

   5. Applied egg-rr 60.5%

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

   if 8.5000000000000001e-22 < y

   1. Initial program 42.7%

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

      1. add-sqr-sqrt 42.7%

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

      2. difference-of-squares 42.7%

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

      3. *-commutative 42.7%

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

      4. associate-*r* 42.7%

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

      5. sqrt-prod 42.7%

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

      6. sqrt-unprod 42.4%

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

      7. add-sqr-sqrt 42.7%

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

      8. metadata-eval 42.7%

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

      9. *-commutative 42.7%

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

      10. associate-*r* 42.7%

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

      11. sqrt-prod 42.7%

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

      12. sqrt-unprod 42.4%

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

      13. add-sqr-sqrt 42.7%

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

      14. metadata-eval 42.7%

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

   4. Applied egg-rr 42.7%

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

      1. add-sqr-sqrt 42.7%

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

      2. times-frac 44.5%

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

      3. +-commutative 44.5%

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

      4. fma-define 44.5%

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

      5. add-sqr-sqrt 44.5%

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

      6. hypot-define 44.5%

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

      7. *-commutative 44.5%

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

      8. associate-*r* 44.5%

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

      9. metadata-eval 44.5%

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

      10. swap-sqr 44.5%

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

      11. sqrt-unprod 44.1%

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

      12. add-sqr-sqrt 44.5%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around inf 81.8%

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

   8. Taylor expanded in x around 0 81.5%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-22}:\\ \;\;\;\;1 + \frac{y}{x} \cdot \left(\frac{y}{x} \cdot -8\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y} \cdot 0.5\right) \cdot \left(-1 + \frac{x}{y} \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-20}:\\ \;\;\;\;1 + \frac{y}{x} \cdot \left(\frac{y}{x} \cdot -8\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.8e-20) (+ 1.0 (* (/ y x) (* (/ y x) -8.0))) -1.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.8e-20) {
		tmp = 1.0 + ((y / x) * ((y / x) * -8.0));
	} 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 (y <= 4.8d-20) then
        tmp = 1.0d0 + ((y / x) * ((y / x) * (-8.0d0)))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.8e-20) {
		tmp = 1.0 + ((y / x) * ((y / x) * -8.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.8e-20:
		tmp = 1.0 + ((y / x) * ((y / x) * -8.0))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.8e-20)
		tmp = Float64(1.0 + Float64(Float64(y / x) * Float64(Float64(y / x) * -8.0)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.8e-20)
		tmp = 1.0 + ((y / x) * ((y / x) * -8.0));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.8e-20], N[(1.0 + N[(N[(y / x), $MachinePrecision] * N[(N[(y / x), $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 4.79999999999999986e-20

   1. Initial program 54.9%

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

   3. Taylor expanded in y around 0 54.4%

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

      1. *-commutative 54.4%

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

      2. pow2 54.4%

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

      3. add-sqr-sqrt 54.4%

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

      4. associate-*l* 54.4%

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

      5. sqrt-div 54.4%

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

      6. sqrt-pow1 53.1%

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

      7. metadata-eval 53.1%

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

      8. pow1 53.1%

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

      9. sqrt-prod 31.9%

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

      10. add-sqr-sqrt 53.7%

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

      11. sqrt-div 53.7%

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

      12. sqrt-pow1 54.9%

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

      13. metadata-eval 54.9%

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

      14. pow1 54.9%

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

      15. sqrt-prod 34.2%

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

      16. add-sqr-sqrt 60.7%

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

   5. Applied egg-rr 60.7%

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

   if 4.79999999999999986e-20 < y

   1. Initial program 43.2%

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

   3. Taylor expanded in x around 0 81.9%

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

Alternative 8: 61.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-19}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 7e-19) 1.0 -1.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7e-19) {
		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 (y <= 7d-19) 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 (y <= 7e-19) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7e-19:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7e-19)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7e-19)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7e-19], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 7.00000000000000031e-19

   1. Initial program 54.9%

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

   3. Taylor expanded in x around inf 59.0%

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

   if 7.00000000000000031e-19 < y

   1. Initial program 43.2%

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

   3. Taylor expanded in x around 0 81.9%

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

Alternative 9: 50.8% 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 51.5%

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

3. Taylor expanded in x around 0 53.2%

   \[\leadsto \color{blue}{-1} \]
4. Add Preprocessing

Developer target: 50.6% 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 2024096 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (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))))