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

Percentage Accurate: 50.2% → 80.2%

Time: 10.2s

Alternatives: 5

Speedup: 19.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ t_1 := \frac{\frac{x}{y}}{\frac{y}{x}}\\ \mathbf{if}\;y \leq 4.8 \cdot 10^{-68}:\\ \;\;\;\;1 + \frac{-8}{t_1}\\ \mathbf{elif}\;y \leq 3.05 \cdot 10^{+81}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot 0.5 + -1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))) (t_1 (/ (/ x y) (/ y x))))
   (if (<= y 4.8e-68)
     (+ 1.0 (/ -8.0 t_1))
     (if (<= y 3.05e+81)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (+ (* t_1 0.5) -1.0)))))

C

y = abs(y);
double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = (x / y) / (y / x);
	double tmp;
	if (y <= 4.8e-68) {
		tmp = 1.0 + (-8.0 / t_1);
	} else if (y <= 3.05e+81) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = (t_1 * 0.5) + -1.0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
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) / (y / x)
    if (y <= 4.8d-68) then
        tmp = 1.0d0 + ((-8.0d0) / t_1)
    else if (y <= 3.05d+81) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else
        tmp = (t_1 * 0.5d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double t_1 = (x / y) / (y / x);
	double tmp;
	if (y <= 4.8e-68) {
		tmp = 1.0 + (-8.0 / t_1);
	} else if (y <= 3.05e+81) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = (t_1 * 0.5) + -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	t_0 = y * (y * 4.0)
	t_1 = (x / y) / (y / x)
	tmp = 0
	if y <= 4.8e-68:
		tmp = 1.0 + (-8.0 / t_1)
	elif y <= 3.05e+81:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	else:
		tmp = (t_1 * 0.5) + -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	t_1 = Float64(Float64(x / y) / Float64(y / x))
	tmp = 0.0
	if (y <= 4.8e-68)
		tmp = Float64(1.0 + Float64(-8.0 / t_1));
	elseif (y <= 3.05e+81)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = Float64(Float64(t_1 * 0.5) + -1.0);
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	t_1 = (x / y) / (y / x);
	tmp = 0.0;
	if (y <= 4.8e-68)
		tmp = 1.0 + (-8.0 / t_1);
	elseif (y <= 3.05e+81)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	else
		tmp = (t_1 * 0.5) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 4.8e-68], N[(1.0 + N[(-8.0 / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.05e+81], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$1 * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.79999999999999982e-68

   1. Initial program 49.1%

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

      1. frac-2neg 49.1%

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

      2. div-inv 48.4%

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

      3. *-commutative 48.4%

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

      4. neg-sub0 48.4%

         \[\leadsto \frac{1}{\color{blue}{0 - \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}} \cdot \left(-\left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)\right) \]

      5. +-commutative 48.4%

         \[\leadsto \frac{1}{0 - \color{blue}{\left(\left(y \cdot 4\right) \cdot y + x \cdot x\right)}} \cdot \left(-\left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)\right) \]

      6. associate--r+ 48.4%

         \[\leadsto \frac{1}{\color{blue}{\left(0 - \left(y \cdot 4\right) \cdot y\right) - x \cdot x}} \cdot \left(-\left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)\right) \]

      7. neg-sub0 48.4%

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

      8. distribute-lft-neg-in 48.4%

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

      9. *-commutative 48.4%

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

      10. distribute-rgt-neg-in 48.4%

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

      11. metadata-eval 48.4%

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

      12. pow2 48.4%

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

      13. pow2 48.4%

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

      14. *-commutative 48.4%

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

      15. associate-*l* 48.4%

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

      16. pow2 48.4%

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

   3. Applied egg-rr 48.4%

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

      1. unpow2 48.4%

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

      2. add-sqr-sqrt 48.4%

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

      3. difference-of-squares 48.4%

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

      4. *-commutative 48.4%

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

      5. sqrt-prod 48.4%

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

      6. unpow2 48.4%

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

      7. sqrt-prod 20.5%

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

      8. add-sqr-sqrt 29.0%

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

      9. metadata-eval 29.0%

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

      10. *-commutative 29.0%

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

      11. sqrt-prod 29.0%

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

      12. unpow2 29.0%

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

      13. sqrt-prod 20.7%

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

      14. add-sqr-sqrt 48.4%

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

      15. metadata-eval 48.4%

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

   5. Applied egg-rr 48.4%

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

   6. Taylor expanded in y around 0 50.6%

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

      1. associate-*r/ 50.6%

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

      2. associate-/l* 50.6%

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

      3. unpow2 50.6%

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

      4. unpow2 50.6%

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

      5. times-frac 57.3%

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

      6. unpow2 57.3%

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

   8. Simplified 57.3%

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

      1. unpow2 57.3%

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

      2. associate-*r/ 57.3%

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

      3. associate-/l* 57.3%

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

   10. Applied egg-rr 57.3%

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

   if 4.79999999999999982e-68 < y < 3.05000000000000019e81

   1. Initial program 87.0%

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

   if 3.05000000000000019e81 < y

   1. Initial program 19.2%

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

   2. Taylor expanded in x around 0 71.2%

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

      1. unpow2 71.2%

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

      2. unpow2 71.2%

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

      3. times-frac 71.2%

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

   4. Applied egg-rr 71.2%

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

   5. Taylor expanded in x around 0 79.0%

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

      1. unpow2 79.0%

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

      2. unpow2 79.0%

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

      3. times-frac 85.5%

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

      4. unpow2 85.5%

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

   7. Simplified 85.5%

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

      1. unpow2 18.5%

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

      2. associate-*r/ 18.5%

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

      3. associate-/l* 18.5%

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

   9. Applied egg-rr 85.5%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.8 \cdot 10^{-68}:\\ \;\;\;\;1 + \frac{-8}{\frac{\frac{x}{y}}{\frac{y}{x}}}\\ \mathbf{elif}\;y \leq 3.05 \cdot 10^{+81}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}} \cdot 0.5 + -1\\ \end{array} \]

Alternative 2: 73.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-68} \lor \neg \left(y \leq 0.000162\right) \land y \leq 3.25 \cdot 10^{+35}:\\ \;\;\;\;1 + \frac{-8}{\frac{\frac{x}{y}}{\frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (or (<= y 4.6e-68) (and (not (<= y 0.000162)) (<= y 3.25e+35)))
   (+ 1.0 (/ -8.0 (/ (/ x y) (/ y x))))
   -1.0))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if ((y <= 4.6e-68) || (!(y <= 0.000162) && (y <= 3.25e+35))) {
		tmp = 1.0 + (-8.0 / ((x / y) / (y / x)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= 4.6d-68) .or. (.not. (y <= 0.000162d0)) .and. (y <= 3.25d+35)) then
        tmp = 1.0d0 + ((-8.0d0) / ((x / y) / (y / x)))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if ((y <= 4.6e-68) || (!(y <= 0.000162) && (y <= 3.25e+35))) {
		tmp = 1.0 + (-8.0 / ((x / y) / (y / x)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if (y <= 4.6e-68) or (not (y <= 0.000162) and (y <= 3.25e+35)):
		tmp = 1.0 + (-8.0 / ((x / y) / (y / x)))
	else:
		tmp = -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if ((y <= 4.6e-68) || (!(y <= 0.000162) && (y <= 3.25e+35)))
		tmp = Float64(1.0 + Float64(-8.0 / Float64(Float64(x / y) / Float64(y / x))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 4.6e-68) || (~((y <= 0.000162)) && (y <= 3.25e+35)))
		tmp = 1.0 + (-8.0 / ((x / y) / (y / x)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[Or[LessEqual[y, 4.6e-68], And[N[Not[LessEqual[y, 0.000162]], $MachinePrecision], LessEqual[y, 3.25e+35]]], N[(1.0 + N[(-8.0 / N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 4.59999999999999994e-68 or 1.62000000000000007e-4 < y < 3.2500000000000002e35

   1. Initial program 49.4%

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

      1. frac-2neg 49.4%

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

      2. div-inv 48.8%

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

      3. *-commutative 48.8%

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

      4. neg-sub0 48.8%

         \[\leadsto \frac{1}{\color{blue}{0 - \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}} \cdot \left(-\left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)\right) \]

      5. +-commutative 48.8%

         \[\leadsto \frac{1}{0 - \color{blue}{\left(\left(y \cdot 4\right) \cdot y + x \cdot x\right)}} \cdot \left(-\left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)\right) \]

      6. associate--r+ 48.8%

         \[\leadsto \frac{1}{\color{blue}{\left(0 - \left(y \cdot 4\right) \cdot y\right) - x \cdot x}} \cdot \left(-\left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)\right) \]

      7. neg-sub0 48.8%

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

      8. distribute-lft-neg-in 48.8%

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

      9. *-commutative 48.8%

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

      10. distribute-rgt-neg-in 48.8%

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

      11. metadata-eval 48.8%

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

      12. pow2 48.8%

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

      13. pow2 48.8%

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

      14. *-commutative 48.8%

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

      15. associate-*l* 48.8%

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

      16. pow2 48.8%

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

   3. Applied egg-rr 48.8%

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

      1. unpow2 48.8%

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

      2. add-sqr-sqrt 48.8%

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

      3. difference-of-squares 48.8%

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

      4. *-commutative 48.8%

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

      5. sqrt-prod 48.8%

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

      6. unpow2 48.8%

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

      7. sqrt-prod 21.7%

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

      8. add-sqr-sqrt 29.8%

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

      9. metadata-eval 29.8%

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

      10. *-commutative 29.8%

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

      11. sqrt-prod 29.8%

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

      12. unpow2 29.8%

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

      13. sqrt-prod 21.8%

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

      14. add-sqr-sqrt 48.8%

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

      15. metadata-eval 48.8%

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

   5. Applied egg-rr 48.8%

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

   6. Taylor expanded in y around 0 51.4%

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

      1. associate-*r/ 51.4%

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

      2. associate-/l* 51.4%

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

      3. unpow2 51.4%

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

      4. unpow2 51.4%

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

      5. times-frac 58.0%

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

      6. unpow2 58.0%

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

   8. Simplified 58.0%

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

      1. unpow2 58.0%

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

      2. associate-*r/ 57.9%

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

      3. associate-/l* 58.0%

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

   10. Applied egg-rr 58.0%

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

   if 4.59999999999999994e-68 < y < 1.62000000000000007e-4 or 3.2500000000000002e35 < y

   1. Initial program 43.6%

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

   2. Taylor expanded in x around 0 81.0%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-68} \lor \neg \left(y \leq 0.000162\right) \land y \leq 3.25 \cdot 10^{+35}:\\ \;\;\;\;1 + \frac{-8}{\frac{\frac{x}{y}}{\frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3: 74.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \frac{\frac{x}{y}}{\frac{y}{x}}\\ \mathbf{if}\;y \leq 1.65 \cdot 10^{-67}:\\ \;\;\;\;1 + \frac{-8}{t_0}\\ \mathbf{elif}\;y \leq 0.000175 \lor \neg \left(y \leq 4000\right):\\ \;\;\;\;t_0 \cdot 0.5 + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (/ x y) (/ y x))))
   (if (<= y 1.65e-67)
     (+ 1.0 (/ -8.0 t_0))
     (if (or (<= y 0.000175) (not (<= y 4000.0))) (+ (* t_0 0.5) -1.0) 1.0))))

C

y = abs(y);
double code(double x, double y) {
	double t_0 = (x / y) / (y / x);
	double tmp;
	if (y <= 1.65e-67) {
		tmp = 1.0 + (-8.0 / t_0);
	} else if ((y <= 0.000175) || !(y <= 4000.0)) {
		tmp = (t_0 * 0.5) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x / y) / (y / x)
    if (y <= 1.65d-67) then
        tmp = 1.0d0 + ((-8.0d0) / t_0)
    else if ((y <= 0.000175d0) .or. (.not. (y <= 4000.0d0))) then
        tmp = (t_0 * 0.5d0) + (-1.0d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double t_0 = (x / y) / (y / x);
	double tmp;
	if (y <= 1.65e-67) {
		tmp = 1.0 + (-8.0 / t_0);
	} else if ((y <= 0.000175) || !(y <= 4000.0)) {
		tmp = (t_0 * 0.5) + -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	t_0 = (x / y) / (y / x)
	tmp = 0
	if y <= 1.65e-67:
		tmp = 1.0 + (-8.0 / t_0)
	elif (y <= 0.000175) or not (y <= 4000.0):
		tmp = (t_0 * 0.5) + -1.0
	else:
		tmp = 1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	t_0 = Float64(Float64(x / y) / Float64(y / x))
	tmp = 0.0
	if (y <= 1.65e-67)
		tmp = Float64(1.0 + Float64(-8.0 / t_0));
	elseif ((y <= 0.000175) || !(y <= 4000.0))
		tmp = Float64(Float64(t_0 * 0.5) + -1.0);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	t_0 = (x / y) / (y / x);
	tmp = 0.0;
	if (y <= 1.65e-67)
		tmp = 1.0 + (-8.0 / t_0);
	elseif ((y <= 0.000175) || ~((y <= 4000.0)))
		tmp = (t_0 * 0.5) + -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 1.65e-67], N[(1.0 + N[(-8.0 / t$95$0), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, 0.000175], N[Not[LessEqual[y, 4000.0]], $MachinePrecision]], N[(N[(t$95$0 * 0.5), $MachinePrecision] + -1.0), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.6500000000000001e-67

   1. Initial program 49.1%

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

      1. frac-2neg 49.1%

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

      2. div-inv 48.4%

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

      3. *-commutative 48.4%

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

      4. neg-sub0 48.4%

         \[\leadsto \frac{1}{\color{blue}{0 - \left(x \cdot x + \left(y \cdot 4\right) \cdot y\right)}} \cdot \left(-\left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)\right) \]

      5. +-commutative 48.4%

         \[\leadsto \frac{1}{0 - \color{blue}{\left(\left(y \cdot 4\right) \cdot y + x \cdot x\right)}} \cdot \left(-\left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)\right) \]

      6. associate--r+ 48.4%

         \[\leadsto \frac{1}{\color{blue}{\left(0 - \left(y \cdot 4\right) \cdot y\right) - x \cdot x}} \cdot \left(-\left(x \cdot x - \left(y \cdot 4\right) \cdot y\right)\right) \]

      7. neg-sub0 48.4%

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

      8. distribute-lft-neg-in 48.4%

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

      9. *-commutative 48.4%

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

      10. distribute-rgt-neg-in 48.4%

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

      11. metadata-eval 48.4%

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

      12. pow2 48.4%

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

      13. pow2 48.4%

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

      14. *-commutative 48.4%

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

      15. associate-*l* 48.4%

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

      16. pow2 48.4%

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

   3. Applied egg-rr 48.4%

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

      1. unpow2 48.4%

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

      2. add-sqr-sqrt 48.4%

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

      3. difference-of-squares 48.4%

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

      4. *-commutative 48.4%

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

      5. sqrt-prod 48.4%

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

      6. unpow2 48.4%

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

      7. sqrt-prod 20.5%

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

      8. add-sqr-sqrt 29.0%

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

      9. metadata-eval 29.0%

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

      10. *-commutative 29.0%

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

      11. sqrt-prod 29.0%

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

      12. unpow2 29.0%

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

      13. sqrt-prod 20.7%

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

      14. add-sqr-sqrt 48.4%

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

      15. metadata-eval 48.4%

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

   5. Applied egg-rr 48.4%

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

   6. Taylor expanded in y around 0 50.6%

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

      1. associate-*r/ 50.6%

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

      2. associate-/l* 50.6%

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

      3. unpow2 50.6%

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

      4. unpow2 50.6%

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

      5. times-frac 57.3%

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

      6. unpow2 57.3%

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

   8. Simplified 57.3%

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

      1. unpow2 57.3%

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

      2. associate-*r/ 57.3%

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

      3. associate-/l* 57.3%

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

   10. Applied egg-rr 57.3%

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

   if 1.6500000000000001e-67 < y < 1.74999999999999998e-4 or 4e3 < y

   1. Initial program 44.4%

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

   2. Taylor expanded in x around 0 71.7%

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

      1. unpow2 71.7%

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

      2. unpow2 71.7%

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

      3. times-frac 71.7%

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

   4. Applied egg-rr 71.7%

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

   5. Taylor expanded in x around 0 76.9%

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

      1. unpow2 76.9%

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

      2. unpow2 76.9%

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

      3. times-frac 81.1%

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

      4. unpow2 81.1%

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

   7. Simplified 81.1%

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

      1. unpow2 22.8%

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

      2. associate-*r/ 22.8%

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

      3. associate-/l* 22.8%

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

   9. Applied egg-rr 81.1%

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

   if 1.74999999999999998e-4 < y < 4e3

   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 inf 100.0%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-67}:\\ \;\;\;\;1 + \frac{-8}{\frac{\frac{x}{y}}{\frac{y}{x}}}\\ \mathbf{elif}\;y \leq 0.000175 \lor \neg \left(y \leq 4000\right):\\ \;\;\;\;\frac{\frac{x}{y}}{\frac{y}{x}} \cdot 0.5 + -1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 73.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-67}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-5}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+35}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 1.6e-67) 1.0 (if (<= y 8e-5) -1.0 (if (<= y 3.9e+35) 1.0 -1.0))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.6e-67) {
		tmp = 1.0;
	} else if (y <= 8e-5) {
		tmp = -1.0;
	} else if (y <= 3.9e+35) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.6d-67) then
        tmp = 1.0d0
    else if (y <= 8d-5) then
        tmp = -1.0d0
    else if (y <= 3.9d+35) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.6e-67) {
		tmp = 1.0;
	} else if (y <= 8e-5) {
		tmp = -1.0;
	} else if (y <= 3.9e+35) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 1.6e-67:
		tmp = 1.0
	elif y <= 8e-5:
		tmp = -1.0
	elif y <= 3.9e+35:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 1.6e-67)
		tmp = 1.0;
	elseif (y <= 8e-5)
		tmp = -1.0;
	elseif (y <= 3.9e+35)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.6e-67)
		tmp = 1.0;
	elseif (y <= 8e-5)
		tmp = -1.0;
	elseif (y <= 3.9e+35)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 1.6e-67], 1.0, If[LessEqual[y, 8e-5], -1.0, If[LessEqual[y, 3.9e+35], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.60000000000000011e-67 or 8.00000000000000065e-5 < y < 3.8999999999999999e35

   1. Initial program 49.4%

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

   2. Taylor expanded in x around inf 56.7%

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

   if 1.60000000000000011e-67 < y < 8.00000000000000065e-5 or 3.8999999999999999e35 < y

   1. Initial program 43.6%

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

   2. Taylor expanded in x around 0 81.0%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-67}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-5}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+35}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5: 50.5% accurate, 19.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ -1 \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y) :precision binary64 -1.0)

C

y = abs(y);
double code(double x, double y) {
	return -1.0;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	return -1.0;
}

Python

y = abs(y)
def code(x, y):
	return -1.0

Julia

y = abs(y)
function code(x, y)
	return -1.0
end

MATLAB

y = abs(y)
function tmp = code(x, y)
	tmp = -1.0;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := -1.0

Derivation

1. Initial program 47.6%

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

2. Taylor expanded in x around 0 55.6%

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

3. Final simplification 55.6%

   \[\leadsto -1 \]

Developer target: 50.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot 4\\ t_1 := x \cdot x + t_0\\ t_2 := \frac{t_0}{t_1}\\ t_3 := \left(y \cdot 4\right) \cdot y\\ \mathbf{if}\;\frac{x \cdot x - t_3}{x \cdot x + t_3} < 0.9743233849626781:\\ \;\;\;\;\frac{x \cdot x}{t_1} - t_2\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{t_1}}\right)}^{2} - t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y y) 4.0))
        (t_1 (+ (* x x) t_0))
        (t_2 (/ t_0 t_1))
        (t_3 (* (* y 4.0) y)))
   (if (< (/ (- (* x x) t_3) (+ (* x x) t_3)) 0.9743233849626781)
     (- (/ (* x x) t_1) t_2)
     (- (pow (/ x (sqrt t_1)) 2.0) t_2))))

C

double code(double x, double y) {
	double t_0 = (y * y) * 4.0;
	double t_1 = (x * x) + t_0;
	double t_2 = t_0 / t_1;
	double t_3 = (y * 4.0) * y;
	double tmp;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781) {
		tmp = ((x * x) / t_1) - t_2;
	} else {
		tmp = pow((x / sqrt(t_1)), 2.0) - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (y * y) * 4.0d0
    t_1 = (x * x) + t_0
    t_2 = t_0 / t_1
    t_3 = (y * 4.0d0) * y
    if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781d0) then
        tmp = ((x * x) / t_1) - t_2
    else
        tmp = ((x / sqrt(t_1)) ** 2.0d0) - t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * y) * 4.0;
	double t_1 = (x * x) + t_0;
	double t_2 = t_0 / t_1;
	double t_3 = (y * 4.0) * y;
	double tmp;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781) {
		tmp = ((x * x) / t_1) - t_2;
	} else {
		tmp = Math.pow((x / Math.sqrt(t_1)), 2.0) - t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) * 4.0
	t_1 = (x * x) + t_0
	t_2 = t_0 / t_1
	t_3 = (y * 4.0) * y
	tmp = 0
	if (((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781:
		tmp = ((x * x) / t_1) - t_2
	else:
		tmp = math.pow((x / math.sqrt(t_1)), 2.0) - t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * 4.0)
	t_1 = Float64(Float64(x * x) + t_0)
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(Float64(y * 4.0) * y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * x) - t_3) / Float64(Float64(x * x) + t_3)) < 0.9743233849626781)
		tmp = Float64(Float64(Float64(x * x) / t_1) - t_2);
	else
		tmp = Float64((Float64(x / sqrt(t_1)) ^ 2.0) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) * 4.0;
	t_1 = (x * x) + t_0;
	t_2 = t_0 / t_1;
	t_3 = (y * 4.0) * y;
	tmp = 0.0;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781)
		tmp = ((x * x) / t_1) - t_2;
	else
		tmp = ((x / sqrt(t_1)) ^ 2.0) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[Less[N[(N[(N[(x * x), $MachinePrecision] - t$95$3), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], 0.9743233849626781], N[(N[(N[(x * x), $MachinePrecision] / t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[Power[N[(x / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]

Reproduce

herbie shell --seed 2023301 
(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))))