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

Percentage Accurate: 50.5% → 80.8%

Time: 10.1s

Alternatives: 5

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.5% 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, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;1 + \frac{-8}{\frac{\frac{x}{y}}{\frac{y}{x}}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+149}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{t\_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 0.0)
     (+ 1.0 (/ -8.0 (/ (/ x y) (/ y x))))
     (if (<= t_0 5e+149)
       (/ (- (* x x) t_0) (+ t_0 (* x x)))
       (+ (* 0.5 (* (/ x y) (/ x y))) -1.0)))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 + (-8.0 / ((x / y) / (y / x)));
	} else if (t_0 <= 5e+149) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    if (t_0 <= 0.0d0) then
        tmp = 1.0d0 + ((-8.0d0) / ((x / y) / (y / x)))
    else if (t_0 <= 5d+149) then
        tmp = ((x * x) - t_0) / (t_0 + (x * x))
    else
        tmp = (0.5d0 * ((x / y) * (x / y))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = 1.0 + (-8.0 / ((x / y) / (y / x)));
	} else if (t_0 <= 5e+149) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if t_0 <= 0.0:
		tmp = 1.0 + (-8.0 / ((x / y) / (y / x)))
	elif t_0 <= 5e+149:
		tmp = ((x * x) - t_0) / (t_0 + (x * x))
	else:
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(1.0 + Float64(-8.0 / Float64(Float64(x / y) / Float64(y / x))));
	elseif (t_0 <= 5e+149)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(t_0 + Float64(x * x)));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = 1.0 + (-8.0 / ((x / y) / (y / x)));
	elseif (t_0 <= 5e+149)
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	else
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[(1.0 + N[(-8.0 / N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+149], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 0.0

   1. Initial program 45.7%

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

      1. sub-neg 45.7%

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

      2. add-sqr-sqrt 14.2%

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

      3. associate-*r* 14.2%

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

      4. add-sqr-sqrt 14.2%

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

      5. fma-def 14.2%

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

      6. add-sqr-sqrt 14.2%

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

      7. *-commutative 14.2%

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

      8. distribute-rgt-neg-in 14.2%

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

      9. distribute-rgt-neg-in 14.2%

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

      10. metadata-eval 14.2%

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

   4. Applied egg-rr 14.2%

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

   5. Taylor expanded in y around 0 71.4%

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

      1. associate-*r/ 71.4%

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

      2. associate-/l* 71.4%

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

      3. unpow2 71.4%

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

      4. unpow2 71.4%

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

      5. times-frac 89.7%

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

      6. unpow2 89.7%

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

   7. Simplified 89.7%

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

      1. pow2 89.7%

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

      2. clear-num 89.7%

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

      3. un-div-inv 89.7%

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

   9. Applied egg-rr 89.7%

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

   if 0.0 < (*.f64 (*.f64 y 4) y) < 4.9999999999999999e149

   1. Initial program 77.8%

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

   if 4.9999999999999999e149 < (*.f64 (*.f64 y 4) y)

   1. Initial program 23.1%

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

   3. Taylor expanded in x around 0 81.5%

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

      1. unpow2 81.5%

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

      2. pow2 81.5%

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

      3. times-frac 87.6%

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

   5. Applied egg-rr 87.6%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 0:\\ \;\;\;\;1 + \frac{-8}{\frac{\frac{x}{y}}{\frac{y}{x}}}\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+149}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-14} \lor \neg \left(y \leq 6.8 \cdot 10^{-6}\right) \land y \leq 1.15 \cdot 10^{+37}:\\ \;\;\;\;1 + \frac{-8}{\frac{\frac{x}{y}}{\frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 7e-14) (and (not (<= y 6.8e-6)) (<= y 1.15e+37)))
   (+ 1.0 (/ -8.0 (/ (/ x y) (/ y x))))
   -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 7e-14) || (!(y <= 6.8e-6) && (y <= 1.15e+37))) {
		tmp = 1.0 + (-8.0 / ((x / y) / (y / x)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= 7d-14) .or. (.not. (y <= 6.8d-6)) .and. (y <= 1.15d+37)) then
        tmp = 1.0d0 + ((-8.0d0) / ((x / y) / (y / x)))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 7e-14) || (!(y <= 6.8e-6) && (y <= 1.15e+37))) {
		tmp = 1.0 + (-8.0 / ((x / y) / (y / x)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 7e-14) or (not (y <= 6.8e-6) and (y <= 1.15e+37)):
		tmp = 1.0 + (-8.0 / ((x / y) / (y / x)))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 7e-14) || (!(y <= 6.8e-6) && (y <= 1.15e+37)))
		tmp = Float64(1.0 + Float64(-8.0 / Float64(Float64(x / y) / Float64(y / x))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 7e-14) || (~((y <= 6.8e-6)) && (y <= 1.15e+37)))
		tmp = 1.0 + (-8.0 / ((x / y) / (y / x)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 7e-14], And[N[Not[LessEqual[y, 6.8e-6]], $MachinePrecision], LessEqual[y, 1.15e+37]]], 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 < 7.0000000000000005e-14 or 6.80000000000000012e-6 < y < 1.15000000000000001e37

   1. Initial program 57.3%

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

      1. sub-neg 57.3%

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

      2. add-sqr-sqrt 24.9%

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

      3. associate-*r* 24.9%

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

      4. add-sqr-sqrt 24.8%

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

      5. fma-def 24.8%

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

      6. add-sqr-sqrt 24.9%

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

      7. *-commutative 24.9%

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

      8. distribute-rgt-neg-in 24.9%

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

      9. distribute-rgt-neg-in 24.9%

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

      10. metadata-eval 24.9%

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

   4. Applied egg-rr 24.9%

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

   5. Taylor expanded in y around 0 55.8%

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

      1. associate-*r/ 55.8%

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

      2. associate-/l* 55.8%

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

      3. unpow2 55.8%

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

      4. unpow2 55.8%

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

      5. times-frac 64.9%

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

      6. unpow2 64.9%

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

   7. Simplified 64.9%

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

      1. pow2 64.9%

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

      2. clear-num 64.9%

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

      3. un-div-inv 64.9%

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

   9. Applied egg-rr 64.9%

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

   if 7.0000000000000005e-14 < y < 6.80000000000000012e-6 or 1.15000000000000001e37 < y

   1. Initial program 26.5%

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

   3. Taylor expanded in x around 0 90.5%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{-14} \lor \neg \left(y \leq 6.8 \cdot 10^{-6}\right) \land y \leq 1.15 \cdot 10^{+37}:\\ \;\;\;\;1 + \frac{-8}{\frac{\frac{x}{y}}{\frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{-24} \lor \neg \left(y \leq 0.000205\right) \land y \leq 6.5 \cdot 10^{+35}:\\ \;\;\;\;1 + \frac{-8}{\frac{\frac{x}{y}}{\frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= y 5.4e-24) (and (not (<= y 0.000205)) (<= y 6.5e+35)))
   (+ 1.0 (/ -8.0 (/ (/ x y) (/ y x))))
   (+ (* 0.5 (* (/ x y) (/ x y))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y <= 5.4e-24) || (!(y <= 0.000205) && (y <= 6.5e+35))) {
		tmp = 1.0 + (-8.0 / ((x / y) / (y / x)));
	} else {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((y <= 5.4d-24) .or. (.not. (y <= 0.000205d0)) .and. (y <= 6.5d+35)) then
        tmp = 1.0d0 + ((-8.0d0) / ((x / y) / (y / x)))
    else
        tmp = (0.5d0 * ((x / y) * (x / y))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((y <= 5.4e-24) || (!(y <= 0.000205) && (y <= 6.5e+35))) {
		tmp = 1.0 + (-8.0 / ((x / y) / (y / x)));
	} else {
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y <= 5.4e-24) or (not (y <= 0.000205) and (y <= 6.5e+35)):
		tmp = 1.0 + (-8.0 / ((x / y) / (y / x)))
	else:
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((y <= 5.4e-24) || (!(y <= 0.000205) && (y <= 6.5e+35)))
		tmp = Float64(1.0 + Float64(-8.0 / Float64(Float64(x / y) / Float64(y / x))));
	else
		tmp = Float64(Float64(0.5 * Float64(Float64(x / y) * Float64(x / y))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y <= 5.4e-24) || (~((y <= 0.000205)) && (y <= 6.5e+35)))
		tmp = 1.0 + (-8.0 / ((x / y) / (y / x)));
	else
		tmp = (0.5 * ((x / y) * (x / y))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[y, 5.4e-24], And[N[Not[LessEqual[y, 0.000205]], $MachinePrecision], LessEqual[y, 6.5e+35]]], N[(1.0 + N[(-8.0 / N[(N[(x / y), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.40000000000000014e-24 or 2.05e-4 < y < 6.5000000000000003e35

   1. Initial program 57.0%

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

      1. sub-neg 57.0%

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

      2. add-sqr-sqrt 25.0%

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

      3. associate-*r* 25.0%

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

      4. add-sqr-sqrt 25.0%

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

      5. fma-def 25.0%

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

      6. add-sqr-sqrt 25.0%

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

      7. *-commutative 25.0%

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

      8. distribute-rgt-neg-in 25.0%

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

      9. distribute-rgt-neg-in 25.0%

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

      10. metadata-eval 25.0%

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

   4. Applied egg-rr 25.0%

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

   5. Taylor expanded in y around 0 55.5%

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

      1. associate-*r/ 55.5%

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

      2. associate-/l* 55.5%

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

      3. unpow2 55.5%

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

      4. unpow2 55.5%

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

      5. times-frac 64.8%

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

      6. unpow2 64.8%

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

   7. Simplified 64.8%

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

      1. pow2 64.8%

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

      2. clear-num 64.8%

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

      3. un-div-inv 64.8%

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

   9. Applied egg-rr 64.8%

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

   if 5.40000000000000014e-24 < y < 2.05e-4 or 6.5000000000000003e35 < y

   1. Initial program 27.7%

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

   3. Taylor expanded in x around 0 85.0%

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

      1. unpow2 85.0%

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

      2. pow2 85.0%

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

      3. times-frac 89.8%

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

   5. Applied egg-rr 89.8%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{-24} \lor \neg \left(y \leq 0.000205\right) \land y \leq 6.5 \cdot 10^{+35}:\\ \;\;\;\;1 + \frac{-8}{\frac{\frac{x}{y}}{\frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\frac{x}{y} \cdot \frac{x}{y}\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-20}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 0.2:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 10^{+36}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4e-20) 1.0 (if (<= y 0.2) -1.0 (if (<= y 1e+36) 1.0 -1.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4e-20) {
		tmp = 1.0;
	} else if (y <= 0.2) {
		tmp = -1.0;
	} else if (y <= 1e+36) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4d-20) then
        tmp = 1.0d0
    else if (y <= 0.2d0) then
        tmp = -1.0d0
    else if (y <= 1d+36) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4e-20) {
		tmp = 1.0;
	} else if (y <= 0.2) {
		tmp = -1.0;
	} else if (y <= 1e+36) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4e-20:
		tmp = 1.0
	elif y <= 0.2:
		tmp = -1.0
	elif y <= 1e+36:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4e-20)
		tmp = 1.0;
	elseif (y <= 0.2)
		tmp = -1.0;
	elseif (y <= 1e+36)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4e-20)
		tmp = 1.0;
	elseif (y <= 0.2)
		tmp = -1.0;
	elseif (y <= 1e+36)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4e-20], 1.0, If[LessEqual[y, 0.2], -1.0, If[LessEqual[y, 1e+36], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if y < 3.99999999999999978e-20 or 0.20000000000000001 < y < 1.00000000000000004e36

   1. Initial program 57.3%

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

   3. Taylor expanded in x around inf 63.6%

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

   if 3.99999999999999978e-20 < y < 0.20000000000000001 or 1.00000000000000004e36 < y

   1. Initial program 26.5%

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

   3. Taylor expanded in x around 0 90.5%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-20}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 0.2:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 10^{+36}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.3% 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 49.6%

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

3. Taylor expanded in x around 0 50.0%

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

4. Final simplification 50.0%

   \[\leadsto -1 \]
5. Add Preprocessing

Developer target: 50.9% 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 2024029 
(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))))