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

Percentage Accurate: 50.8% → 80.8%

Time: 7.7s

Alternatives: 7

Speedup: 2.1×

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: 50.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-284}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+182}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 5e-284)
     (fma 0.5 (* (/ x y) (/ x y)) -1.0)
     (if (<= (* x x) 2e+182)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (+ 1.0 (* -4.0 (/ (/ y x) (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 5e-284) {
		tmp = fma(0.5, ((x / y) * (x / y)), -1.0);
	} else if ((x * x) <= 2e+182) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 5e-284)
		tmp = fma(0.5, Float64(Float64(x / y) * Float64(x / y)), -1.0);
	elseif (Float64(x * x) <= 2e+182)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = Float64(1.0 + Float64(-4.0 * Float64(Float64(y / x) / Float64(x / y))));
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e-284], N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+182], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-4.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 4.99999999999999973e-284

   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 75.1%

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

      1. fma-neg 75.1%

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

      2. unpow2 75.1%

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

      3. unpow2 75.1%

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

      4. times-frac 87.2%

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

      5. metadata-eval 87.2%

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

   4. Simplified 87.2%

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

   if 4.99999999999999973e-284 < (*.f64 x x) < 2.0000000000000001e182

   1. Initial program 74.2%

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

   if 2.0000000000000001e182 < (*.f64 x x)

   1. Initial program 18.1%

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

   2. Taylor expanded in x around inf 18.3%

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

      1. unpow2 18.3%

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

   4. Simplified 18.3%

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

   5. Taylor expanded in x around inf 76.1%

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

      1. unpow2 76.1%

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

      2. unpow2 76.1%

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

      3. times-frac 86.2%

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

      4. unpow2 86.2%

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

   7. Simplified 86.2%

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

      1. unpow2 86.2%

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

      2. clear-num 86.2%

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

      3. un-div-inv 86.2%

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

   9. Applied egg-rr 86.2%

      \[\leadsto 1 + -4 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-284}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+182}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \end{array} \]

Alternative 2: 80.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-284}:\\ \;\;\;\;-1 + \frac{\frac{x}{\frac{y}{x}}}{y \cdot 4}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+182}:\\ \;\;\;\;\frac{x \cdot x - t_0}{x \cdot x + t_0}\\ \mathbf{else}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 5e-284)
     (+ -1.0 (/ (/ x (/ y x)) (* y 4.0)))
     (if (<= (* x x) 2e+182)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (+ 1.0 (* -4.0 (/ (/ y x) (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 5e-284) {
		tmp = -1.0 + ((x / (y / x)) / (y * 4.0));
	} else if ((x * x) <= 2e+182) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    if ((x * x) <= 5d-284) then
        tmp = (-1.0d0) + ((x / (y / x)) / (y * 4.0d0))
    else if ((x * x) <= 2d+182) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else
        tmp = 1.0d0 + ((-4.0d0) * ((y / x) / (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 5e-284) {
		tmp = -1.0 + ((x / (y / x)) / (y * 4.0));
	} else if ((x * x) <= 2e+182) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if (x * x) <= 5e-284:
		tmp = -1.0 + ((x / (y / x)) / (y * 4.0))
	elif (x * x) <= 2e+182:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	else:
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 5e-284)
		tmp = Float64(-1.0 + Float64(Float64(x / Float64(y / x)) / Float64(y * 4.0)));
	elseif (Float64(x * x) <= 2e+182)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = Float64(1.0 + Float64(-4.0 * Float64(Float64(y / x) / Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if ((x * x) <= 5e-284)
		tmp = -1.0 + ((x / (y / x)) / (y * 4.0));
	elseif ((x * x) <= 2e+182)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	else
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 5e-284], N[(-1.0 + N[(N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision] / N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+182], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(-4.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 4.99999999999999973e-284

   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 46.4%

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

      1. *-commutative 46.4%

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

      2. unpow2 46.4%

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

      3. associate-*r* 46.4%

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

   4. Simplified 46.4%

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

      1. div-sub 46.3%

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

      2. associate-*r* 46.3%

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

      3. times-frac 46.3%

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

      4. *-commutative 46.3%

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

      5. pow1 46.3%

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

      6. pow1 46.3%

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

      7. pow-div 75.5%

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

      8. metadata-eval 75.5%

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

      9. metadata-eval 75.5%

         \[\leadsto \frac{x}{y \cdot y} \cdot \frac{x}{4} - \color{blue}{1} \]

   6. Applied egg-rr 75.5%

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

      1. *-commutative 75.5%

         \[\leadsto \color{blue}{\frac{x}{4} \cdot \frac{x}{y \cdot y}} - 1 \]

      2. associate-/r* 86.9%

         \[\leadsto \frac{x}{4} \cdot \color{blue}{\frac{\frac{x}{y}}{y}} - 1 \]

      3. frac-times 87.0%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{x}{y}}{4 \cdot y}} - 1 \]

      4. clear-num 87.0%

         \[\leadsto \frac{x \cdot \color{blue}{\frac{1}{\frac{y}{x}}}}{4 \cdot y} - 1 \]

      5. div-inv 87.0%

         \[\leadsto \frac{\color{blue}{\frac{x}{\frac{y}{x}}}}{4 \cdot y} - 1 \]

   8. Applied egg-rr 87.0%

      \[\leadsto \color{blue}{\frac{\frac{x}{\frac{y}{x}}}{4 \cdot y}} - 1 \]

   if 4.99999999999999973e-284 < (*.f64 x x) < 2.0000000000000001e182

   1. Initial program 74.2%

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

   if 2.0000000000000001e182 < (*.f64 x x)

   1. Initial program 18.1%

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

   2. Taylor expanded in x around inf 18.3%

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

      1. unpow2 18.3%

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

   4. Simplified 18.3%

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

   5. Taylor expanded in x around inf 76.1%

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

      1. unpow2 76.1%

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

      2. unpow2 76.1%

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

      3. times-frac 86.2%

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

      4. unpow2 86.2%

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

   7. Simplified 86.2%

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

      1. unpow2 86.2%

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

      2. clear-num 86.2%

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

      3. un-div-inv 86.2%

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

   9. Applied egg-rr 86.2%

      \[\leadsto 1 + -4 \cdot \color{blue}{\frac{\frac{y}{x}}{\frac{x}{y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-284}:\\ \;\;\;\;-1 + \frac{\frac{x}{\frac{y}{x}}}{y \cdot 4}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+182}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \end{array} \]

Alternative 3: 74.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-40}:\\ \;\;\;\;-1 + \frac{\frac{x}{\frac{y}{x}}}{y \cdot 4}\\ \mathbf{elif}\;x \cdot x \leq 0.001 \lor \neg \left(x \cdot x \leq 10^{+73}\right):\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y \cdot y} \cdot \frac{x}{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 4e-40)
   (+ -1.0 (/ (/ x (/ y x)) (* y 4.0)))
   (if (or (<= (* x x) 0.001) (not (<= (* x x) 1e+73)))
     (+ 1.0 (* -4.0 (/ (/ y x) (/ x y))))
     (+ -1.0 (* (/ x (* y y)) (/ x 4.0))))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 4e-40) {
		tmp = -1.0 + ((x / (y / x)) / (y * 4.0));
	} else if (((x * x) <= 0.001) || !((x * x) <= 1e+73)) {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	} else {
		tmp = -1.0 + ((x / (y * y)) * (x / 4.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x * x) <= 4d-40) then
        tmp = (-1.0d0) + ((x / (y / x)) / (y * 4.0d0))
    else if (((x * x) <= 0.001d0) .or. (.not. ((x * x) <= 1d+73))) then
        tmp = 1.0d0 + ((-4.0d0) * ((y / x) / (x / y)))
    else
        tmp = (-1.0d0) + ((x / (y * y)) * (x / 4.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * x) <= 4e-40) {
		tmp = -1.0 + ((x / (y / x)) / (y * 4.0));
	} else if (((x * x) <= 0.001) || !((x * x) <= 1e+73)) {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	} else {
		tmp = -1.0 + ((x / (y * y)) * (x / 4.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 4e-40:
		tmp = -1.0 + ((x / (y / x)) / (y * 4.0))
	elif ((x * x) <= 0.001) or not ((x * x) <= 1e+73):
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)))
	else:
		tmp = -1.0 + ((x / (y * y)) * (x / 4.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 4e-40)
		tmp = Float64(-1.0 + Float64(Float64(x / Float64(y / x)) / Float64(y * 4.0)));
	elseif ((Float64(x * x) <= 0.001) || !(Float64(x * x) <= 1e+73))
		tmp = Float64(1.0 + Float64(-4.0 * Float64(Float64(y / x) / Float64(x / y))));
	else
		tmp = Float64(-1.0 + Float64(Float64(x / Float64(y * y)) * Float64(x / 4.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 4e-40)
		tmp = -1.0 + ((x / (y / x)) / (y * 4.0));
	elseif (((x * x) <= 0.001) || ~(((x * x) <= 1e+73)))
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	else
		tmp = -1.0 + ((x / (y * y)) * (x / 4.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 4e-40], N[(-1.0 + N[(N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision] / N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[N[(x * x), $MachinePrecision], 0.001], N[Not[LessEqual[N[(x * x), $MachinePrecision], 1e+73]], $MachinePrecision]], N[(1.0 + N[(-4.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(x / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 3.9999999999999997e-40

   1. Initial program 60.2%

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

   2. Taylor expanded in x around 0 45.2%

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

      1. *-commutative 45.2%

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

      2. unpow2 45.2%

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

      3. associate-*r* 45.2%

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

   4. Simplified 45.2%

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

      1. div-sub 44.9%

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

      2. associate-*r* 44.9%

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

      3. times-frac 44.9%

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

      4. *-commutative 44.9%

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

      5. pow1 44.9%

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

      6. pow1 44.9%

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

      7. pow-div 72.0%

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

      8. metadata-eval 72.0%

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

      9. metadata-eval 72.0%

         \[\leadsto \frac{x}{y \cdot y} \cdot \frac{x}{4} - \color{blue}{1} \]

   6. Applied egg-rr 72.0%

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

      1. *-commutative 72.0%

         \[\leadsto \color{blue}{\frac{x}{4} \cdot \frac{x}{y \cdot y}} - 1 \]

      2. associate-/r* 79.2%

         \[\leadsto \frac{x}{4} \cdot \color{blue}{\frac{\frac{x}{y}}{y}} - 1 \]

      3. frac-times 79.2%

         \[\leadsto \color{blue}{\frac{x \cdot \frac{x}{y}}{4 \cdot y}} - 1 \]

      4. clear-num 79.2%

         \[\leadsto \frac{x \cdot \color{blue}{\frac{1}{\frac{y}{x}}}}{4 \cdot y} - 1 \]

      5. div-inv 79.2%

         \[\leadsto \frac{\color{blue}{\frac{x}{\frac{y}{x}}}}{4 \cdot y} - 1 \]

   8. Applied egg-rr 79.2%

      \[\leadsto \color{blue}{\frac{\frac{x}{\frac{y}{x}}}{4 \cdot y}} - 1 \]

   if 3.9999999999999997e-40 < (*.f64 x x) < 1e-3 or 9.99999999999999983e72 < (*.f64 x x)

   1. Initial program 33.6%

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

   2. Taylor expanded in x around inf 31.0%

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

      1. unpow2 31.0%

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

   4. Simplified 31.0%

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

   5. Taylor expanded in x around inf 73.6%

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

      1. unpow2 73.6%

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

      2. unpow2 73.6%

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

      3. times-frac 81.1%

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

      4. unpow2 81.1%

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

   7. Simplified 81.1%

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

      1. unpow2 81.1%

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

      2. clear-num 81.1%

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

      3. un-div-inv 81.1%

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

   9. Applied egg-rr 81.1%

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

   if 1e-3 < (*.f64 x x) < 9.99999999999999983e72

   1. Initial program 60.0%

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

   2. Taylor expanded in x around 0 24.6%

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

      1. *-commutative 24.6%

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

      2. unpow2 24.6%

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

      3. associate-*r* 24.6%

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

   4. Simplified 24.6%

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

      1. div-sub 23.9%

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

      2. associate-*r* 23.9%

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

      3. times-frac 23.9%

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

      4. *-commutative 23.9%

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

      5. pow1 23.9%

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

      6. pow1 23.9%

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

      7. pow-div 64.6%

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

      8. metadata-eval 64.6%

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

      9. metadata-eval 64.6%

         \[\leadsto \frac{x}{y \cdot y} \cdot \frac{x}{4} - \color{blue}{1} \]

   6. Applied egg-rr 64.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-40}:\\ \;\;\;\;-1 + \frac{\frac{x}{\frac{y}{x}}}{y \cdot 4}\\ \mathbf{elif}\;x \cdot x \leq 0.001 \lor \neg \left(x \cdot x \leq 10^{+73}\right):\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y \cdot y} \cdot \frac{x}{4}\\ \end{array} \]

Alternative 4: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.72 \cdot 10^{+115} \lor \neg \left(x \leq -5 \cdot 10^{+86} \lor \neg \left(x \leq -5.2 \cdot 10^{+38}\right) \land x \leq 2.9 \cdot 10^{-80}\right):\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x -1.72e+115)
         (not (or (<= x -5e+86) (and (not (<= x -5.2e+38)) (<= x 2.9e-80)))))
   (+ 1.0 (* -4.0 (/ (/ y x) (/ x y))))
   -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x <= -1.72e+115) || !((x <= -5e+86) || (!(x <= -5.2e+38) && (x <= 2.9e-80)))) {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x <= (-1.72d+115)) .or. (.not. (x <= (-5d+86)) .or. (.not. (x <= (-5.2d+38))) .and. (x <= 2.9d-80))) then
        tmp = 1.0d0 + ((-4.0d0) * ((y / x) / (x / y)))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= -1.72e+115) || !((x <= -5e+86) || (!(x <= -5.2e+38) && (x <= 2.9e-80)))) {
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= -1.72e+115) or not ((x <= -5e+86) or (not (x <= -5.2e+38) and (x <= 2.9e-80))):
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= -1.72e+115) || !((x <= -5e+86) || (!(x <= -5.2e+38) && (x <= 2.9e-80))))
		tmp = Float64(1.0 + Float64(-4.0 * Float64(Float64(y / x) / Float64(x / y))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= -1.72e+115) || ~(((x <= -5e+86) || (~((x <= -5.2e+38)) && (x <= 2.9e-80)))))
		tmp = 1.0 + (-4.0 * ((y / x) / (x / y)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, -1.72e+115], N[Not[Or[LessEqual[x, -5e+86], And[N[Not[LessEqual[x, -5.2e+38]], $MachinePrecision], LessEqual[x, 2.9e-80]]]], $MachinePrecision]], N[(1.0 + N[(-4.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if x < -1.72e115 or -4.9999999999999998e86 < x < -5.1999999999999998e38 or 2.89999999999999998e-80 < x

   1. Initial program 37.7%

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

   2. Taylor expanded in x around inf 33.7%

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

      1. unpow2 33.7%

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

   4. Simplified 33.7%

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

   5. Taylor expanded in x around inf 73.2%

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

      1. unpow2 73.2%

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

      2. unpow2 73.2%

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

      3. times-frac 80.0%

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

      4. unpow2 80.0%

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

   7. Simplified 80.0%

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

      1. unpow2 80.0%

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

      2. clear-num 80.0%

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

      3. un-div-inv 80.0%

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

   9. Applied egg-rr 80.0%

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

   if -1.72e115 < x < -4.9999999999999998e86 or -5.1999999999999998e38 < x < 2.89999999999999998e-80

   1. Initial program 58.2%

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

   2. Taylor expanded in x around 0 79.6%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.72 \cdot 10^{+115} \lor \neg \left(x \leq -5 \cdot 10^{+86} \lor \neg \left(x \leq -5.2 \cdot 10^{+38}\right) \land x \leq 2.9 \cdot 10^{-80}\right):\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5: 72.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{if}\;x \leq -1.72 \cdot 10^{+115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{+87}:\\ \;\;\;\;-1 + \frac{x}{y \cdot y} \cdot \frac{x}{4}\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{+38} \lor \neg \left(x \leq 1.35 \cdot 10^{-79}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* -4.0 (/ (/ y x) (/ x y))))))
   (if (<= x -1.72e+115)
     t_0
     (if (<= x -6.5e+87)
       (+ -1.0 (* (/ x (* y y)) (/ x 4.0)))
       (if (or (<= x -4.7e+38) (not (<= x 1.35e-79))) t_0 -1.0)))))

C

double code(double x, double y) {
	double t_0 = 1.0 + (-4.0 * ((y / x) / (x / y)));
	double tmp;
	if (x <= -1.72e+115) {
		tmp = t_0;
	} else if (x <= -6.5e+87) {
		tmp = -1.0 + ((x / (y * y)) * (x / 4.0));
	} else if ((x <= -4.7e+38) || !(x <= 1.35e-79)) {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((-4.0d0) * ((y / x) / (x / y)))
    if (x <= (-1.72d+115)) then
        tmp = t_0
    else if (x <= (-6.5d+87)) then
        tmp = (-1.0d0) + ((x / (y * y)) * (x / 4.0d0))
    else if ((x <= (-4.7d+38)) .or. (.not. (x <= 1.35d-79))) then
        tmp = t_0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = 1.0 + (-4.0 * ((y / x) / (x / y)));
	double tmp;
	if (x <= -1.72e+115) {
		tmp = t_0;
	} else if (x <= -6.5e+87) {
		tmp = -1.0 + ((x / (y * y)) * (x / 4.0));
	} else if ((x <= -4.7e+38) || !(x <= 1.35e-79)) {
		tmp = t_0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = 1.0 + (-4.0 * ((y / x) / (x / y)))
	tmp = 0
	if x <= -1.72e+115:
		tmp = t_0
	elif x <= -6.5e+87:
		tmp = -1.0 + ((x / (y * y)) * (x / 4.0))
	elif (x <= -4.7e+38) or not (x <= 1.35e-79):
		tmp = t_0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(1.0 + Float64(-4.0 * Float64(Float64(y / x) / Float64(x / y))))
	tmp = 0.0
	if (x <= -1.72e+115)
		tmp = t_0;
	elseif (x <= -6.5e+87)
		tmp = Float64(-1.0 + Float64(Float64(x / Float64(y * y)) * Float64(x / 4.0)));
	elseif ((x <= -4.7e+38) || !(x <= 1.35e-79))
		tmp = t_0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = 1.0 + (-4.0 * ((y / x) / (x / y)));
	tmp = 0.0;
	if (x <= -1.72e+115)
		tmp = t_0;
	elseif (x <= -6.5e+87)
		tmp = -1.0 + ((x / (y * y)) * (x / 4.0));
	elseif ((x <= -4.7e+38) || ~((x <= 1.35e-79)))
		tmp = t_0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(1.0 + N[(-4.0 * N[(N[(y / x), $MachinePrecision] / N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.72e+115], t$95$0, If[LessEqual[x, -6.5e+87], N[(-1.0 + N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(x / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x, -4.7e+38], N[Not[LessEqual[x, 1.35e-79]], $MachinePrecision]], t$95$0, -1.0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.72e115 or -6.5000000000000002e87 < x < -4.6999999999999999e38 or 1.3500000000000001e-79 < x

   1. Initial program 37.7%

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

   2. Taylor expanded in x around inf 33.7%

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

      1. unpow2 33.7%

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

   4. Simplified 33.7%

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

   5. Taylor expanded in x around inf 73.2%

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

      1. unpow2 73.2%

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

      2. unpow2 73.2%

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

      3. times-frac 80.0%

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

      4. unpow2 80.0%

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

   7. Simplified 80.0%

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

      1. unpow2 80.0%

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

      2. clear-num 80.0%

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

      3. un-div-inv 80.0%

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

   9. Applied egg-rr 80.0%

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

   if -1.72e115 < x < -6.5000000000000002e87

   1. Initial program 42.9%

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

   2. Taylor expanded in x around 0 29.0%

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

      1. *-commutative 29.0%

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

      2. unpow2 29.0%

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

      3. associate-*r* 29.0%

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

   4. Simplified 29.0%

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

      1. div-sub 28.6%

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

      2. associate-*r* 28.6%

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

      3. times-frac 28.6%

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

      4. *-commutative 28.6%

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

      5. pow1 28.6%

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

      6. pow1 28.6%

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

      7. pow-div 86.2%

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

      8. metadata-eval 86.2%

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

      9. metadata-eval 86.2%

         \[\leadsto \frac{x}{y \cdot y} \cdot \frac{x}{4} - \color{blue}{1} \]

   6. Applied egg-rr 86.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot y} \cdot \frac{x}{4} - 1} \]

   if -4.6999999999999999e38 < x < 1.3500000000000001e-79

   1. Initial program 59.1%

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

   2. Taylor expanded in x around 0 79.3%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.72 \cdot 10^{+115}:\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{+87}:\\ \;\;\;\;-1 + \frac{x}{y \cdot y} \cdot \frac{x}{4}\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{+38} \lor \neg \left(x \leq 1.35 \cdot 10^{-79}\right):\\ \;\;\;\;1 + -4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6: 72.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.72 \cdot 10^{+115}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -5 \cdot 10^{+88}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{+38}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-79}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -1.72e+115)
   1.0
   (if (<= x -5e+88)
     -1.0
     (if (<= x -4.7e+38) 1.0 (if (<= x 1.5e-79) -1.0 1.0)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -1.72e+115) {
		tmp = 1.0;
	} else if (x <= -5e+88) {
		tmp = -1.0;
	} else if (x <= -4.7e+38) {
		tmp = 1.0;
	} else if (x <= 1.5e-79) {
		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 (x <= (-1.72d+115)) then
        tmp = 1.0d0
    else if (x <= (-5d+88)) then
        tmp = -1.0d0
    else if (x <= (-4.7d+38)) then
        tmp = 1.0d0
    else if (x <= 1.5d-79) 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 (x <= -1.72e+115) {
		tmp = 1.0;
	} else if (x <= -5e+88) {
		tmp = -1.0;
	} else if (x <= -4.7e+38) {
		tmp = 1.0;
	} else if (x <= 1.5e-79) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -1.72e+115:
		tmp = 1.0
	elif x <= -5e+88:
		tmp = -1.0
	elif x <= -4.7e+38:
		tmp = 1.0
	elif x <= 1.5e-79:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -1.72e+115)
		tmp = 1.0;
	elseif (x <= -5e+88)
		tmp = -1.0;
	elseif (x <= -4.7e+38)
		tmp = 1.0;
	elseif (x <= 1.5e-79)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -1.72e+115)
		tmp = 1.0;
	elseif (x <= -5e+88)
		tmp = -1.0;
	elseif (x <= -4.7e+38)
		tmp = 1.0;
	elseif (x <= 1.5e-79)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -1.72e+115], 1.0, If[LessEqual[x, -5e+88], -1.0, If[LessEqual[x, -4.7e+38], 1.0, If[LessEqual[x, 1.5e-79], -1.0, 1.0]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.72e115 or -4.99999999999999997e88 < x < -4.6999999999999999e38 or 1.5e-79 < x

   1. Initial program 37.7%

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

   2. Taylor expanded in x around inf 79.5%

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

   if -1.72e115 < x < -4.99999999999999997e88 or -4.6999999999999999e38 < x < 1.5e-79

   1. Initial program 58.2%

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

   2. Taylor expanded in x around 0 79.6%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.72 \cdot 10^{+115}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -5 \cdot 10^{+88}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq -4.7 \cdot 10^{+38}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-79}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 50.1% 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 48.4%

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

2. Taylor expanded in x around 0 51.9%

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

3. Final simplification 51.9%

   \[\leadsto -1 \]

Developer target: 51.3% 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 2023174 
(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))))