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

Percentage Accurate: 49.8% → 99.9%

Time: 8.7s

Alternatives: 10

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: 49.8% 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

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

FPCore

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

C

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

Java

public static double code(double x, double y) {
	double t_0 = Math.hypot((y * 2.0), x);
	return (((y * 2.0) + x) / t_0) * ((x + (y * -2.0)) / t_0);
}

Python

def code(x, y):
	t_0 = math.hypot((y * 2.0), x)
	return (((y * 2.0) + x) / t_0) * ((x + (y * -2.0)) / t_0)

Julia

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

MATLAB

function tmp = code(x, y)
	t_0 = hypot((y * 2.0), x);
	tmp = (((y * 2.0) + x) / t_0) * ((x + (y * -2.0)) / t_0);
end

Wolfram

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

Derivation

1. Initial program 52.3%

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

   1. add-sqr-sqrt 52.3%

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

   2. difference-of-squares 52.3%

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

   3. *-commutative 52.3%

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

   4. associate-*r* 52.3%

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

   5. sqrt-prod 52.3%

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

   6. sqrt-unprod 30.7%

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

   7. add-sqr-sqrt 42.4%

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

   8. metadata-eval 42.4%

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

   9. *-commutative 42.4%

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

   10. associate-*r* 42.4%

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

   11. sqrt-prod 42.4%

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

   12. sqrt-unprod 30.7%

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

   13. add-sqr-sqrt 52.3%

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

   14. metadata-eval 52.3%

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

4. Applied egg-rr 52.3%

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

   1. add-sqr-sqrt 52.3%

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

   2. times-frac 53.8%

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

   3. +-commutative 53.8%

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

   4. fma-define 53.8%

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

   5. +-commutative 53.8%

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

   6. add-sqr-sqrt 53.8%

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

   7. hypot-define 53.8%

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

   8. *-commutative 53.8%

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

   9. sqrt-prod 31.4%

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

   10. sqrt-prod 31.4%

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

   11. metadata-eval 31.4%

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

   12. associate-*l* 31.4%

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

   13. add-sqr-sqrt 53.8%

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

6. Applied egg-rr 99.6%

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

   1. fma-undefine 99.6%

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

8. Applied egg-rr 99.6%

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

Alternative 2: 80.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-261}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+206}:\\ \;\;\;\;\frac{\left(y \cdot 2 + x\right) \cdot \left(x - y \cdot 2\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-8, {\left(\frac{y}{x}\right)}^{2}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 1e-261)
   (+ (* 0.5 (* (/ x y) (/ x y))) -1.0)
   (if (<= (* x x) 2e+206)
     (/ (* (+ (* y 2.0) x) (- x (* y 2.0))) (+ (* x x) (* y (* y 4.0))))
     (fma -8.0 (pow (/ y x) 2.0) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1e-261) {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	} else if ((x * x) <= 2e+206) {
		tmp = (((y * 2.0) + x) * (x - (y * 2.0))) / ((x * x) + (y * (y * 4.0)));
	} else {
		tmp = fma(-8.0, pow((y / x), 2.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 1e-261)
		tmp = Float64(Float64(0.5 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	elseif (Float64(x * x) <= 2e+206)
		tmp = Float64(Float64(Float64(Float64(y * 2.0) + x) * Float64(x - Float64(y * 2.0))) / Float64(Float64(x * x) + Float64(y * Float64(y * 4.0))));
	else
		tmp = fma(-8.0, (Float64(y / x) ^ 2.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-261], N[(N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+206], N[(N[(N[(N[(y * 2.0), $MachinePrecision] + x), $MachinePrecision] * N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-8.0 * N[Power[N[(y / x), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 9.99999999999999984e-262

   1. Initial program 57.6%

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

      1. fma-neg 57.6%

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

      2. *-commutative 57.6%

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

      3. distribute-rgt-neg-in 57.6%

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

      4. distribute-rgt-neg-in 57.6%

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

      5. metadata-eval 57.6%

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

      6. fma-define 57.6%

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

      7. *-commutative 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in x around 0 80.4%

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

      1. pow2 80.4%

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

      2. unpow2 80.4%

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

      3. times-frac 85.8%

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

   7. Applied egg-rr 85.8%

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

   if 9.99999999999999984e-262 < (*.f64 x x) < 2.0000000000000001e206

   1. Initial program 79.7%

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

      1. add-sqr-sqrt 79.7%

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

      2. difference-of-squares 79.7%

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

      3. *-commutative 79.7%

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

      4. associate-*r* 79.7%

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

      5. sqrt-prod 79.7%

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

      6. sqrt-unprod 49.8%

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

      7. add-sqr-sqrt 70.4%

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

      8. metadata-eval 70.4%

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

      9. *-commutative 70.4%

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

      10. associate-*r* 70.4%

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

      11. sqrt-prod 70.4%

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

      12. sqrt-unprod 49.8%

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

      13. add-sqr-sqrt 79.7%

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

      14. metadata-eval 79.7%

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

   4. Applied egg-rr 79.7%

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

   if 2.0000000000000001e206 < (*.f64 x x)

   1. Initial program 15.1%

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

      1. add-sqr-sqrt 15.1%

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

      2. difference-of-squares 15.1%

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

      3. *-commutative 15.1%

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

      4. associate-*r* 15.1%

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

      5. sqrt-prod 15.1%

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

      6. sqrt-unprod 7.0%

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

      7. add-sqr-sqrt 15.1%

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

      8. metadata-eval 15.1%

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

      9. *-commutative 15.1%

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

      10. associate-*r* 15.1%

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

      11. sqrt-prod 15.1%

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

      12. sqrt-unprod 7.0%

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

      13. add-sqr-sqrt 15.1%

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

      14. metadata-eval 15.1%

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

   4. Applied egg-rr 15.1%

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

      1. add-sqr-sqrt 15.1%

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

      2. times-frac 17.8%

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

      3. +-commutative 17.8%

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

      4. fma-define 17.8%

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

      5. +-commutative 17.8%

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

      6. add-sqr-sqrt 17.8%

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

      7. hypot-define 17.8%

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

      8. *-commutative 17.8%

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

      9. sqrt-prod 8.5%

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

      10. sqrt-prod 8.5%

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

      11. metadata-eval 8.5%

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

      12. associate-*l* 8.5%

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

      13. add-sqr-sqrt 17.8%

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

   6. Applied egg-rr 99.9%

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

      1. fma-undefine 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in y around 0 68.9%

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

       1. +-commutative 68.9%

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

       2. fma-define 68.9%

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

       3. unpow2 68.9%

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

       4. unpow2 68.9%

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

       5. times-frac 86.3%

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

       6. unpow2 86.3%

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

   11. Simplified 86.3%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-261}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+206}:\\ \;\;\;\;\frac{\left(y \cdot 2 + x\right) \cdot \left(x - y \cdot 2\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-8, {\left(\frac{y}{x}\right)}^{2}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 67.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot 2 + x\\ \mathbf{if}\;x \cdot x \leq 10^{-261}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+206}:\\ \;\;\;\;\frac{t\_0 \cdot \left(x - y \cdot 2\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\mathsf{hypot}\left(y \cdot 2, x\right)} \cdot \left(1 + -2 \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ (* y 2.0) x)))
   (if (<= (* x x) 1e-261)
     (+ (* 0.5 (* (/ x y) (/ x y))) -1.0)
     (if (<= (* x x) 2e+206)
       (/ (* t_0 (- x (* y 2.0))) (+ (* x x) (* y (* y 4.0))))
       (* (/ t_0 (hypot (* y 2.0) x)) (+ 1.0 (* -2.0 (/ y x))))))))

C

double code(double x, double y) {
	double t_0 = (y * 2.0) + x;
	double tmp;
	if ((x * x) <= 1e-261) {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	} else if ((x * x) <= 2e+206) {
		tmp = (t_0 * (x - (y * 2.0))) / ((x * x) + (y * (y * 4.0)));
	} else {
		tmp = (t_0 / hypot((y * 2.0), x)) * (1.0 + (-2.0 * (y / x)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (y * 2.0) + x;
	double tmp;
	if ((x * x) <= 1e-261) {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	} else if ((x * x) <= 2e+206) {
		tmp = (t_0 * (x - (y * 2.0))) / ((x * x) + (y * (y * 4.0)));
	} else {
		tmp = (t_0 / Math.hypot((y * 2.0), x)) * (1.0 + (-2.0 * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * 2.0) + x
	tmp = 0
	if (x * x) <= 1e-261:
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0
	elif (x * x) <= 2e+206:
		tmp = (t_0 * (x - (y * 2.0))) / ((x * x) + (y * (y * 4.0)))
	else:
		tmp = (t_0 / math.hypot((y * 2.0), x)) * (1.0 + (-2.0 * (y / x)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * 2.0) + x)
	tmp = 0.0
	if (Float64(x * x) <= 1e-261)
		tmp = Float64(Float64(0.5 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	elseif (Float64(x * x) <= 2e+206)
		tmp = Float64(Float64(t_0 * Float64(x - Float64(y * 2.0))) / Float64(Float64(x * x) + Float64(y * Float64(y * 4.0))));
	else
		tmp = Float64(Float64(t_0 / hypot(Float64(y * 2.0), x)) * Float64(1.0 + Float64(-2.0 * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * 2.0) + x;
	tmp = 0.0;
	if ((x * x) <= 1e-261)
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	elseif ((x * x) <= 2e+206)
		tmp = (t_0 * (x - (y * 2.0))) / ((x * x) + (y * (y * 4.0)));
	else
		tmp = (t_0 / hypot((y * 2.0), x)) * (1.0 + (-2.0 * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * 2.0), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 1e-261], N[(N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+206], N[(N[(t$95$0 * N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$0 / N[Sqrt[N[(y * 2.0), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(-2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 9.99999999999999984e-262

   1. Initial program 57.6%

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

      1. fma-neg 57.6%

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

      2. *-commutative 57.6%

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

      3. distribute-rgt-neg-in 57.6%

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

      4. distribute-rgt-neg-in 57.6%

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

      5. metadata-eval 57.6%

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

      6. fma-define 57.6%

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

      7. *-commutative 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in x around 0 80.4%

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

      1. pow2 80.4%

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

      2. unpow2 80.4%

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

      3. times-frac 85.8%

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

   7. Applied egg-rr 85.8%

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

   if 9.99999999999999984e-262 < (*.f64 x x) < 2.0000000000000001e206

   1. Initial program 79.7%

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

      1. add-sqr-sqrt 79.7%

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

      2. difference-of-squares 79.7%

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

      3. *-commutative 79.7%

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

      4. associate-*r* 79.7%

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

      5. sqrt-prod 79.7%

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

      6. sqrt-unprod 49.8%

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

      7. add-sqr-sqrt 70.4%

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

      8. metadata-eval 70.4%

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

      9. *-commutative 70.4%

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

      10. associate-*r* 70.4%

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

      11. sqrt-prod 70.4%

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

      12. sqrt-unprod 49.8%

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

      13. add-sqr-sqrt 79.7%

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

      14. metadata-eval 79.7%

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

   4. Applied egg-rr 79.7%

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

   if 2.0000000000000001e206 < (*.f64 x x)

   1. Initial program 15.1%

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

      1. add-sqr-sqrt 15.1%

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

      2. difference-of-squares 15.1%

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

      3. *-commutative 15.1%

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

      4. associate-*r* 15.1%

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

      5. sqrt-prod 15.1%

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

      6. sqrt-unprod 7.0%

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

      7. add-sqr-sqrt 15.1%

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

      8. metadata-eval 15.1%

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

      9. *-commutative 15.1%

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

      10. associate-*r* 15.1%

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

      11. sqrt-prod 15.1%

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

      12. sqrt-unprod 7.0%

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

      13. add-sqr-sqrt 15.1%

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

      14. metadata-eval 15.1%

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

   4. Applied egg-rr 15.1%

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

      1. add-sqr-sqrt 15.1%

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

      2. times-frac 17.8%

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

      3. +-commutative 17.8%

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

      4. fma-define 17.8%

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

      5. +-commutative 17.8%

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

      6. add-sqr-sqrt 17.8%

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

      7. hypot-define 17.8%

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

      8. *-commutative 17.8%

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

      9. sqrt-prod 8.5%

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

      10. sqrt-prod 8.5%

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

      11. metadata-eval 8.5%

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

      12. associate-*l* 8.5%

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

      13. add-sqr-sqrt 17.8%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 45.3%

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

      1. fma-undefine 99.9%

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

   9. Applied egg-rr 45.3%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-261}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+206}:\\ \;\;\;\;\frac{\left(y \cdot 2 + x\right) \cdot \left(x - y \cdot 2\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot 2 + x}{\mathsf{hypot}\left(y \cdot 2, x\right)} \cdot \left(1 + -2 \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-261}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+206}:\\ \;\;\;\;\frac{\left(y \cdot 2 + x\right) \cdot \left(x - y \cdot 2\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot -2}{\mathsf{hypot}\left(y \cdot 2, x\right)} \cdot \left(1 + 2 \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 1e-261)
   (+ (* 0.5 (* (/ x y) (/ x y))) -1.0)
   (if (<= (* x x) 2e+206)
     (/ (* (+ (* y 2.0) x) (- x (* y 2.0))) (+ (* x x) (* y (* y 4.0))))
     (* (/ (+ x (* y -2.0)) (hypot (* y 2.0) x)) (+ 1.0 (* 2.0 (/ y x)))))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1e-261) {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	} else if ((x * x) <= 2e+206) {
		tmp = (((y * 2.0) + x) * (x - (y * 2.0))) / ((x * x) + (y * (y * 4.0)));
	} else {
		tmp = ((x + (y * -2.0)) / hypot((y * 2.0), x)) * (1.0 + (2.0 * (y / x)));
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1e-261) {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	} else if ((x * x) <= 2e+206) {
		tmp = (((y * 2.0) + x) * (x - (y * 2.0))) / ((x * x) + (y * (y * 4.0)));
	} else {
		tmp = ((x + (y * -2.0)) / Math.hypot((y * 2.0), x)) * (1.0 + (2.0 * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 1e-261:
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0
	elif (x * x) <= 2e+206:
		tmp = (((y * 2.0) + x) * (x - (y * 2.0))) / ((x * x) + (y * (y * 4.0)))
	else:
		tmp = ((x + (y * -2.0)) / math.hypot((y * 2.0), x)) * (1.0 + (2.0 * (y / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 1e-261)
		tmp = Float64(Float64(0.5 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	elseif (Float64(x * x) <= 2e+206)
		tmp = Float64(Float64(Float64(Float64(y * 2.0) + x) * Float64(x - Float64(y * 2.0))) / Float64(Float64(x * x) + Float64(y * Float64(y * 4.0))));
	else
		tmp = Float64(Float64(Float64(x + Float64(y * -2.0)) / hypot(Float64(y * 2.0), x)) * Float64(1.0 + Float64(2.0 * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 1e-261)
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	elseif ((x * x) <= 2e+206)
		tmp = (((y * 2.0) + x) * (x - (y * 2.0))) / ((x * x) + (y * (y * 4.0)));
	else
		tmp = ((x + (y * -2.0)) / hypot((y * 2.0), x)) * (1.0 + (2.0 * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-261], N[(N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+206], N[(N[(N[(N[(y * 2.0), $MachinePrecision] + x), $MachinePrecision] * N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + N[(y * -2.0), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(y * 2.0), $MachinePrecision] ^ 2 + x ^ 2], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 9.99999999999999984e-262

   1. Initial program 57.6%

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

      1. fma-neg 57.6%

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

      2. *-commutative 57.6%

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

      3. distribute-rgt-neg-in 57.6%

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

      4. distribute-rgt-neg-in 57.6%

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

      5. metadata-eval 57.6%

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

      6. fma-define 57.6%

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

      7. *-commutative 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in x around 0 80.4%

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

      1. pow2 80.4%

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

      2. unpow2 80.4%

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

      3. times-frac 85.8%

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

   7. Applied egg-rr 85.8%

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

   if 9.99999999999999984e-262 < (*.f64 x x) < 2.0000000000000001e206

   1. Initial program 79.7%

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

      1. add-sqr-sqrt 79.7%

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

      2. difference-of-squares 79.7%

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

      3. *-commutative 79.7%

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

      4. associate-*r* 79.7%

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

      5. sqrt-prod 79.7%

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

      6. sqrt-unprod 49.8%

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

      7. add-sqr-sqrt 70.4%

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

      8. metadata-eval 70.4%

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

      9. *-commutative 70.4%

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

      10. associate-*r* 70.4%

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

      11. sqrt-prod 70.4%

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

      12. sqrt-unprod 49.8%

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

      13. add-sqr-sqrt 79.7%

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

      14. metadata-eval 79.7%

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

   4. Applied egg-rr 79.7%

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

   if 2.0000000000000001e206 < (*.f64 x x)

   1. Initial program 15.1%

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

      1. add-sqr-sqrt 15.1%

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

      2. difference-of-squares 15.1%

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

      3. *-commutative 15.1%

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

      4. associate-*r* 15.1%

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

      5. sqrt-prod 15.1%

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

      6. sqrt-unprod 7.0%

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

      7. add-sqr-sqrt 15.1%

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

      8. metadata-eval 15.1%

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

      9. *-commutative 15.1%

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

      10. associate-*r* 15.1%

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

      11. sqrt-prod 15.1%

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

      12. sqrt-unprod 7.0%

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

      13. add-sqr-sqrt 15.1%

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

      14. metadata-eval 15.1%

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

   4. Applied egg-rr 15.1%

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

      1. add-sqr-sqrt 15.1%

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

      2. times-frac 17.8%

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

      3. +-commutative 17.8%

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

      4. fma-define 17.8%

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

      5. +-commutative 17.8%

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

      6. add-sqr-sqrt 17.8%

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

      7. hypot-define 17.8%

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

      8. *-commutative 17.8%

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

      9. sqrt-prod 8.5%

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

      10. sqrt-prod 8.5%

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

      11. metadata-eval 8.5%

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

      12. associate-*l* 8.5%

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

      13. add-sqr-sqrt 17.8%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0 45.3%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-261}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+206}:\\ \;\;\;\;\frac{\left(y \cdot 2 + x\right) \cdot \left(x - y \cdot 2\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot -2}{\mathsf{hypot}\left(y \cdot 2, x\right)} \cdot \left(1 + 2 \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-261}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+206}:\\ \;\;\;\;\frac{\left(y \cdot 2 + x\right) \cdot \left(x - y \cdot 2\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -2 \cdot \frac{y}{x}\right) \cdot \left(1 + 2 \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 1e-261)
   (+ (* 0.5 (* (/ x y) (/ x y))) -1.0)
   (if (<= (* x x) 2e+206)
     (/ (* (+ (* y 2.0) x) (- x (* y 2.0))) (+ (* x x) (* y (* y 4.0))))
     (* (+ 1.0 (* -2.0 (/ y x))) (+ 1.0 (* 2.0 (/ y x)))))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1e-261) {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	} else if ((x * x) <= 2e+206) {
		tmp = (((y * 2.0) + x) * (x - (y * 2.0))) / ((x * x) + (y * (y * 4.0)));
	} else {
		tmp = (1.0 + (-2.0 * (y / x))) * (1.0 + (2.0 * (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x * x) <= 1d-261) then
        tmp = (0.5d0 * ((x / y) * (x / y))) + (-1.0d0)
    else if ((x * x) <= 2d+206) then
        tmp = (((y * 2.0d0) + x) * (x - (y * 2.0d0))) / ((x * x) + (y * (y * 4.0d0)))
    else
        tmp = (1.0d0 + ((-2.0d0) * (y / x))) * (1.0d0 + (2.0d0 * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1e-261) {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	} else if ((x * x) <= 2e+206) {
		tmp = (((y * 2.0) + x) * (x - (y * 2.0))) / ((x * x) + (y * (y * 4.0)));
	} else {
		tmp = (1.0 + (-2.0 * (y / x))) * (1.0 + (2.0 * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 1e-261:
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0
	elif (x * x) <= 2e+206:
		tmp = (((y * 2.0) + x) * (x - (y * 2.0))) / ((x * x) + (y * (y * 4.0)))
	else:
		tmp = (1.0 + (-2.0 * (y / x))) * (1.0 + (2.0 * (y / x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 1e-261)
		tmp = Float64(Float64(0.5 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	elseif (Float64(x * x) <= 2e+206)
		tmp = Float64(Float64(Float64(Float64(y * 2.0) + x) * Float64(x - Float64(y * 2.0))) / Float64(Float64(x * x) + Float64(y * Float64(y * 4.0))));
	else
		tmp = Float64(Float64(1.0 + Float64(-2.0 * Float64(y / x))) * Float64(1.0 + Float64(2.0 * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 1e-261)
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	elseif ((x * x) <= 2e+206)
		tmp = (((y * 2.0) + x) * (x - (y * 2.0))) / ((x * x) + (y * (y * 4.0)));
	else
		tmp = (1.0 + (-2.0 * (y / x))) * (1.0 + (2.0 * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-261], N[(N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+206], N[(N[(N[(N[(y * 2.0), $MachinePrecision] + x), $MachinePrecision] * N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 9.99999999999999984e-262

   1. Initial program 57.6%

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

      1. fma-neg 57.6%

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

      2. *-commutative 57.6%

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

      3. distribute-rgt-neg-in 57.6%

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

      4. distribute-rgt-neg-in 57.6%

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

      5. metadata-eval 57.6%

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

      6. fma-define 57.6%

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

      7. *-commutative 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in x around 0 80.4%

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

      1. pow2 80.4%

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

      2. unpow2 80.4%

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

      3. times-frac 85.8%

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

   7. Applied egg-rr 85.8%

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

   if 9.99999999999999984e-262 < (*.f64 x x) < 2.0000000000000001e206

   1. Initial program 79.7%

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

      1. add-sqr-sqrt 79.7%

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

      2. difference-of-squares 79.7%

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

      3. *-commutative 79.7%

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

      4. associate-*r* 79.7%

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

      5. sqrt-prod 79.7%

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

      6. sqrt-unprod 49.8%

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

      7. add-sqr-sqrt 70.4%

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

      8. metadata-eval 70.4%

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

      9. *-commutative 70.4%

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

      10. associate-*r* 70.4%

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

      11. sqrt-prod 70.4%

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

      12. sqrt-unprod 49.8%

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

      13. add-sqr-sqrt 79.7%

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

      14. metadata-eval 79.7%

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

   4. Applied egg-rr 79.7%

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

   if 2.0000000000000001e206 < (*.f64 x x)

   1. Initial program 15.1%

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

      1. add-sqr-sqrt 15.1%

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

      2. difference-of-squares 15.1%

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

      3. *-commutative 15.1%

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

      4. associate-*r* 15.1%

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

      5. sqrt-prod 15.1%

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

      6. sqrt-unprod 7.0%

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

      7. add-sqr-sqrt 15.1%

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

      8. metadata-eval 15.1%

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

      9. *-commutative 15.1%

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

      10. associate-*r* 15.1%

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

      11. sqrt-prod 15.1%

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

      12. sqrt-unprod 7.0%

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

      13. add-sqr-sqrt 15.1%

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

      14. metadata-eval 15.1%

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

   4. Applied egg-rr 15.1%

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

      1. add-sqr-sqrt 15.1%

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

      2. times-frac 17.8%

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

      3. +-commutative 17.8%

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

      4. fma-define 17.8%

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

      5. +-commutative 17.8%

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

      6. add-sqr-sqrt 17.8%

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

      7. hypot-define 17.8%

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

      8. *-commutative 17.8%

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

      9. sqrt-prod 8.5%

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

      10. sqrt-prod 8.5%

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

      11. metadata-eval 8.5%

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

      12. associate-*l* 8.5%

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

      13. add-sqr-sqrt 17.8%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 45.3%

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

   8. Taylor expanded in y around 0 85.6%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-261}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+206}:\\ \;\;\;\;\frac{\left(y \cdot 2 + x\right) \cdot \left(x - y \cdot 2\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -2 \cdot \frac{y}{x}\right) \cdot \left(1 + 2 \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 10^{-261}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+206}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -2 \cdot \frac{y}{x}\right) \cdot \left(1 + 2 \cdot \frac{y}{x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 1e-261)
     (+ (* 0.5 (* (/ x y) (/ x y))) -1.0)
     (if (<= (* x x) 2e+206)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (* (+ 1.0 (* -2.0 (/ y x))) (+ 1.0 (* 2.0 (/ y x))))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 1e-261) {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	} else if ((x * x) <= 2e+206) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = (1.0 + (-2.0 * (y / x))) * (1.0 + (2.0 * (y / x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    if ((x * x) <= 1d-261) then
        tmp = (0.5d0 * ((x / y) * (x / y))) + (-1.0d0)
    else if ((x * x) <= 2d+206) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else
        tmp = (1.0d0 + ((-2.0d0) * (y / x))) * (1.0d0 + (2.0d0 * (y / x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 1e-261) {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	} else if ((x * x) <= 2e+206) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = (1.0 + (-2.0 * (y / x))) * (1.0 + (2.0 * (y / x)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if (x * x) <= 1e-261:
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0
	elif (x * x) <= 2e+206:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	else:
		tmp = (1.0 + (-2.0 * (y / x))) * (1.0 + (2.0 * (y / x)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 1e-261)
		tmp = Float64(Float64(0.5 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	elseif (Float64(x * x) <= 2e+206)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = Float64(Float64(1.0 + Float64(-2.0 * Float64(y / x))) * Float64(1.0 + Float64(2.0 * Float64(y / x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if ((x * x) <= 1e-261)
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	elseif ((x * x) <= 2e+206)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	else
		tmp = (1.0 + (-2.0 * (y / x))) * (1.0 + (2.0 * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 1e-261], N[(N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+206], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(-2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 9.99999999999999984e-262

   1. Initial program 57.6%

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

      1. fma-neg 57.6%

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

      2. *-commutative 57.6%

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

      3. distribute-rgt-neg-in 57.6%

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

      4. distribute-rgt-neg-in 57.6%

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

      5. metadata-eval 57.6%

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

      6. fma-define 57.6%

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

      7. *-commutative 57.6%

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

   3. Simplified 57.6%

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

   5. Taylor expanded in x around 0 80.4%

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

      1. pow2 80.4%

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

      2. unpow2 80.4%

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

      3. times-frac 85.8%

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

   7. Applied egg-rr 85.8%

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

   if 9.99999999999999984e-262 < (*.f64 x x) < 2.0000000000000001e206

   1. Initial program 79.7%

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

   if 2.0000000000000001e206 < (*.f64 x x)

   1. Initial program 15.1%

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

      1. add-sqr-sqrt 15.1%

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

      2. difference-of-squares 15.1%

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

      3. *-commutative 15.1%

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

      4. associate-*r* 15.1%

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

      5. sqrt-prod 15.1%

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

      6. sqrt-unprod 7.0%

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

      7. add-sqr-sqrt 15.1%

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

      8. metadata-eval 15.1%

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

      9. *-commutative 15.1%

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

      10. associate-*r* 15.1%

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

      11. sqrt-prod 15.1%

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

      12. sqrt-unprod 7.0%

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

      13. add-sqr-sqrt 15.1%

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

      14. metadata-eval 15.1%

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

   4. Applied egg-rr 15.1%

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

      1. add-sqr-sqrt 15.1%

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

      2. times-frac 17.8%

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

      3. +-commutative 17.8%

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

      4. fma-define 17.8%

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

      5. +-commutative 17.8%

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

      6. add-sqr-sqrt 17.8%

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

      7. hypot-define 17.8%

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

      8. *-commutative 17.8%

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

      9. sqrt-prod 8.5%

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

      10. sqrt-prod 8.5%

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

      11. metadata-eval 8.5%

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

      12. associate-*l* 8.5%

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

      13. add-sqr-sqrt 17.8%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 45.3%

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

   8. Taylor expanded in y around 0 85.6%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-261}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+206}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + -2 \cdot \frac{y}{x}\right) \cdot \left(1 + 2 \cdot \frac{y}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{+74}:\\ \;\;\;\;\left(1 + -2 \cdot \frac{y}{x}\right) \cdot \left(1 + 2 \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1e+74)
   (* (+ 1.0 (* -2.0 (/ y x))) (+ 1.0 (* 2.0 (/ y x))))
   (+ (* 0.5 (* (/ x y) (/ x y))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1e+74) {
		tmp = (1.0 + (-2.0 * (y / x))) * (1.0 + (2.0 * (y / x)));
	} else {
		tmp = (0.5 * ((x / y) * (x / y))) + -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 <= 1d+74) then
        tmp = (1.0d0 + ((-2.0d0) * (y / x))) * (1.0d0 + (2.0d0 * (y / x)))
    else
        tmp = (0.5d0 * ((x / y) * (x / y))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1e+74) {
		tmp = (1.0 + (-2.0 * (y / x))) * (1.0 + (2.0 * (y / x)));
	} else {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1e+74:
		tmp = (1.0 + (-2.0 * (y / x))) * (1.0 + (2.0 * (y / x)))
	else:
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1e+74)
		tmp = Float64(Float64(1.0 + Float64(-2.0 * Float64(y / x))) * Float64(1.0 + Float64(2.0 * Float64(y / x))));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1e+74)
		tmp = (1.0 + (-2.0 * (y / x))) * (1.0 + (2.0 * (y / x)));
	else
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1e+74], N[(N[(1.0 + N[(-2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(2.0 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.99999999999999952e73

   1. Initial program 55.5%

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

      1. add-sqr-sqrt 55.5%

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

      2. difference-of-squares 55.5%

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

      3. *-commutative 55.5%

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

      4. associate-*r* 55.5%

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

      5. sqrt-prod 55.5%

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

      6. sqrt-unprod 27.9%

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

      7. add-sqr-sqrt 42.8%

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

      8. metadata-eval 42.8%

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

      9. *-commutative 42.8%

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

      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 27.9%

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

      13. add-sqr-sqrt 55.5%

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

      14. metadata-eval 55.5%

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

   4. Applied egg-rr 55.5%

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

      1. add-sqr-sqrt 55.5%

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

      2. times-frac 56.9%

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

      3. +-commutative 56.9%

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

      4. fma-define 56.9%

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

      5. +-commutative 56.9%

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

      6. add-sqr-sqrt 56.9%

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

      7. hypot-define 56.9%

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

      8. *-commutative 56.9%

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

      9. sqrt-prod 28.3%

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

      10. sqrt-prod 28.3%

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

      11. metadata-eval 28.3%

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

      12. associate-*l* 28.3%

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

      13. add-sqr-sqrt 56.9%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around inf 32.1%

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

   8. Taylor expanded in y around 0 63.6%

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

   if 9.99999999999999952e73 < y

   1. Initial program 41.0%

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

      1. fma-neg 41.0%

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

      2. *-commutative 41.0%

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

      3. distribute-rgt-neg-in 41.0%

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

      4. distribute-rgt-neg-in 41.0%

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

      5. metadata-eval 41.0%

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

      6. fma-define 41.0%

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

      7. *-commutative 41.0%

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

   3. Simplified 41.0%

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

   5. Taylor expanded in x around 0 77.1%

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

      1. pow2 77.1%

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

      2. unpow2 77.1%

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

      3. times-frac 83.3%

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

   7. Applied egg-rr 83.3%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{+74}:\\ \;\;\;\;\left(1 + -2 \cdot \frac{y}{x}\right) \cdot \left(1 + 2 \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+41}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6.2e+41) 1.0 (+ (* 0.5 (* (/ x y) (/ x y))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6.2e+41) {
		tmp = 1.0;
	} else {
		tmp = (0.5 * ((x / y) * (x / y))) + -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 <= 6.2d+41) then
        tmp = 1.0d0
    else
        tmp = (0.5d0 * ((x / y) * (x / y))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6.2e+41) {
		tmp = 1.0;
	} else {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6.2e+41:
		tmp = 1.0
	else:
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6.2e+41)
		tmp = 1.0;
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.2e+41)
		tmp = 1.0;
	else
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6.2e+41], 1.0, N[(N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.2e41

   1. Initial program 55.4%

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

      1. fma-neg 55.4%

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

      2. *-commutative 55.4%

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

      3. distribute-rgt-neg-in 55.4%

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

      4. distribute-rgt-neg-in 55.4%

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

      5. metadata-eval 55.4%

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

      6. fma-define 55.4%

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

      7. *-commutative 55.4%

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

   3. Simplified 55.4%

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

   5. Taylor expanded in x around inf 62.7%

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

   if 6.2e41 < y

   1. Initial program 42.6%

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

      1. fma-neg 42.6%

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

      2. *-commutative 42.6%

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

      3. distribute-rgt-neg-in 42.6%

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

      4. distribute-rgt-neg-in 42.6%

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

      5. metadata-eval 42.6%

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

      6. fma-define 42.6%

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

      7. *-commutative 42.6%

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

   3. Simplified 42.6%

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

   5. Taylor expanded in x around 0 74.3%

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

      1. pow2 74.3%

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

      2. unpow2 74.3%

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

      3. times-frac 80.0%

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

   7. Applied egg-rr 80.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+41}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{+73}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.65e+73) 1.0 -1.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.65e+73) {
		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 <= 1.65d+73) 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 <= 1.65e+73) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.65e+73:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.65e+73)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.65e+73)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.65e+73], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.65000000000000015e73

   1. Initial program 55.5%

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

      1. fma-neg 55.5%

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

      2. *-commutative 55.5%

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

      3. distribute-rgt-neg-in 55.5%

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

      4. distribute-rgt-neg-in 55.5%

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

      5. metadata-eval 55.5%

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

      6. fma-define 55.5%

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

      7. *-commutative 55.5%

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

   3. Simplified 55.5%

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

   5. Taylor expanded in x around inf 62.6%

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

   if 1.65000000000000015e73 < y

   1. Initial program 41.0%

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

      1. fma-neg 41.0%

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

      2. *-commutative 41.0%

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

      3. distribute-rgt-neg-in 41.0%

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

      4. distribute-rgt-neg-in 41.0%

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

      5. metadata-eval 41.0%

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

      6. fma-define 41.0%

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

      7. *-commutative 41.0%

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

   3. Simplified 41.0%

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

   5. Taylor expanded in x around 0 81.4%

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

Alternative 10: 50.0% 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 52.3%

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

   1. fma-neg 52.3%

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

   2. *-commutative 52.3%

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

   3. distribute-rgt-neg-in 52.3%

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

   4. distribute-rgt-neg-in 52.3%

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

   5. metadata-eval 52.3%

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

   6. fma-define 52.3%

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

   7. *-commutative 52.3%

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

3. Simplified 52.3%

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

5. Taylor expanded in x around 0 47.7%

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

Developer Target 1: 50.1% 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 2024146 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ (- (* x x) (* (* y 4) y)) (+ (* x x) (* (* y 4) y))) 9743233849626781/10000000000000000) (- (/ (* x x) (+ (* x x) (* (* y y) 4))) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4)))) 2) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4))))))

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