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

Percentage Accurate: 50.4% → 80.3%

Time: 7.7s

Alternatives: 6

Speedup: 2.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.4% 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: 80.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-241}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\frac{x \cdot x - t_0}{t_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(\frac{x}{y}\right)}^{2} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 2e-241)
     (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))
     (if (<= t_0 5e+143)
       (/ (- (* x x) t_0) (+ t_0 (* x x)))
       (+ (* 0.5 (pow (/ x y) 2.0)) -1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 2e-241) {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	} else if (t_0 <= 5e+143) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = (0.5 * pow((x / y), 2.0)) + -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) :: tmp
    t_0 = y * (y * 4.0d0)
    if (t_0 <= 2d-241) then
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) * (y / x)))
    else if (t_0 <= 5d+143) then
        tmp = ((x * x) - t_0) / (t_0 + (x * x))
    else
        tmp = (0.5d0 * ((x / y) ** 2.0d0)) + (-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 tmp;
	if (t_0 <= 2e-241) {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	} else if (t_0 <= 5e+143) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = (0.5 * Math.pow((x / y), 2.0)) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if t_0 <= 2e-241:
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	elif t_0 <= 5e+143:
		tmp = ((x * x) - t_0) / (t_0 + (x * x))
	else:
		tmp = (0.5 * math.pow((x / y), 2.0)) + -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 2e-241)
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	elseif (t_0 <= 5e+143)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(t_0 + Float64(x * x)));
	else
		tmp = Float64(Float64(0.5 * (Float64(x / y) ^ 2.0)) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if (t_0 <= 2e-241)
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	elseif (t_0 <= 5e+143)
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	else
		tmp = (0.5 * ((x / y) ^ 2.0)) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-241], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+143], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[Power[N[(x / y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 1.9999999999999999e-241

   1. Initial program 58.8%

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

      1. div-inv 57.3%

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

      2. sub-neg 57.3%

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

      3. +-commutative 57.3%

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

      4. *-commutative 57.3%

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

      5. distribute-rgt-neg-in 57.3%

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

      6. fma-def 57.3%

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

      7. distribute-rgt-neg-in 57.3%

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

      8. metadata-eval 57.3%

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

      9. fma-def 57.3%

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

      10. *-commutative 57.3%

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

   3. Applied egg-rr 57.3%

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

   4. Taylor expanded in y around 0 79.0%

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

      1. unpow2 79.0%

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

      2. unpow2 79.0%

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

      3. times-frac 84.8%

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

   6. Simplified 84.8%

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

   if 1.9999999999999999e-241 < (*.f64 (*.f64 y 4) y) < 5.00000000000000012e143

   1. Initial program 80.0%

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

   if 5.00000000000000012e143 < (*.f64 (*.f64 y 4) 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. Step-by-step derivation

      1. div-inv 23.5%

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

      2. sub-neg 23.5%

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

      3. +-commutative 23.5%

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

      4. *-commutative 23.5%

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

      5. distribute-rgt-neg-in 23.5%

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

      6. fma-def 23.5%

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

      7. distribute-rgt-neg-in 23.5%

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

      8. metadata-eval 23.5%

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

      9. fma-def 23.5%

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

      10. *-commutative 23.5%

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

   3. Applied egg-rr 23.5%

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

   4. Taylor expanded in x around 0 64.2%

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

      1. associate--l+ 64.2%

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

      2. fma-def 64.2%

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

      3. metadata-eval 64.2%

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

      4. pow-sqr 64.2%

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

      5. unpow2 64.2%

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

      6. unpow2 64.2%

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

      7. metadata-eval 64.2%

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

      8. pow-sqr 64.2%

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

      9. unpow2 64.2%

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

      10. unpow2 64.2%

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

      11. times-frac 75.5%

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

      12. times-frac 75.5%

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

      13. unpow2 75.5%

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

      14. times-frac 75.5%

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

      15. unpow2 75.5%

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

      16. pow-sqr 75.5%

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

      17. metadata-eval 75.5%

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

      18. fma-neg 75.5%

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

      19. unpow2 75.5%

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

      20. unpow2 75.5%

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

      21. times-frac 82.2%

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

      22. unpow2 82.2%

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

      23. metadata-eval 82.2%

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

   6. Simplified 82.2%

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

      1. fma-udef 82.2%

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

      2. fma-udef 82.2%

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

      3. associate-+r+ 82.2%

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

   8. Applied egg-rr 82.2%

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

   9. Taylor expanded in x around 0 75.8%

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

       1. unpow2 75.8%

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

       2. unpow2 75.8%

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

       3. times-frac 82.9%

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

       4. unpow2 82.9%

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

   11. Simplified 82.9%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-241}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(\frac{x}{y}\right)}^{2} + -1\\ \end{array} \]

Alternative 2: 80.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-241}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+232}:\\ \;\;\;\;\frac{x \cdot x - t_0}{t_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 2e-241)
     (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))
     (if (<= t_0 2e+232) (/ (- (* x x) t_0) (+ t_0 (* x x))) -1.0))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 2e-241) {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	} else if (t_0 <= 2e+232) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} 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) :: tmp
    t_0 = y * (y * 4.0d0)
    if (t_0 <= 2d-241) then
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) * (y / x)))
    else if (t_0 <= 2d+232) then
        tmp = ((x * x) - t_0) / (t_0 + (x * x))
    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 tmp;
	if (t_0 <= 2e-241) {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	} else if (t_0 <= 2e+232) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if t_0 <= 2e-241:
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	elif t_0 <= 2e+232:
		tmp = ((x * x) - t_0) / (t_0 + (x * x))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 2e-241)
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	elseif (t_0 <= 2e+232)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(t_0 + Float64(x * x)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if (t_0 <= 2e-241)
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	elseif (t_0 <= 2e+232)
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	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]}, If[LessEqual[t$95$0, 2e-241], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+232], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 1.9999999999999999e-241

   1. Initial program 58.8%

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

      1. div-inv 57.3%

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

      2. sub-neg 57.3%

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

      3. +-commutative 57.3%

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

      4. *-commutative 57.3%

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

      5. distribute-rgt-neg-in 57.3%

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

      6. fma-def 57.3%

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

      7. distribute-rgt-neg-in 57.3%

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

      8. metadata-eval 57.3%

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

      9. fma-def 57.3%

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

      10. *-commutative 57.3%

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

   3. Applied egg-rr 57.3%

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

   4. Taylor expanded in y around 0 79.0%

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

      1. unpow2 79.0%

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

      2. unpow2 79.0%

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

      3. times-frac 84.8%

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

   6. Simplified 84.8%

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

   if 1.9999999999999999e-241 < (*.f64 (*.f64 y 4) y) < 2.00000000000000011e232

   1. Initial program 75.5%

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

   if 2.00000000000000011e232 < (*.f64 (*.f64 y 4) y)

   1. Initial program 15.1%

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

   2. Taylor expanded in x around 0 87.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-241}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{+232}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3: 74.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := 1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{if}\;t_0 \leq 2 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 4 \cdot 10^{+25}:\\ \;\;\;\;-1 + 0.5 \cdot \frac{x \cdot x}{y \cdot y}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+98}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))) (t_1 (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))))
   (if (<= t_0 2e-82)
     t_1
     (if (<= t_0 4e+25)
       (+ -1.0 (* 0.5 (/ (* x x) (* y y))))
       (if (<= t_0 5e+98) t_1 -1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = 1.0 + (-8.0 * ((y / x) * (y / x)));
	double tmp;
	if (t_0 <= 2e-82) {
		tmp = t_1;
	} else if (t_0 <= 4e+25) {
		tmp = -1.0 + (0.5 * ((x * x) / (y * y)));
	} else if (t_0 <= 5e+98) {
		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 = 1.0d0 + ((-8.0d0) * ((y / x) * (y / x)))
    if (t_0 <= 2d-82) then
        tmp = t_1
    else if (t_0 <= 4d+25) then
        tmp = (-1.0d0) + (0.5d0 * ((x * x) / (y * y)))
    else if (t_0 <= 5d+98) 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 = 1.0 + (-8.0 * ((y / x) * (y / x)));
	double tmp;
	if (t_0 <= 2e-82) {
		tmp = t_1;
	} else if (t_0 <= 4e+25) {
		tmp = -1.0 + (0.5 * ((x * x) / (y * y)));
	} else if (t_0 <= 5e+98) {
		tmp = t_1;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = 1.0 + (-8.0 * ((y / x) * (y / x)))
	tmp = 0
	if t_0 <= 2e-82:
		tmp = t_1
	elif t_0 <= 4e+25:
		tmp = -1.0 + (0.5 * ((x * x) / (y * y)))
	elif t_0 <= 5e+98:
		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(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))))
	tmp = 0.0
	if (t_0 <= 2e-82)
		tmp = t_1;
	elseif (t_0 <= 4e+25)
		tmp = Float64(-1.0 + Float64(0.5 * Float64(Float64(x * x) / Float64(y * y))));
	elseif (t_0 <= 5e+98)
		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 = 1.0 + (-8.0 * ((y / x) * (y / x)));
	tmp = 0.0;
	if (t_0 <= 2e-82)
		tmp = t_1;
	elseif (t_0 <= 4e+25)
		tmp = -1.0 + (0.5 * ((x * x) / (y * y)));
	elseif (t_0 <= 5e+98)
		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[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2e-82], t$95$1, If[LessEqual[t$95$0, 4e+25], N[(-1.0 + N[(0.5 * N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+98], t$95$1, -1.0]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 2e-82 or 4.00000000000000036e25 < (*.f64 (*.f64 y 4) y) < 4.9999999999999998e98

   1. Initial program 64.8%

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

      1. div-inv 63.8%

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

      2. sub-neg 63.8%

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

      3. +-commutative 63.8%

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

      4. *-commutative 63.8%

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

      5. distribute-rgt-neg-in 63.8%

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

      6. fma-def 63.8%

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

      7. distribute-rgt-neg-in 63.8%

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

      8. metadata-eval 63.8%

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

      9. fma-def 63.8%

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

      10. *-commutative 63.8%

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

   3. Applied egg-rr 63.8%

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

   4. Taylor expanded in y around 0 77.8%

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

      1. unpow2 77.8%

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

      2. unpow2 77.8%

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

      3. times-frac 81.5%

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

   6. Simplified 81.5%

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

   if 2e-82 < (*.f64 (*.f64 y 4) y) < 4.00000000000000036e25

   1. Initial program 83.3%

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

      1. div-inv 83.0%

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

      2. sub-neg 83.0%

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

      3. +-commutative 83.0%

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

      4. *-commutative 83.0%

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

      5. distribute-rgt-neg-in 83.0%

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

      6. fma-def 83.0%

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

      7. distribute-rgt-neg-in 83.0%

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

      8. metadata-eval 83.0%

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

      9. fma-def 83.0%

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

      10. *-commutative 83.0%

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

   3. Applied egg-rr 83.0%

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

   4. Taylor expanded in x around 0 66.7%

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

      1. associate--l+ 66.7%

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

      2. fma-def 66.7%

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

      3. metadata-eval 66.7%

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

      4. pow-sqr 66.7%

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

      5. unpow2 66.7%

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

      6. unpow2 66.7%

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

      7. metadata-eval 66.7%

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

      8. pow-sqr 66.7%

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

      9. unpow2 66.7%

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

      10. unpow2 66.7%

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

      11. times-frac 66.7%

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

      12. times-frac 66.7%

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

      13. unpow2 66.7%

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

      14. times-frac 66.7%

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

      15. unpow2 66.7%

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

      16. pow-sqr 66.7%

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

      17. metadata-eval 66.7%

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

      18. fma-neg 66.7%

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

      19. unpow2 66.7%

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

      20. unpow2 66.7%

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

      21. times-frac 66.7%

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

      22. unpow2 66.7%

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

      23. metadata-eval 66.7%

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

   6. Simplified 66.7%

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

      1. fma-udef 66.7%

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

      2. fma-udef 66.7%

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

      3. associate-+r+ 66.7%

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

   8. Applied egg-rr 66.7%

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

   9. Taylor expanded in x around 0 68.0%

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

       1. *-commutative 68.0%

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

       2. unpow2 68.0%

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

       3. unpow2 68.0%

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

   11. Simplified 68.0%

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

   if 4.9999999999999998e98 < (*.f64 (*.f64 y 4) y)

   1. Initial program 28.3%

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

   2. Taylor expanded in x around 0 81.4%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 2 \cdot 10^{-82}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 4 \cdot 10^{+25}:\\ \;\;\;\;-1 + 0.5 \cdot \frac{x \cdot x}{y \cdot y}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+98}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 4: 64.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{-38} \lor \neg \left(y \leq 5600000000000\right) \land y \leq 1.05 \cdot 10^{+49}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 8.6e-38) (and (not (<= y 5600000000000.0)) (<= y 1.05e+49)))
   (+ 1.0 (* -8.0 (* (/ y x) (/ y x))))
   -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 8.6e-38) || (!(y <= 5600000000000.0) && (y <= 1.05e+49))) {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	} 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 <= 8.6d-38) .or. (.not. (y <= 5600000000000.0d0)) .and. (y <= 1.05d+49)) then
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) * (y / x)))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 8.6e-38) || (!(y <= 5600000000000.0) && (y <= 1.05e+49))) {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 8.6e-38) or (not (y <= 5600000000000.0) and (y <= 1.05e+49)):
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 8.6e-38) || (!(y <= 5600000000000.0) && (y <= 1.05e+49)))
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 8.6e-38) || (~((y <= 5600000000000.0)) && (y <= 1.05e+49)))
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 8.6e-38], And[N[Not[LessEqual[y, 5600000000000.0]], $MachinePrecision], LessEqual[y, 1.05e+49]]], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 8.6000000000000004e-38 or 5.6e12 < y < 1.05000000000000005e49

   1. Initial program 54.0%

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

      1. div-inv 53.4%

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

      2. sub-neg 53.4%

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

      3. +-commutative 53.4%

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

      4. *-commutative 53.4%

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

      5. distribute-rgt-neg-in 53.4%

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

      6. fma-def 53.4%

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

      7. distribute-rgt-neg-in 53.4%

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

      8. metadata-eval 53.4%

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

      9. fma-def 53.4%

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

      10. *-commutative 53.4%

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

   3. Applied egg-rr 53.4%

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

   4. Taylor expanded in y around 0 56.1%

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

      1. unpow2 56.1%

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

      2. unpow2 56.1%

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

      3. times-frac 60.9%

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

   6. Simplified 60.9%

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

   if 8.6000000000000004e-38 < y < 5.6e12 or 1.05000000000000005e49 < y

   1. Initial program 39.4%

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

   2. Taylor expanded in x around 0 74.5%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.6 \cdot 10^{-38} \lor \neg \left(y \leq 5600000000000\right) \land y \leq 1.05 \cdot 10^{+49}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5: 63.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 10^{-41}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5000000000000:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+49}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1e-41)
   1.0
   (if (<= y 5000000000000.0) -1.0 (if (<= y 1.06e+49) 1.0 -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1e-41) {
		tmp = 1.0;
	} else if (y <= 5000000000000.0) {
		tmp = -1.0;
	} else if (y <= 1.06e+49) {
		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 <= 1d-41) then
        tmp = 1.0d0
    else if (y <= 5000000000000.0d0) then
        tmp = -1.0d0
    else if (y <= 1.06d+49) 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 <= 1e-41) {
		tmp = 1.0;
	} else if (y <= 5000000000000.0) {
		tmp = -1.0;
	} else if (y <= 1.06e+49) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1e-41:
		tmp = 1.0
	elif y <= 5000000000000.0:
		tmp = -1.0
	elif y <= 1.06e+49:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1e-41)
		tmp = 1.0;
	elseif (y <= 5000000000000.0)
		tmp = -1.0;
	elseif (y <= 1.06e+49)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1e-41)
		tmp = 1.0;
	elseif (y <= 5000000000000.0)
		tmp = -1.0;
	elseif (y <= 1.06e+49)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1e-41], 1.0, If[LessEqual[y, 5000000000000.0], -1.0, If[LessEqual[y, 1.06e+49], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.00000000000000001e-41 or 5e12 < y < 1.06e49

   1. Initial program 54.0%

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

   2. Taylor expanded in x around inf 59.7%

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

   if 1.00000000000000001e-41 < y < 5e12 or 1.06e49 < y

   1. Initial program 39.4%

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

   2. Taylor expanded in x around 0 74.5%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 10^{-41}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5000000000000:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{+49}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6: 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 50.0%

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

2. Taylor expanded in x around 0 50.2%

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

3. Final simplification 50.2%

   \[\leadsto -1 \]

Developer target: 50.8% 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 2023285 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

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

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