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

Percentage Accurate: 51.6% → 81.6%

Time: 9.8s

Alternatives: 12

Speedup: 1.2×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \frac{x \cdot x - t\_0}{x \cdot x + t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot 4\right) \cdot y\\ \frac{x \cdot x - t\_0}{x \cdot x + t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{\mathsf{fma}\left(y \cdot -4, y, {x}^{2}\right)}{t\_0 + x \cdot x}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-273}:\\ \;\;\;\;1 + -8 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 200000:\\ \;\;\;\;1 + \left(-4 \cdot \frac{-8 \cdot {y}^{4}}{{x}^{4}} + -8 \cdot \frac{{y}^{2}}{{x}^{2}}\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+301}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0)))
        (t_1 (/ (fma (* y -4.0) y (pow x 2.0)) (+ t_0 (* x x)))))
   (if (<= t_0 4e-273)
     (+ 1.0 (* -8.0 (/ (/ y x) (/ x y))))
     (if (<= t_0 2e-20)
       t_1
       (if (<= t_0 200000.0)
         (+
          1.0
          (+
           (* -4.0 (/ (* -8.0 (pow y 4.0)) (pow x 4.0)))
           (* -8.0 (/ (pow y 2.0) (pow x 2.0)))))
         (if (<= t_0 1e+301) t_1 -1.0))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = fma((y * -4.0), y, pow(x, 2.0)) / (t_0 + (x * x));
	double tmp;
	if (t_0 <= 4e-273) {
		tmp = 1.0 + (-8.0 * ((y / x) / (x / y)));
	} else if (t_0 <= 2e-20) {
		tmp = t_1;
	} else if (t_0 <= 200000.0) {
		tmp = 1.0 + ((-4.0 * ((-8.0 * pow(y, 4.0)) / pow(x, 4.0))) + (-8.0 * (pow(y, 2.0) / pow(x, 2.0))));
	} else if (t_0 <= 1e+301) {
		tmp = t_1;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(fma(Float64(y * -4.0), y, (x ^ 2.0)) / Float64(t_0 + Float64(x * x)))
	tmp = 0.0
	if (t_0 <= 4e-273)
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) / Float64(x / y))));
	elseif (t_0 <= 2e-20)
		tmp = t_1;
	elseif (t_0 <= 200000.0)
		tmp = Float64(1.0 + Float64(Float64(-4.0 * Float64(Float64(-8.0 * (y ^ 4.0)) / (x ^ 4.0))) + Float64(-8.0 * Float64((y ^ 2.0) / (x ^ 2.0)))));
	elseif (t_0 <= 1e+301)
		tmp = t_1;
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y * -4.0), $MachinePrecision] * y + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-273], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-20], t$95$1, If[LessEqual[t$95$0, 200000.0], N[(1.0 + N[(N[(-4.0 * N[(N[(-8.0 * N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-8.0 * N[(N[Power[y, 2.0], $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+301], t$95$1, -1.0]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4e-273

   1. Initial program 49.3%

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

   3. Taylor expanded in x around inf 53.3%

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

      1. associate--l+ 53.3%

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

      2. associate--l+ 53.3%

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

      3. distribute-rgt-out-- 53.3%

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

      4. metadata-eval 53.3%

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

      5. associate-*r* 53.3%

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

      6. pow-sqr 53.3%

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

      7. metadata-eval 53.3%

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

      8. distribute-rgt-out-- 53.3%

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

      9. metadata-eval 53.3%

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

   5. Simplified 53.3%

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

   6. Taylor expanded in y around 0 69.3%

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

      1. unpow2 69.3%

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

      2. unpow2 69.3%

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

      3. times-frac 82.9%

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

      4. unpow2 82.9%

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

   8. Simplified 82.9%

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

      1. pow2 82.9%

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

      2. clear-num 82.9%

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

      3. un-div-inv 82.9%

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

   10. Applied egg-rr 82.9%

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

   if 4e-273 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.99999999999999989e-20 or 2e5 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.00000000000000005e301

   1. Initial program 79.3%

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

      1. sub-neg 79.3%

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

      2. +-commutative 79.3%

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

      3. distribute-lft-neg-in 79.3%

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

      4. fma-define 79.4%

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

      5. distribute-rgt-neg-in 79.4%

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

      6. metadata-eval 79.4%

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

      7. pow2 79.4%

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

   4. Applied egg-rr 79.4%

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

   if 1.99999999999999989e-20 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 2e5

   1. Initial program 16.7%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate--l+ 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-*r* 100.0%

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

      6. pow-sqr 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-out-- 100.0%

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

      9. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   if 1.00000000000000005e301 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0 85.2%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 4 \cdot 10^{-273}:\\ \;\;\;\;1 + -8 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot -4, y, {x}^{2}\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 200000:\\ \;\;\;\;1 + \left(-4 \cdot \frac{-8 \cdot {y}^{4}}{{x}^{4}} + -8 \cdot \frac{{y}^{2}}{{x}^{2}}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{+301}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot -4, y, {x}^{2}\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.6% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{\mathsf{fma}\left(y \cdot -4, y, {x}^{2}\right)}{t\_0 + x \cdot x}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-273}:\\ \;\;\;\;1 + -8 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 200000:\\ \;\;\;\;1 + \left(32 \cdot \left(-1 + e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{4}\right)}\right) + -8 \cdot {\left(\frac{y}{x}\right)}^{2}\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+301}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0)))
        (t_1 (/ (fma (* y -4.0) y (pow x 2.0)) (+ t_0 (* x x)))))
   (if (<= t_0 4e-273)
     (+ 1.0 (* -8.0 (/ (/ y x) (/ x y))))
     (if (<= t_0 2e-20)
       t_1
       (if (<= t_0 200000.0)
         (+
          1.0
          (+
           (* 32.0 (+ -1.0 (exp (log1p (pow (/ y x) 4.0)))))
           (* -8.0 (pow (/ y x) 2.0))))
         (if (<= t_0 1e+301) t_1 -1.0))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = fma((y * -4.0), y, pow(x, 2.0)) / (t_0 + (x * x));
	double tmp;
	if (t_0 <= 4e-273) {
		tmp = 1.0 + (-8.0 * ((y / x) / (x / y)));
	} else if (t_0 <= 2e-20) {
		tmp = t_1;
	} else if (t_0 <= 200000.0) {
		tmp = 1.0 + ((32.0 * (-1.0 + exp(log1p(pow((y / x), 4.0))))) + (-8.0 * pow((y / x), 2.0)));
	} else if (t_0 <= 1e+301) {
		tmp = t_1;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(fma(Float64(y * -4.0), y, (x ^ 2.0)) / Float64(t_0 + Float64(x * x)))
	tmp = 0.0
	if (t_0 <= 4e-273)
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) / Float64(x / y))));
	elseif (t_0 <= 2e-20)
		tmp = t_1;
	elseif (t_0 <= 200000.0)
		tmp = Float64(1.0 + Float64(Float64(32.0 * Float64(-1.0 + exp(log1p((Float64(y / x) ^ 4.0))))) + Float64(-8.0 * (Float64(y / x) ^ 2.0))));
	elseif (t_0 <= 1e+301)
		tmp = t_1;
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y * -4.0), $MachinePrecision] * y + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-273], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-20], t$95$1, If[LessEqual[t$95$0, 200000.0], N[(1.0 + N[(N[(32.0 * N[(-1.0 + N[Exp[N[Log[1 + N[Power[N[(y / x), $MachinePrecision], 4.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-8.0 * N[Power[N[(y / x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+301], t$95$1, -1.0]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4e-273

   1. Initial program 49.3%

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

   3. Taylor expanded in x around inf 53.3%

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

      1. associate--l+ 53.3%

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

      2. associate--l+ 53.3%

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

      3. distribute-rgt-out-- 53.3%

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

      4. metadata-eval 53.3%

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

      5. associate-*r* 53.3%

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

      6. pow-sqr 53.3%

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

      7. metadata-eval 53.3%

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

      8. distribute-rgt-out-- 53.3%

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

      9. metadata-eval 53.3%

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

   5. Simplified 53.3%

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

   6. Taylor expanded in y around 0 69.3%

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

      1. unpow2 69.3%

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

      2. unpow2 69.3%

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

      3. times-frac 82.9%

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

      4. unpow2 82.9%

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

   8. Simplified 82.9%

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

      1. pow2 82.9%

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

      2. clear-num 82.9%

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

      3. un-div-inv 82.9%

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

   10. Applied egg-rr 82.9%

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

   if 4e-273 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.99999999999999989e-20 or 2e5 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.00000000000000005e301

   1. Initial program 79.3%

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

      1. sub-neg 79.3%

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

      2. +-commutative 79.3%

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

      3. distribute-lft-neg-in 79.3%

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

      4. fma-define 79.4%

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

      5. distribute-rgt-neg-in 79.4%

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

      6. metadata-eval 79.4%

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

      7. pow2 79.4%

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

   4. Applied egg-rr 79.4%

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

   if 1.99999999999999989e-20 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 2e5

   1. Initial program 16.7%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate--l+ 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-*r* 100.0%

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

      6. pow-sqr 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-out-- 100.0%

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

      9. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

      1. unpow2 100.0%

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

      2. pow2 100.0%

         \[\leadsto 1 + \left(-4 \cdot \frac{{y}^{4} \cdot -8}{{x}^{4}} + \frac{y \cdot y}{\color{blue}{x \cdot x}} \cdot -8\right) \]

      3. times-frac 100.0%

         \[\leadsto 1 + \left(-4 \cdot \frac{{y}^{4} \cdot -8}{{x}^{4}} + \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \cdot -8\right) \]

   7. Applied egg-rr 100.0%

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

      1. *-un-lft-identity 100.0%

         \[\leadsto 1 + \color{blue}{1 \cdot \left(-4 \cdot \frac{{y}^{4} \cdot -8}{{x}^{4}} + \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot -8\right)} \]

      2. fma-define 100.0%

         \[\leadsto 1 + 1 \cdot \color{blue}{\mathsf{fma}\left(-4, \frac{{y}^{4} \cdot -8}{{x}^{4}}, \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot -8\right)} \]

   9. Applied egg-rr 100.0%

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

       1. *-lft-identity 100.0%

          \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(-4, -8 \cdot {\left(\frac{y}{x}\right)}^{4}, -8 \cdot {\left(\frac{y}{x}\right)}^{2}\right)} \]

       2. fma-undefine 100.0%

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

       3. associate-*r* 100.0%

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

       4. metadata-eval 100.0%

          \[\leadsto 1 + \left(\color{blue}{32} \cdot {\left(\frac{y}{x}\right)}^{4} + -8 \cdot {\left(\frac{y}{x}\right)}^{2}\right) \]

   11. Simplified 100.0%

       \[\leadsto 1 + \color{blue}{\left(32 \cdot {\left(\frac{y}{x}\right)}^{4} + -8 \cdot {\left(\frac{y}{x}\right)}^{2}\right)} \]
   12. Step-by-step derivation

       1. expm1-log1p-u 100.0%

          \[\leadsto 1 + \left(32 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{4}\right)\right)} + -8 \cdot {\left(\frac{y}{x}\right)}^{2}\right) \]

       2. expm1-undefine 100.0%

          \[\leadsto 1 + \left(32 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{4}\right)} - 1\right)} + -8 \cdot {\left(\frac{y}{x}\right)}^{2}\right) \]

   13. Applied egg-rr 100.0%

       \[\leadsto 1 + \left(32 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{4}\right)} - 1\right)} + -8 \cdot {\left(\frac{y}{x}\right)}^{2}\right) \]

   if 1.00000000000000005e301 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0 85.2%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 4 \cdot 10^{-273}:\\ \;\;\;\;1 + -8 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot -4, y, {x}^{2}\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 200000:\\ \;\;\;\;1 + \left(32 \cdot \left(-1 + e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{4}\right)}\right) + -8 \cdot {\left(\frac{y}{x}\right)}^{2}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{+301}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot -4, y, {x}^{2}\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{\mathsf{fma}\left(y \cdot -4, y, {x}^{2}\right)}{t\_0 + x \cdot x}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-273}:\\ \;\;\;\;1 + -8 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 200000:\\ \;\;\;\;1 + \left(-4 \cdot \frac{-8 \cdot {y}^{4}}{{x}^{4}} + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+301}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0)))
        (t_1 (/ (fma (* y -4.0) y (pow x 2.0)) (+ t_0 (* x x)))))
   (if (<= t_0 4e-273)
     (+ 1.0 (* -8.0 (/ (/ y x) (/ x y))))
     (if (<= t_0 2e-20)
       t_1
       (if (<= t_0 200000.0)
         (+
          1.0
          (+
           (* -4.0 (/ (* -8.0 (pow y 4.0)) (pow x 4.0)))
           (* -8.0 (* (/ y x) (/ y x)))))
         (if (<= t_0 1e+301) t_1 -1.0))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = fma((y * -4.0), y, pow(x, 2.0)) / (t_0 + (x * x));
	double tmp;
	if (t_0 <= 4e-273) {
		tmp = 1.0 + (-8.0 * ((y / x) / (x / y)));
	} else if (t_0 <= 2e-20) {
		tmp = t_1;
	} else if (t_0 <= 200000.0) {
		tmp = 1.0 + ((-4.0 * ((-8.0 * pow(y, 4.0)) / pow(x, 4.0))) + (-8.0 * ((y / x) * (y / x))));
	} else if (t_0 <= 1e+301) {
		tmp = t_1;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(fma(Float64(y * -4.0), y, (x ^ 2.0)) / Float64(t_0 + Float64(x * x)))
	tmp = 0.0
	if (t_0 <= 4e-273)
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) / Float64(x / y))));
	elseif (t_0 <= 2e-20)
		tmp = t_1;
	elseif (t_0 <= 200000.0)
		tmp = Float64(1.0 + Float64(Float64(-4.0 * Float64(Float64(-8.0 * (y ^ 4.0)) / (x ^ 4.0))) + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x)))));
	elseif (t_0 <= 1e+301)
		tmp = t_1;
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y * -4.0), $MachinePrecision] * y + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-273], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-20], t$95$1, If[LessEqual[t$95$0, 200000.0], N[(1.0 + N[(N[(-4.0 * N[(N[(-8.0 * N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision] / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+301], t$95$1, -1.0]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4e-273

   1. Initial program 49.3%

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

   3. Taylor expanded in x around inf 53.3%

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

      1. associate--l+ 53.3%

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

      2. associate--l+ 53.3%

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

      3. distribute-rgt-out-- 53.3%

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

      4. metadata-eval 53.3%

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

      5. associate-*r* 53.3%

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

      6. pow-sqr 53.3%

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

      7. metadata-eval 53.3%

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

      8. distribute-rgt-out-- 53.3%

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

      9. metadata-eval 53.3%

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

   5. Simplified 53.3%

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

   6. Taylor expanded in y around 0 69.3%

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

      1. unpow2 69.3%

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

      2. unpow2 69.3%

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

      3. times-frac 82.9%

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

      4. unpow2 82.9%

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

   8. Simplified 82.9%

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

      1. pow2 82.9%

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

      2. clear-num 82.9%

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

      3. un-div-inv 82.9%

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

   10. Applied egg-rr 82.9%

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

   if 4e-273 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.99999999999999989e-20 or 2e5 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.00000000000000005e301

   1. Initial program 79.3%

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

      1. sub-neg 79.3%

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

      2. +-commutative 79.3%

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

      3. distribute-lft-neg-in 79.3%

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

      4. fma-define 79.4%

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

      5. distribute-rgt-neg-in 79.4%

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

      6. metadata-eval 79.4%

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

      7. pow2 79.4%

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

   4. Applied egg-rr 79.4%

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

   if 1.99999999999999989e-20 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 2e5

   1. Initial program 16.7%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate--l+ 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-*r* 100.0%

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

      6. pow-sqr 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-out-- 100.0%

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

      9. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

      1. unpow2 100.0%

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

      2. pow2 100.0%

         \[\leadsto 1 + \left(-4 \cdot \frac{{y}^{4} \cdot -8}{{x}^{4}} + \frac{y \cdot y}{\color{blue}{x \cdot x}} \cdot -8\right) \]

      3. times-frac 100.0%

         \[\leadsto 1 + \left(-4 \cdot \frac{{y}^{4} \cdot -8}{{x}^{4}} + \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \cdot -8\right) \]

   7. Applied egg-rr 100.0%

      \[\leadsto 1 + \left(-4 \cdot \frac{{y}^{4} \cdot -8}{{x}^{4}} + \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \cdot -8\right) \]

   if 1.00000000000000005e301 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0 85.2%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 4 \cdot 10^{-273}:\\ \;\;\;\;1 + -8 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot -4, y, {x}^{2}\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 200000:\\ \;\;\;\;1 + \left(-4 \cdot \frac{-8 \cdot {y}^{4}}{{x}^{4}} + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{+301}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot -4, y, {x}^{2}\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{\mathsf{fma}\left(y \cdot -4, y, {x}^{2}\right)}{t\_0 + x \cdot x}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-273}:\\ \;\;\;\;1 + -8 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 200000:\\ \;\;\;\;1 + \left(-8 \cdot {\left(\frac{y}{x}\right)}^{2} + 32 \cdot {\left(\frac{y}{x}\right)}^{4}\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+301}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0)))
        (t_1 (/ (fma (* y -4.0) y (pow x 2.0)) (+ t_0 (* x x)))))
   (if (<= t_0 4e-273)
     (+ 1.0 (* -8.0 (/ (/ y x) (/ x y))))
     (if (<= t_0 2e-20)
       t_1
       (if (<= t_0 200000.0)
         (+ 1.0 (+ (* -8.0 (pow (/ y x) 2.0)) (* 32.0 (pow (/ y x) 4.0))))
         (if (<= t_0 1e+301) t_1 -1.0))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = fma((y * -4.0), y, pow(x, 2.0)) / (t_0 + (x * x));
	double tmp;
	if (t_0 <= 4e-273) {
		tmp = 1.0 + (-8.0 * ((y / x) / (x / y)));
	} else if (t_0 <= 2e-20) {
		tmp = t_1;
	} else if (t_0 <= 200000.0) {
		tmp = 1.0 + ((-8.0 * pow((y / x), 2.0)) + (32.0 * pow((y / x), 4.0)));
	} else if (t_0 <= 1e+301) {
		tmp = t_1;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(fma(Float64(y * -4.0), y, (x ^ 2.0)) / Float64(t_0 + Float64(x * x)))
	tmp = 0.0
	if (t_0 <= 4e-273)
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) / Float64(x / y))));
	elseif (t_0 <= 2e-20)
		tmp = t_1;
	elseif (t_0 <= 200000.0)
		tmp = Float64(1.0 + Float64(Float64(-8.0 * (Float64(y / x) ^ 2.0)) + Float64(32.0 * (Float64(y / x) ^ 4.0))));
	elseif (t_0 <= 1e+301)
		tmp = t_1;
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(y * -4.0), $MachinePrecision] * y + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-273], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-20], t$95$1, If[LessEqual[t$95$0, 200000.0], N[(1.0 + N[(N[(-8.0 * N[Power[N[(y / x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(32.0 * N[Power[N[(y / x), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+301], t$95$1, -1.0]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4e-273

   1. Initial program 49.3%

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

   3. Taylor expanded in x around inf 53.3%

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

      1. associate--l+ 53.3%

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

      2. associate--l+ 53.3%

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

      3. distribute-rgt-out-- 53.3%

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

      4. metadata-eval 53.3%

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

      5. associate-*r* 53.3%

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

      6. pow-sqr 53.3%

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

      7. metadata-eval 53.3%

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

      8. distribute-rgt-out-- 53.3%

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

      9. metadata-eval 53.3%

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

   5. Simplified 53.3%

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

   6. Taylor expanded in y around 0 69.3%

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

      1. unpow2 69.3%

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

      2. unpow2 69.3%

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

      3. times-frac 82.9%

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

      4. unpow2 82.9%

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

   8. Simplified 82.9%

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

      1. pow2 82.9%

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

      2. clear-num 82.9%

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

      3. un-div-inv 82.9%

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

   10. Applied egg-rr 82.9%

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

   if 4e-273 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.99999999999999989e-20 or 2e5 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.00000000000000005e301

   1. Initial program 79.3%

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

      1. sub-neg 79.3%

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

      2. +-commutative 79.3%

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

      3. distribute-lft-neg-in 79.3%

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

      4. fma-define 79.4%

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

      5. distribute-rgt-neg-in 79.4%

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

      6. metadata-eval 79.4%

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

      7. pow2 79.4%

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

   4. Applied egg-rr 79.4%

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

   if 1.99999999999999989e-20 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 2e5

   1. Initial program 16.7%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate--l+ 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-*r* 100.0%

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

      6. pow-sqr 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-out-- 100.0%

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

      9. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

      1. unpow2 100.0%

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

      2. pow2 100.0%

         \[\leadsto 1 + \left(-4 \cdot \frac{{y}^{4} \cdot -8}{{x}^{4}} + \frac{y \cdot y}{\color{blue}{x \cdot x}} \cdot -8\right) \]

      3. times-frac 100.0%

         \[\leadsto 1 + \left(-4 \cdot \frac{{y}^{4} \cdot -8}{{x}^{4}} + \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \cdot -8\right) \]

   7. Applied egg-rr 100.0%

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

      1. *-un-lft-identity 100.0%

         \[\leadsto 1 + \color{blue}{1 \cdot \left(-4 \cdot \frac{{y}^{4} \cdot -8}{{x}^{4}} + \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot -8\right)} \]

      2. fma-define 100.0%

         \[\leadsto 1 + 1 \cdot \color{blue}{\mathsf{fma}\left(-4, \frac{{y}^{4} \cdot -8}{{x}^{4}}, \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot -8\right)} \]

   9. Applied egg-rr 100.0%

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

       1. *-lft-identity 100.0%

          \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(-4, -8 \cdot {\left(\frac{y}{x}\right)}^{4}, -8 \cdot {\left(\frac{y}{x}\right)}^{2}\right)} \]

       2. fma-undefine 100.0%

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

       3. associate-*r* 100.0%

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

       4. metadata-eval 100.0%

          \[\leadsto 1 + \left(\color{blue}{32} \cdot {\left(\frac{y}{x}\right)}^{4} + -8 \cdot {\left(\frac{y}{x}\right)}^{2}\right) \]

   11. Simplified 100.0%

       \[\leadsto 1 + \color{blue}{\left(32 \cdot {\left(\frac{y}{x}\right)}^{4} + -8 \cdot {\left(\frac{y}{x}\right)}^{2}\right)} \]

   if 1.00000000000000005e301 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0 85.2%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 4 \cdot 10^{-273}:\\ \;\;\;\;1 + -8 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot -4, y, {x}^{2}\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 200000:\\ \;\;\;\;1 + \left(-8 \cdot {\left(\frac{y}{x}\right)}^{2} + 32 \cdot {\left(\frac{y}{x}\right)}^{4}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{+301}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot -4, y, {x}^{2}\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{t\_0 + x \cdot x}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-273}:\\ \;\;\;\;1 + -8 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 200000:\\ \;\;\;\;1 + \left(-8 \cdot {\left(\frac{y}{x}\right)}^{2} + 32 \cdot {\left(\frac{y}{x}\right)}^{4}\right)\\ \mathbf{elif}\;t\_0 \leq 10^{+301}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0)))
        (t_1 (/ (* (+ x (* y 2.0)) (- x (* y 2.0))) (+ t_0 (* x x)))))
   (if (<= t_0 4e-273)
     (+ 1.0 (* -8.0 (/ (/ y x) (/ x y))))
     (if (<= t_0 2e-20)
       t_1
       (if (<= t_0 200000.0)
         (+ 1.0 (+ (* -8.0 (pow (/ y x) 2.0)) (* 32.0 (pow (/ y x) 4.0))))
         (if (<= t_0 1e+301) t_1 -1.0))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_0 + (x * x));
	double tmp;
	if (t_0 <= 4e-273) {
		tmp = 1.0 + (-8.0 * ((y / x) / (x / y)));
	} else if (t_0 <= 2e-20) {
		tmp = t_1;
	} else if (t_0 <= 200000.0) {
		tmp = 1.0 + ((-8.0 * pow((y / x), 2.0)) + (32.0 * pow((y / x), 4.0)));
	} else if (t_0 <= 1e+301) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    t_1 = ((x + (y * 2.0d0)) * (x - (y * 2.0d0))) / (t_0 + (x * x))
    if (t_0 <= 4d-273) then
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) / (x / y)))
    else if (t_0 <= 2d-20) then
        tmp = t_1
    else if (t_0 <= 200000.0d0) then
        tmp = 1.0d0 + (((-8.0d0) * ((y / x) ** 2.0d0)) + (32.0d0 * ((y / x) ** 4.0d0)))
    else if (t_0 <= 1d+301) then
        tmp = t_1
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_0 + (x * x));
	double tmp;
	if (t_0 <= 4e-273) {
		tmp = 1.0 + (-8.0 * ((y / x) / (x / y)));
	} else if (t_0 <= 2e-20) {
		tmp = t_1;
	} else if (t_0 <= 200000.0) {
		tmp = 1.0 + ((-8.0 * Math.pow((y / x), 2.0)) + (32.0 * Math.pow((y / x), 4.0)));
	} else if (t_0 <= 1e+301) {
		tmp = t_1;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_0 + (x * x))
	tmp = 0
	if t_0 <= 4e-273:
		tmp = 1.0 + (-8.0 * ((y / x) / (x / y)))
	elif t_0 <= 2e-20:
		tmp = t_1
	elif t_0 <= 200000.0:
		tmp = 1.0 + ((-8.0 * math.pow((y / x), 2.0)) + (32.0 * math.pow((y / x), 4.0)))
	elif t_0 <= 1e+301:
		tmp = t_1
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(Float64(Float64(x + Float64(y * 2.0)) * Float64(x - Float64(y * 2.0))) / Float64(t_0 + Float64(x * x)))
	tmp = 0.0
	if (t_0 <= 4e-273)
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) / Float64(x / y))));
	elseif (t_0 <= 2e-20)
		tmp = t_1;
	elseif (t_0 <= 200000.0)
		tmp = Float64(1.0 + Float64(Float64(-8.0 * (Float64(y / x) ^ 2.0)) + Float64(32.0 * (Float64(y / x) ^ 4.0))));
	elseif (t_0 <= 1e+301)
		tmp = t_1;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	t_1 = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_0 + (x * x));
	tmp = 0.0;
	if (t_0 <= 4e-273)
		tmp = 1.0 + (-8.0 * ((y / x) / (x / y)));
	elseif (t_0 <= 2e-20)
		tmp = t_1;
	elseif (t_0 <= 200000.0)
		tmp = 1.0 + ((-8.0 * ((y / x) ^ 2.0)) + (32.0 * ((y / x) ^ 4.0)));
	elseif (t_0 <= 1e+301)
		tmp = t_1;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision] * N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-273], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-20], t$95$1, If[LessEqual[t$95$0, 200000.0], N[(1.0 + N[(N[(-8.0 * N[Power[N[(y / x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[(32.0 * N[Power[N[(y / x), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+301], t$95$1, -1.0]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4e-273

   1. Initial program 49.3%

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

   3. Taylor expanded in x around inf 53.3%

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

      1. associate--l+ 53.3%

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

      2. associate--l+ 53.3%

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

      3. distribute-rgt-out-- 53.3%

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

      4. metadata-eval 53.3%

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

      5. associate-*r* 53.3%

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

      6. pow-sqr 53.3%

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

      7. metadata-eval 53.3%

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

      8. distribute-rgt-out-- 53.3%

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

      9. metadata-eval 53.3%

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

   5. Simplified 53.3%

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

   6. Taylor expanded in y around 0 69.3%

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

      1. unpow2 69.3%

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

      2. unpow2 69.3%

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

      3. times-frac 82.9%

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

      4. unpow2 82.9%

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

   8. Simplified 82.9%

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

      1. pow2 82.9%

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

      2. clear-num 82.9%

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

      3. un-div-inv 82.9%

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

   10. Applied egg-rr 82.9%

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

   if 4e-273 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.99999999999999989e-20 or 2e5 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.00000000000000005e301

   1. Initial program 79.3%

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

      1. sub-neg 79.3%

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

      2. +-commutative 79.3%

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

      3. distribute-lft-neg-in 79.3%

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

      4. fma-define 79.4%

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

      5. distribute-rgt-neg-in 79.4%

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

      6. metadata-eval 79.4%

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

      7. pow2 79.4%

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

   4. Applied egg-rr 79.4%

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

      1. fma-undefine 79.3%

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

      2. metadata-eval 79.3%

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

      3. distribute-rgt-neg-in 79.3%

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

      4. distribute-lft-neg-in 79.3%

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

      5. +-commutative 79.3%

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

      6. sub-neg 79.3%

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

      7. pow2 79.3%

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

      8. add-sqr-sqrt 79.3%

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

      9. difference-of-squares 79.3%

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

      10. *-commutative 79.3%

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

      11. associate-*r* 79.3%

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

      12. unpow2 79.3%

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

      13. *-commutative 79.3%

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

      14. sqrt-prod 79.3%

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

      15. sqrt-pow1 49.8%

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

      16. metadata-eval 49.8%

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

      17. pow1 49.8%

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

      18. metadata-eval 49.8%

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

      19. *-commutative 49.8%

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

      20. associate-*r* 49.8%

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

      21. unpow2 49.8%

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

   6. Applied egg-rr 79.3%

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

   if 1.99999999999999989e-20 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 2e5

   1. Initial program 16.7%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate--l+ 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-*r* 100.0%

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

      6. pow-sqr 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-out-- 100.0%

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

      9. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

      1. unpow2 100.0%

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

      2. pow2 100.0%

         \[\leadsto 1 + \left(-4 \cdot \frac{{y}^{4} \cdot -8}{{x}^{4}} + \frac{y \cdot y}{\color{blue}{x \cdot x}} \cdot -8\right) \]

      3. times-frac 100.0%

         \[\leadsto 1 + \left(-4 \cdot \frac{{y}^{4} \cdot -8}{{x}^{4}} + \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \cdot -8\right) \]

   7. Applied egg-rr 100.0%

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

      1. *-un-lft-identity 100.0%

         \[\leadsto 1 + \color{blue}{1 \cdot \left(-4 \cdot \frac{{y}^{4} \cdot -8}{{x}^{4}} + \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot -8\right)} \]

      2. fma-define 100.0%

         \[\leadsto 1 + 1 \cdot \color{blue}{\mathsf{fma}\left(-4, \frac{{y}^{4} \cdot -8}{{x}^{4}}, \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot -8\right)} \]

   9. Applied egg-rr 100.0%

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

       1. *-lft-identity 100.0%

          \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(-4, -8 \cdot {\left(\frac{y}{x}\right)}^{4}, -8 \cdot {\left(\frac{y}{x}\right)}^{2}\right)} \]

       2. fma-undefine 100.0%

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

       3. associate-*r* 100.0%

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

       4. metadata-eval 100.0%

          \[\leadsto 1 + \left(\color{blue}{32} \cdot {\left(\frac{y}{x}\right)}^{4} + -8 \cdot {\left(\frac{y}{x}\right)}^{2}\right) \]

   11. Simplified 100.0%

       \[\leadsto 1 + \color{blue}{\left(32 \cdot {\left(\frac{y}{x}\right)}^{4} + -8 \cdot {\left(\frac{y}{x}\right)}^{2}\right)} \]

   if 1.00000000000000005e301 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0 85.2%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 4 \cdot 10^{-273}:\\ \;\;\;\;1 + -8 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 200000:\\ \;\;\;\;1 + \left(-8 \cdot {\left(\frac{y}{x}\right)}^{2} + 32 \cdot {\left(\frac{y}{x}\right)}^{4}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{+301}:\\ \;\;\;\;\frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -8 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ t_1 := y \cdot \left(y \cdot 4\right)\\ t_2 := \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{t\_1 + x \cdot x}\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-273}:\\ \;\;\;\;1 + t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-20}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 200000:\\ \;\;\;\;1 + \left(t\_0 + 32 \cdot {\left(\frac{y}{x}\right)}^{4}\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+301}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* -8.0 (/ (/ y x) (/ x y))))
        (t_1 (* y (* y 4.0)))
        (t_2 (/ (* (+ x (* y 2.0)) (- x (* y 2.0))) (+ t_1 (* x x)))))
   (if (<= t_1 4e-273)
     (+ 1.0 t_0)
     (if (<= t_1 2e-20)
       t_2
       (if (<= t_1 200000.0)
         (+ 1.0 (+ t_0 (* 32.0 (pow (/ y x) 4.0))))
         (if (<= t_1 1e+301) t_2 -1.0))))))

C

double code(double x, double y) {
	double t_0 = -8.0 * ((y / x) / (x / y));
	double t_1 = y * (y * 4.0);
	double t_2 = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_1 + (x * x));
	double tmp;
	if (t_1 <= 4e-273) {
		tmp = 1.0 + t_0;
	} else if (t_1 <= 2e-20) {
		tmp = t_2;
	} else if (t_1 <= 200000.0) {
		tmp = 1.0 + (t_0 + (32.0 * pow((y / x), 4.0)));
	} else if (t_1 <= 1e+301) {
		tmp = t_2;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (-8.0d0) * ((y / x) / (x / y))
    t_1 = y * (y * 4.0d0)
    t_2 = ((x + (y * 2.0d0)) * (x - (y * 2.0d0))) / (t_1 + (x * x))
    if (t_1 <= 4d-273) then
        tmp = 1.0d0 + t_0
    else if (t_1 <= 2d-20) then
        tmp = t_2
    else if (t_1 <= 200000.0d0) then
        tmp = 1.0d0 + (t_0 + (32.0d0 * ((y / x) ** 4.0d0)))
    else if (t_1 <= 1d+301) then
        tmp = t_2
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = -8.0 * ((y / x) / (x / y));
	double t_1 = y * (y * 4.0);
	double t_2 = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_1 + (x * x));
	double tmp;
	if (t_1 <= 4e-273) {
		tmp = 1.0 + t_0;
	} else if (t_1 <= 2e-20) {
		tmp = t_2;
	} else if (t_1 <= 200000.0) {
		tmp = 1.0 + (t_0 + (32.0 * Math.pow((y / x), 4.0)));
	} else if (t_1 <= 1e+301) {
		tmp = t_2;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = -8.0 * ((y / x) / (x / y))
	t_1 = y * (y * 4.0)
	t_2 = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_1 + (x * x))
	tmp = 0
	if t_1 <= 4e-273:
		tmp = 1.0 + t_0
	elif t_1 <= 2e-20:
		tmp = t_2
	elif t_1 <= 200000.0:
		tmp = 1.0 + (t_0 + (32.0 * math.pow((y / x), 4.0)))
	elif t_1 <= 1e+301:
		tmp = t_2
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(-8.0 * Float64(Float64(y / x) / Float64(x / y)))
	t_1 = Float64(y * Float64(y * 4.0))
	t_2 = Float64(Float64(Float64(x + Float64(y * 2.0)) * Float64(x - Float64(y * 2.0))) / Float64(t_1 + Float64(x * x)))
	tmp = 0.0
	if (t_1 <= 4e-273)
		tmp = Float64(1.0 + t_0);
	elseif (t_1 <= 2e-20)
		tmp = t_2;
	elseif (t_1 <= 200000.0)
		tmp = Float64(1.0 + Float64(t_0 + Float64(32.0 * (Float64(y / x) ^ 4.0))));
	elseif (t_1 <= 1e+301)
		tmp = t_2;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = -8.0 * ((y / x) / (x / y));
	t_1 = y * (y * 4.0);
	t_2 = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_1 + (x * x));
	tmp = 0.0;
	if (t_1 <= 4e-273)
		tmp = 1.0 + t_0;
	elseif (t_1 <= 2e-20)
		tmp = t_2;
	elseif (t_1 <= 200000.0)
		tmp = 1.0 + (t_0 + (32.0 * ((y / x) ^ 4.0)));
	elseif (t_1 <= 1e+301)
		tmp = t_2;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(-8.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision] * N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4e-273], N[(1.0 + t$95$0), $MachinePrecision], If[LessEqual[t$95$1, 2e-20], t$95$2, If[LessEqual[t$95$1, 200000.0], N[(1.0 + N[(t$95$0 + N[(32.0 * N[Power[N[(y / x), $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+301], t$95$2, -1.0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4e-273

   1. Initial program 49.3%

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

   3. Taylor expanded in x around inf 53.3%

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

      1. associate--l+ 53.3%

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

      2. associate--l+ 53.3%

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

      3. distribute-rgt-out-- 53.3%

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

      4. metadata-eval 53.3%

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

      5. associate-*r* 53.3%

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

      6. pow-sqr 53.3%

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

      7. metadata-eval 53.3%

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

      8. distribute-rgt-out-- 53.3%

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

      9. metadata-eval 53.3%

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

   5. Simplified 53.3%

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

   6. Taylor expanded in y around 0 69.3%

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

      1. unpow2 69.3%

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

      2. unpow2 69.3%

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

      3. times-frac 82.9%

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

      4. unpow2 82.9%

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

   8. Simplified 82.9%

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

      1. pow2 82.9%

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

      2. clear-num 82.9%

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

      3. un-div-inv 82.9%

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

   10. Applied egg-rr 82.9%

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

   if 4e-273 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.99999999999999989e-20 or 2e5 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.00000000000000005e301

   1. Initial program 79.3%

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

      1. sub-neg 79.3%

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

      2. +-commutative 79.3%

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

      3. distribute-lft-neg-in 79.3%

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

      4. fma-define 79.4%

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

      5. distribute-rgt-neg-in 79.4%

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

      6. metadata-eval 79.4%

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

      7. pow2 79.4%

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

   4. Applied egg-rr 79.4%

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

      1. fma-undefine 79.3%

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

      2. metadata-eval 79.3%

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

      3. distribute-rgt-neg-in 79.3%

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

      4. distribute-lft-neg-in 79.3%

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

      5. +-commutative 79.3%

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

      6. sub-neg 79.3%

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

      7. pow2 79.3%

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

      8. add-sqr-sqrt 79.3%

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

      9. difference-of-squares 79.3%

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

      10. *-commutative 79.3%

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

      11. associate-*r* 79.3%

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

      12. unpow2 79.3%

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

      13. *-commutative 79.3%

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

      14. sqrt-prod 79.3%

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

      15. sqrt-pow1 49.8%

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

      16. metadata-eval 49.8%

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

      17. pow1 49.8%

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

      18. metadata-eval 49.8%

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

      19. *-commutative 49.8%

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

      20. associate-*r* 49.8%

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

      21. unpow2 49.8%

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

   6. Applied egg-rr 79.3%

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

   if 1.99999999999999989e-20 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 2e5

   1. Initial program 16.7%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate--l+ 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-*r* 100.0%

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

      6. pow-sqr 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-out-- 100.0%

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

      9. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

      1. unpow2 100.0%

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

      2. pow2 100.0%

         \[\leadsto 1 + \left(-4 \cdot \frac{{y}^{4} \cdot -8}{{x}^{4}} + \frac{y \cdot y}{\color{blue}{x \cdot x}} \cdot -8\right) \]

      3. times-frac 100.0%

         \[\leadsto 1 + \left(-4 \cdot \frac{{y}^{4} \cdot -8}{{x}^{4}} + \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \cdot -8\right) \]

   7. Applied egg-rr 100.0%

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

      1. *-un-lft-identity 100.0%

         \[\leadsto 1 + \color{blue}{1 \cdot \left(-4 \cdot \frac{{y}^{4} \cdot -8}{{x}^{4}} + \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot -8\right)} \]

      2. fma-define 100.0%

         \[\leadsto 1 + 1 \cdot \color{blue}{\mathsf{fma}\left(-4, \frac{{y}^{4} \cdot -8}{{x}^{4}}, \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot -8\right)} \]

   9. Applied egg-rr 100.0%

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

       1. *-lft-identity 100.0%

          \[\leadsto 1 + \color{blue}{\mathsf{fma}\left(-4, -8 \cdot {\left(\frac{y}{x}\right)}^{4}, -8 \cdot {\left(\frac{y}{x}\right)}^{2}\right)} \]

       2. fma-undefine 100.0%

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

       3. associate-*r* 100.0%

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

       4. metadata-eval 100.0%

          \[\leadsto 1 + \left(\color{blue}{32} \cdot {\left(\frac{y}{x}\right)}^{4} + -8 \cdot {\left(\frac{y}{x}\right)}^{2}\right) \]

   11. Simplified 100.0%

       \[\leadsto 1 + \color{blue}{\left(32 \cdot {\left(\frac{y}{x}\right)}^{4} + -8 \cdot {\left(\frac{y}{x}\right)}^{2}\right)} \]
   12. Step-by-step derivation

       1. pow2 100.0%

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

       2. clear-num 100.0%

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

       3. un-div-inv 100.0%

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

   13. Applied egg-rr 100.0%

       \[\leadsto 1 + \left(32 \cdot {\left(\frac{y}{x}\right)}^{4} + -8 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}}\right) \]

   if 1.00000000000000005e301 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0 85.2%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 4 \cdot 10^{-273}:\\ \;\;\;\;1 + -8 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 200000:\\ \;\;\;\;1 + \left(-8 \cdot \frac{\frac{y}{x}}{\frac{x}{y}} + 32 \cdot {\left(\frac{y}{x}\right)}^{4}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{+301}:\\ \;\;\;\;\frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{t\_0 + x \cdot x}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-273}:\\ \;\;\;\;1 + -8 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{-20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 200000:\\ \;\;\;\;1 + -8 \cdot {\left(\frac{y}{x}\right)}^{2}\\ \mathbf{elif}\;t\_0 \leq 10^{+301}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0)))
        (t_1 (/ (* (+ x (* y 2.0)) (- x (* y 2.0))) (+ t_0 (* x x)))))
   (if (<= t_0 4e-273)
     (+ 1.0 (* -8.0 (/ (/ y x) (/ x y))))
     (if (<= t_0 2e-20)
       t_1
       (if (<= t_0 200000.0)
         (+ 1.0 (* -8.0 (pow (/ y x) 2.0)))
         (if (<= t_0 1e+301) t_1 -1.0))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_0 + (x * x));
	double tmp;
	if (t_0 <= 4e-273) {
		tmp = 1.0 + (-8.0 * ((y / x) / (x / y)));
	} else if (t_0 <= 2e-20) {
		tmp = t_1;
	} else if (t_0 <= 200000.0) {
		tmp = 1.0 + (-8.0 * pow((y / x), 2.0));
	} else if (t_0 <= 1e+301) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    t_1 = ((x + (y * 2.0d0)) * (x - (y * 2.0d0))) / (t_0 + (x * x))
    if (t_0 <= 4d-273) then
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) / (x / y)))
    else if (t_0 <= 2d-20) then
        tmp = t_1
    else if (t_0 <= 200000.0d0) then
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) ** 2.0d0))
    else if (t_0 <= 1d+301) then
        tmp = t_1
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_0 + (x * x));
	double tmp;
	if (t_0 <= 4e-273) {
		tmp = 1.0 + (-8.0 * ((y / x) / (x / y)));
	} else if (t_0 <= 2e-20) {
		tmp = t_1;
	} else if (t_0 <= 200000.0) {
		tmp = 1.0 + (-8.0 * Math.pow((y / x), 2.0));
	} else if (t_0 <= 1e+301) {
		tmp = t_1;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_0 + (x * x))
	tmp = 0
	if t_0 <= 4e-273:
		tmp = 1.0 + (-8.0 * ((y / x) / (x / y)))
	elif t_0 <= 2e-20:
		tmp = t_1
	elif t_0 <= 200000.0:
		tmp = 1.0 + (-8.0 * math.pow((y / x), 2.0))
	elif t_0 <= 1e+301:
		tmp = t_1
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(Float64(Float64(x + Float64(y * 2.0)) * Float64(x - Float64(y * 2.0))) / Float64(t_0 + Float64(x * x)))
	tmp = 0.0
	if (t_0 <= 4e-273)
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) / Float64(x / y))));
	elseif (t_0 <= 2e-20)
		tmp = t_1;
	elseif (t_0 <= 200000.0)
		tmp = Float64(1.0 + Float64(-8.0 * (Float64(y / x) ^ 2.0)));
	elseif (t_0 <= 1e+301)
		tmp = t_1;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	t_1 = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_0 + (x * x));
	tmp = 0.0;
	if (t_0 <= 4e-273)
		tmp = 1.0 + (-8.0 * ((y / x) / (x / y)));
	elseif (t_0 <= 2e-20)
		tmp = t_1;
	elseif (t_0 <= 200000.0)
		tmp = 1.0 + (-8.0 * ((y / x) ^ 2.0));
	elseif (t_0 <= 1e+301)
		tmp = t_1;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision] * N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-273], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e-20], t$95$1, If[LessEqual[t$95$0, 200000.0], N[(1.0 + N[(-8.0 * N[Power[N[(y / x), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 1e+301], t$95$1, -1.0]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4e-273

   1. Initial program 49.3%

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

   3. Taylor expanded in x around inf 53.3%

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

      1. associate--l+ 53.3%

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

      2. associate--l+ 53.3%

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

      3. distribute-rgt-out-- 53.3%

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

      4. metadata-eval 53.3%

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

      5. associate-*r* 53.3%

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

      6. pow-sqr 53.3%

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

      7. metadata-eval 53.3%

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

      8. distribute-rgt-out-- 53.3%

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

      9. metadata-eval 53.3%

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

   5. Simplified 53.3%

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

   6. Taylor expanded in y around 0 69.3%

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

      1. unpow2 69.3%

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

      2. unpow2 69.3%

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

      3. times-frac 82.9%

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

      4. unpow2 82.9%

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

   8. Simplified 82.9%

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

      1. pow2 82.9%

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

      2. clear-num 82.9%

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

      3. un-div-inv 82.9%

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

   10. Applied egg-rr 82.9%

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

   if 4e-273 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.99999999999999989e-20 or 2e5 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.00000000000000005e301

   1. Initial program 79.3%

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

      1. sub-neg 79.3%

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

      2. +-commutative 79.3%

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

      3. distribute-lft-neg-in 79.3%

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

      4. fma-define 79.4%

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

      5. distribute-rgt-neg-in 79.4%

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

      6. metadata-eval 79.4%

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

      7. pow2 79.4%

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

   4. Applied egg-rr 79.4%

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

      1. fma-undefine 79.3%

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

      2. metadata-eval 79.3%

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

      3. distribute-rgt-neg-in 79.3%

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

      4. distribute-lft-neg-in 79.3%

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

      5. +-commutative 79.3%

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

      6. sub-neg 79.3%

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

      7. pow2 79.3%

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

      8. add-sqr-sqrt 79.3%

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

      9. difference-of-squares 79.3%

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

      10. *-commutative 79.3%

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

      11. associate-*r* 79.3%

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

      12. unpow2 79.3%

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

      13. *-commutative 79.3%

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

      14. sqrt-prod 79.3%

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

      15. sqrt-pow1 49.8%

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

      16. metadata-eval 49.8%

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

      17. pow1 49.8%

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

      18. metadata-eval 49.8%

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

      19. *-commutative 49.8%

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

      20. associate-*r* 49.8%

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

      21. unpow2 49.8%

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

   6. Applied egg-rr 79.3%

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

   if 1.99999999999999989e-20 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 2e5

   1. Initial program 16.7%

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

   3. Taylor expanded in x around inf 100.0%

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

      1. associate--l+ 100.0%

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

      2. associate--l+ 100.0%

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

      3. distribute-rgt-out-- 100.0%

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

      4. metadata-eval 100.0%

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

      5. associate-*r* 100.0%

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

      6. pow-sqr 100.0%

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

      7. metadata-eval 100.0%

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

      8. distribute-rgt-out-- 100.0%

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

      9. metadata-eval 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 100.0%

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

      1. unpow2 100.0%

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

      2. unpow2 100.0%

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

      3. times-frac 100.0%

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

      4. unpow2 100.0%

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

   8. Simplified 100.0%

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

   if 1.00000000000000005e301 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0 85.2%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 4 \cdot 10^{-273}:\\ \;\;\;\;1 + -8 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 200000:\\ \;\;\;\;1 + -8 \cdot {\left(\frac{y}{x}\right)}^{2}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{+301}:\\ \;\;\;\;\frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + -8 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ t_1 := y \cdot \left(y \cdot 4\right)\\ t_2 := \frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{t\_1 + x \cdot x}\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-273}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-20}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 200000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+301}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* -8.0 (/ (/ y x) (/ x y)))))
        (t_1 (* y (* y 4.0)))
        (t_2 (/ (* (+ x (* y 2.0)) (- x (* y 2.0))) (+ t_1 (* x x)))))
   (if (<= t_1 4e-273)
     t_0
     (if (<= t_1 2e-20)
       t_2
       (if (<= t_1 200000.0) t_0 (if (<= t_1 1e+301) t_2 -1.0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (-8.0 * ((y / x) / (x / y)));
	double t_1 = y * (y * 4.0);
	double t_2 = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_1 + (x * x));
	double tmp;
	if (t_1 <= 4e-273) {
		tmp = t_0;
	} else if (t_1 <= 2e-20) {
		tmp = t_2;
	} else if (t_1 <= 200000.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+301) {
		tmp = t_2;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 + ((-8.0d0) * ((y / x) / (x / y)))
    t_1 = y * (y * 4.0d0)
    t_2 = ((x + (y * 2.0d0)) * (x - (y * 2.0d0))) / (t_1 + (x * x))
    if (t_1 <= 4d-273) then
        tmp = t_0
    else if (t_1 <= 2d-20) then
        tmp = t_2
    else if (t_1 <= 200000.0d0) then
        tmp = t_0
    else if (t_1 <= 1d+301) then
        tmp = t_2
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (-8.0 * ((y / x) / (x / y)));
	double t_1 = y * (y * 4.0);
	double t_2 = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_1 + (x * x));
	double tmp;
	if (t_1 <= 4e-273) {
		tmp = t_0;
	} else if (t_1 <= 2e-20) {
		tmp = t_2;
	} else if (t_1 <= 200000.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+301) {
		tmp = t_2;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (-8.0 * ((y / x) / (x / y)))
	t_1 = y * (y * 4.0)
	t_2 = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_1 + (x * x))
	tmp = 0
	if t_1 <= 4e-273:
		tmp = t_0
	elif t_1 <= 2e-20:
		tmp = t_2
	elif t_1 <= 200000.0:
		tmp = t_0
	elif t_1 <= 1e+301:
		tmp = t_2
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) / Float64(x / y))))
	t_1 = Float64(y * Float64(y * 4.0))
	t_2 = Float64(Float64(Float64(x + Float64(y * 2.0)) * Float64(x - Float64(y * 2.0))) / Float64(t_1 + Float64(x * x)))
	tmp = 0.0
	if (t_1 <= 4e-273)
		tmp = t_0;
	elseif (t_1 <= 2e-20)
		tmp = t_2;
	elseif (t_1 <= 200000.0)
		tmp = t_0;
	elseif (t_1 <= 1e+301)
		tmp = t_2;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (-8.0 * ((y / x) / (x / y)));
	t_1 = y * (y * 4.0);
	t_2 = ((x + (y * 2.0)) * (x - (y * 2.0))) / (t_1 + (x * x));
	tmp = 0.0;
	if (t_1 <= 4e-273)
		tmp = t_0;
	elseif (t_1 <= 2e-20)
		tmp = t_2;
	elseif (t_1 <= 200000.0)
		tmp = t_0;
	elseif (t_1 <= 1e+301)
		tmp = t_2;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision] * N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4e-273], t$95$0, If[LessEqual[t$95$1, 2e-20], t$95$2, If[LessEqual[t$95$1, 200000.0], t$95$0, If[LessEqual[t$95$1, 1e+301], t$95$2, -1.0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4e-273 or 1.99999999999999989e-20 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 2e5

   1. Initial program 46.9%

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

   3. Taylor expanded in x around inf 56.8%

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

      1. associate--l+ 56.8%

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

      2. associate--l+ 56.8%

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

      3. distribute-rgt-out-- 56.8%

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

      4. metadata-eval 56.8%

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

      5. associate-*r* 56.8%

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

      6. pow-sqr 56.8%

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

      7. metadata-eval 56.8%

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

      8. distribute-rgt-out-- 56.8%

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

      9. metadata-eval 56.8%

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

   5. Simplified 56.8%

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

   6. Taylor expanded in y around 0 71.6%

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

      1. unpow2 71.6%

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

      2. unpow2 71.6%

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

      3. times-frac 84.2%

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

      4. unpow2 84.2%

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

   8. Simplified 84.2%

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

      1. pow2 84.2%

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

      2. clear-num 84.2%

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

      3. un-div-inv 84.2%

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

   10. Applied egg-rr 84.2%

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

   if 4e-273 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.99999999999999989e-20 or 2e5 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.00000000000000005e301

   1. Initial program 79.3%

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

      1. sub-neg 79.3%

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

      2. +-commutative 79.3%

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

      3. distribute-lft-neg-in 79.3%

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

      4. fma-define 79.4%

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

      5. distribute-rgt-neg-in 79.4%

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

      6. metadata-eval 79.4%

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

      7. pow2 79.4%

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

   4. Applied egg-rr 79.4%

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

      1. fma-undefine 79.3%

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

      2. metadata-eval 79.3%

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

      3. distribute-rgt-neg-in 79.3%

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

      4. distribute-lft-neg-in 79.3%

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

      5. +-commutative 79.3%

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

      6. sub-neg 79.3%

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

      7. pow2 79.3%

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

      8. add-sqr-sqrt 79.3%

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

      9. difference-of-squares 79.3%

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

      10. *-commutative 79.3%

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

      11. associate-*r* 79.3%

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

      12. unpow2 79.3%

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

      13. *-commutative 79.3%

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

      14. sqrt-prod 79.3%

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

      15. sqrt-pow1 49.8%

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

      16. metadata-eval 49.8%

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

      17. pow1 49.8%

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

      18. metadata-eval 49.8%

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

      19. *-commutative 49.8%

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

      20. associate-*r* 49.8%

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

      21. unpow2 49.8%

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

   6. Applied egg-rr 79.3%

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

   if 1.00000000000000005e301 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0 85.2%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 4 \cdot 10^{-273}:\\ \;\;\;\;1 + -8 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 200000:\\ \;\;\;\;1 + -8 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{+301}:\\ \;\;\;\;\frac{\left(x + y \cdot 2\right) \cdot \left(x - y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + -8 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ t_1 := y \cdot \left(y \cdot 4\right)\\ t_2 := \frac{x \cdot x - t\_1}{t\_1 + x \cdot x}\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-273}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-20}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 200000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 10^{+301}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* -8.0 (/ (/ y x) (/ x y)))))
        (t_1 (* y (* y 4.0)))
        (t_2 (/ (- (* x x) t_1) (+ t_1 (* x x)))))
   (if (<= t_1 4e-273)
     t_0
     (if (<= t_1 2e-20)
       t_2
       (if (<= t_1 200000.0) t_0 (if (<= t_1 1e+301) t_2 -1.0))))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (-8.0 * ((y / x) / (x / y)));
	double t_1 = y * (y * 4.0);
	double t_2 = ((x * x) - t_1) / (t_1 + (x * x));
	double tmp;
	if (t_1 <= 4e-273) {
		tmp = t_0;
	} else if (t_1 <= 2e-20) {
		tmp = t_2;
	} else if (t_1 <= 200000.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+301) {
		tmp = t_2;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = 1.0d0 + ((-8.0d0) * ((y / x) / (x / y)))
    t_1 = y * (y * 4.0d0)
    t_2 = ((x * x) - t_1) / (t_1 + (x * x))
    if (t_1 <= 4d-273) then
        tmp = t_0
    else if (t_1 <= 2d-20) then
        tmp = t_2
    else if (t_1 <= 200000.0d0) then
        tmp = t_0
    else if (t_1 <= 1d+301) then
        tmp = t_2
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (-8.0 * ((y / x) / (x / y)));
	double t_1 = y * (y * 4.0);
	double t_2 = ((x * x) - t_1) / (t_1 + (x * x));
	double tmp;
	if (t_1 <= 4e-273) {
		tmp = t_0;
	} else if (t_1 <= 2e-20) {
		tmp = t_2;
	} else if (t_1 <= 200000.0) {
		tmp = t_0;
	} else if (t_1 <= 1e+301) {
		tmp = t_2;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (-8.0 * ((y / x) / (x / y)))
	t_1 = y * (y * 4.0)
	t_2 = ((x * x) - t_1) / (t_1 + (x * x))
	tmp = 0
	if t_1 <= 4e-273:
		tmp = t_0
	elif t_1 <= 2e-20:
		tmp = t_2
	elif t_1 <= 200000.0:
		tmp = t_0
	elif t_1 <= 1e+301:
		tmp = t_2
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) / Float64(x / y))))
	t_1 = Float64(y * Float64(y * 4.0))
	t_2 = Float64(Float64(Float64(x * x) - t_1) / Float64(t_1 + Float64(x * x)))
	tmp = 0.0
	if (t_1 <= 4e-273)
		tmp = t_0;
	elseif (t_1 <= 2e-20)
		tmp = t_2;
	elseif (t_1 <= 200000.0)
		tmp = t_0;
	elseif (t_1 <= 1e+301)
		tmp = t_2;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (-8.0 * ((y / x) / (x / y)));
	t_1 = y * (y * 4.0);
	t_2 = ((x * x) - t_1) / (t_1 + (x * x));
	tmp = 0.0;
	if (t_1 <= 4e-273)
		tmp = t_0;
	elseif (t_1 <= 2e-20)
		tmp = t_2;
	elseif (t_1 <= 200000.0)
		tmp = t_0;
	elseif (t_1 <= 1e+301)
		tmp = t_2;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * x), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(t$95$1 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4e-273], t$95$0, If[LessEqual[t$95$1, 2e-20], t$95$2, If[LessEqual[t$95$1, 200000.0], t$95$0, If[LessEqual[t$95$1, 1e+301], t$95$2, -1.0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4e-273 or 1.99999999999999989e-20 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 2e5

   1. Initial program 46.9%

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

   3. Taylor expanded in x around inf 56.8%

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

      1. associate--l+ 56.8%

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

      2. associate--l+ 56.8%

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

      3. distribute-rgt-out-- 56.8%

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

      4. metadata-eval 56.8%

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

      5. associate-*r* 56.8%

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

      6. pow-sqr 56.8%

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

      7. metadata-eval 56.8%

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

      8. distribute-rgt-out-- 56.8%

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

      9. metadata-eval 56.8%

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

   5. Simplified 56.8%

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

   6. Taylor expanded in y around 0 71.6%

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

      1. unpow2 71.6%

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

      2. unpow2 71.6%

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

      3. times-frac 84.2%

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

      4. unpow2 84.2%

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

   8. Simplified 84.2%

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

      1. pow2 84.2%

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

      2. clear-num 84.2%

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

      3. un-div-inv 84.2%

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

   10. Applied egg-rr 84.2%

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

   if 4e-273 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.99999999999999989e-20 or 2e5 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.00000000000000005e301

   1. Initial program 79.3%

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

   if 1.00000000000000005e301 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0 85.2%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 4 \cdot 10^{-273}:\\ \;\;\;\;1 + -8 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 200000:\\ \;\;\;\;1 + -8 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 10^{+301}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-129} \lor \neg \left(y \leq 5.1 \cdot 10^{-101}\right) \land y \leq 2.5 \cdot 10^{+38}:\\ \;\;\;\;1 + -8 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 2e-129) (and (not (<= y 5.1e-101)) (<= y 2.5e+38)))
   (+ 1.0 (* -8.0 (/ (/ y x) (/ x y))))
   -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 2e-129) || (!(y <= 5.1e-101) && (y <= 2.5e+38))) {
		tmp = 1.0 + (-8.0 * ((y / x) / (x / y)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= 2d-129) .or. (.not. (y <= 5.1d-101)) .and. (y <= 2.5d+38)) then
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) / (x / y)))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 2e-129) || (!(y <= 5.1e-101) && (y <= 2.5e+38))) {
		tmp = 1.0 + (-8.0 * ((y / x) / (x / y)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 2e-129) or (not (y <= 5.1e-101) and (y <= 2.5e+38)):
		tmp = 1.0 + (-8.0 * ((y / x) / (x / y)))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 2e-129) || (!(y <= 5.1e-101) && (y <= 2.5e+38)))
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) / Float64(x / y))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 2e-129) || (~((y <= 5.1e-101)) && (y <= 2.5e+38)))
		tmp = 1.0 + (-8.0 * ((y / x) / (x / y)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 2e-129], And[N[Not[LessEqual[y, 5.1e-101]], $MachinePrecision], LessEqual[y, 2.5e+38]]], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.9999999999999999e-129 or 5.1000000000000002e-101 < y < 2.49999999999999985e38

   1. Initial program 55.6%

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

   3. Taylor expanded in x around inf 43.3%

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

      1. associate--l+ 43.3%

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

      2. associate--l+ 43.3%

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

      3. distribute-rgt-out-- 43.3%

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

      4. metadata-eval 43.3%

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

      5. associate-*r* 43.3%

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

      6. pow-sqr 43.3%

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

      7. metadata-eval 43.3%

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

      8. distribute-rgt-out-- 43.3%

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

      9. metadata-eval 43.3%

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

   5. Simplified 43.3%

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

   6. Taylor expanded in y around 0 53.7%

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

      1. unpow2 53.7%

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

      2. unpow2 53.7%

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

      3. times-frac 61.5%

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

      4. unpow2 61.5%

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

   8. Simplified 61.5%

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

      1. pow2 61.5%

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

      2. clear-num 61.5%

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

      3. un-div-inv 61.5%

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

   10. Applied egg-rr 61.5%

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

   if 1.9999999999999999e-129 < y < 5.1000000000000002e-101 or 2.49999999999999985e38 < y

   1. Initial program 22.5%

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

   3. Taylor expanded in x around 0 76.7%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-129} \lor \neg \left(y \leq 5.1 \cdot 10^{-101}\right) \land y \leq 2.5 \cdot 10^{+38}:\\ \;\;\;\;1 + -8 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-133}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 10^{-100}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+38}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.5e-133) 1.0 (if (<= y 1e-100) -1.0 (if (<= y 2.4e+38) 1.0 -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.5e-133) {
		tmp = 1.0;
	} else if (y <= 1e-100) {
		tmp = -1.0;
	} else if (y <= 2.4e+38) {
		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 <= 2.5d-133) then
        tmp = 1.0d0
    else if (y <= 1d-100) then
        tmp = -1.0d0
    else if (y <= 2.4d+38) 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 <= 2.5e-133) {
		tmp = 1.0;
	} else if (y <= 1e-100) {
		tmp = -1.0;
	} else if (y <= 2.4e+38) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.5e-133:
		tmp = 1.0
	elif y <= 1e-100:
		tmp = -1.0
	elif y <= 2.4e+38:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.5e-133)
		tmp = 1.0;
	elseif (y <= 1e-100)
		tmp = -1.0;
	elseif (y <= 2.4e+38)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.5e-133)
		tmp = 1.0;
	elseif (y <= 1e-100)
		tmp = -1.0;
	elseif (y <= 2.4e+38)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.5e-133], 1.0, If[LessEqual[y, 1e-100], -1.0, If[LessEqual[y, 2.4e+38], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5e-133 or 1e-100 < y < 2.40000000000000017e38

   1. Initial program 55.4%

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

   3. Taylor expanded in x around inf 59.7%

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

   if 2.5e-133 < y < 1e-100 or 2.40000000000000017e38 < y

   1. Initial program 23.6%

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

   3. Taylor expanded in x around 0 75.7%

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

Alternative 12: 49.6% 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 46.5%

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

3. Taylor expanded in x around 0 49.8%

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

Developer target: 52.0% 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 2024103 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

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

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