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

Percentage Accurate: 49.4% → 100.0%

Time: 15.9s

Alternatives: 8

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: 49.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(x, y \cdot 2\right)\\ {\left(\frac{x}{t\_0}\right)}^{2} - {\left(\frac{y \cdot 2}{t\_0}\right)}^{2} \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (hypot x (* y 2.0))))
   (- (pow (/ x t_0) 2.0) (pow (/ (* y 2.0) t_0) 2.0))))

C

double code(double x, double y) {
	double t_0 = hypot(x, (y * 2.0));
	return pow((x / t_0), 2.0) - pow(((y * 2.0) / t_0), 2.0);
}

Java

public static double code(double x, double y) {
	double t_0 = Math.hypot(x, (y * 2.0));
	return Math.pow((x / t_0), 2.0) - Math.pow(((y * 2.0) / t_0), 2.0);
}

Python

def code(x, y):
	t_0 = math.hypot(x, (y * 2.0))
	return math.pow((x / t_0), 2.0) - math.pow(((y * 2.0) / t_0), 2.0)

Julia

function code(x, y)
	t_0 = hypot(x, Float64(y * 2.0))
	return Float64((Float64(x / t_0) ^ 2.0) - (Float64(Float64(y * 2.0) / t_0) ^ 2.0))
end

MATLAB

function tmp = code(x, y)
	t_0 = hypot(x, (y * 2.0));
	tmp = ((x / t_0) ^ 2.0) - (((y * 2.0) / t_0) ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 50.3%

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

   1. div-sub 50.4%

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

   2. associate-/l* 50.8%

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

   3. fma-neg 50.8%

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

   4. +-commutative 50.8%

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

   5. *-commutative 50.8%

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

   6. associate-*l* 50.8%

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

   7. fma-define 50.8%

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

   8. pow2 50.8%

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

   9. pow2 50.8%

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

   10. *-un-lft-identity 50.8%

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

   11. *-commutative 50.8%

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

   12. associate-*l* 50.8%

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

   13. pow2 50.8%

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

   14. *-un-lft-identity 50.8%

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

   15. +-commutative 50.8%

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

4. Applied egg-rr 50.8%

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

   1. distribute-neg-frac 50.8%

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

   2. distribute-lft-neg-in 50.8%

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

   3. metadata-eval 50.8%

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

   4. *-commutative 50.8%

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

6. Simplified 50.8%

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

8. Applied egg-rr 73.0%

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

   1. associate-/l* 72.8%

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

10. Simplified 72.8%

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

    1. *-un-lft-identity 72.8%

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

    2. pow2 72.8%

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

    3. add-sqr-sqrt 72.8%

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

    4. times-frac 72.9%

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

12. Applied egg-rr 99.6%

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

    1. fma-undefine 99.6%

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

    2. frac-times 72.8%

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

    3. *-un-lft-identity 72.8%

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

    4. unpow2 72.8%

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

    5. unsub-neg 72.8%

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

    6. *-un-lft-identity 72.8%

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

    7. unpow2 72.8%

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

    8. frac-times 99.6%

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

    9. associate-*r* 99.6%

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

    10. div-inv 99.7%

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

    11. pow2 99.7%

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

14. Applied egg-rr 99.9%

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

    1. associate-*r/ 99.9%

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

    2. *-commutative 99.9%

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

16. Simplified 99.9%

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

17. Final simplification 99.9%

    \[\leadsto {\left(\frac{x}{\mathsf{hypot}\left(x, y \cdot 2\right)}\right)}^{2} - {\left(\frac{y \cdot 2}{\mathsf{hypot}\left(x, y \cdot 2\right)}\right)}^{2} \]
18. Add Preprocessing

Alternative 2: 80.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-315}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+250}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{\mathsf{fma}\left(x, x, 4 \cdot {y}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot -8\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 2e-315)
   -1.0
   (if (<= (* x x) 2e+250)
     (/ (- (* x x) (* y (* y 4.0))) (fma x x (* 4.0 (pow y 2.0))))
     (+ 1.0 (* (* (/ y x) (/ y x)) -8.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2e-315) {
		tmp = -1.0;
	} else if ((x * x) <= 2e+250) {
		tmp = ((x * x) - (y * (y * 4.0))) / fma(x, x, (4.0 * pow(y, 2.0)));
	} else {
		tmp = 1.0 + (((y / x) * (y / x)) * -8.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 2e-315)
		tmp = -1.0;
	elseif (Float64(x * x) <= 2e+250)
		tmp = Float64(Float64(Float64(x * x) - Float64(y * Float64(y * 4.0))) / fma(x, x, Float64(4.0 * (y ^ 2.0))));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(y / x) * Float64(y / x)) * -8.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-315], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 2e+250], N[(N[(N[(x * x), $MachinePrecision] - N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * x + N[(4.0 * N[Power[y, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 2.0000000019e-315

   1. Initial program 52.5%

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

   3. Taylor expanded in x around 0 93.3%

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

   if 2.0000000019e-315 < (*.f64 x x) < 1.9999999999999998e250

   1. Initial program 74.5%

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

      1. fma-define 74.5%

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

      2. *-commutative 74.5%

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

      3. associate-*l* 74.5%

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

      4. pow2 74.5%

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

   4. Applied egg-rr 74.5%

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

      1. *-commutative 74.5%

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

   6. Simplified 74.5%

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

   if 1.9999999999999998e250 < (*.f64 x x)

   1. Initial program 12.7%

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

   3. Taylor expanded in x around inf 78.6%

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

      1. associate--l+ 78.6%

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

      2. distribute-rgt-out-- 78.6%

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

      3. metadata-eval 78.6%

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

   5. Simplified 78.6%

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

      1. pow2 78.6%

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

      2. unpow2 78.6%

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

      3. times-frac 87.1%

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

   7. Applied egg-rr 87.1%

      \[\leadsto 1 + \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \cdot -8 \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-315}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+250}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{\mathsf{fma}\left(x, x, 4 \cdot {y}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot -8\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-315}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+250}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot -4, y, {x}^{2}\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot -8\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 2e-315)
   -1.0
   (if (<= (* x x) 2e+250)
     (/ (fma (* y -4.0) y (pow x 2.0)) (+ (* x x) (* y (* y 4.0))))
     (+ 1.0 (* (* (/ y x) (/ y x)) -8.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2e-315) {
		tmp = -1.0;
	} else if ((x * x) <= 2e+250) {
		tmp = fma((y * -4.0), y, pow(x, 2.0)) / ((x * x) + (y * (y * 4.0)));
	} else {
		tmp = 1.0 + (((y / x) * (y / x)) * -8.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 2e-315)
		tmp = -1.0;
	elseif (Float64(x * x) <= 2e+250)
		tmp = Float64(fma(Float64(y * -4.0), y, (x ^ 2.0)) / Float64(Float64(x * x) + Float64(y * Float64(y * 4.0))));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(y / x) * Float64(y / x)) * -8.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-315], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 2e+250], N[(N[(N[(y * -4.0), $MachinePrecision] * y + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 2.0000000019e-315

   1. Initial program 52.5%

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

   3. Taylor expanded in x around 0 93.3%

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

   if 2.0000000019e-315 < (*.f64 x x) < 1.9999999999999998e250

   1. Initial program 74.5%

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

      1. sub-neg 74.5%

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

      2. +-commutative 74.5%

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

      3. distribute-lft-neg-in 74.5%

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

      4. fma-define 74.5%

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

      5. distribute-rgt-neg-in 74.5%

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

      6. metadata-eval 74.5%

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

      7. pow2 74.5%

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

   4. Applied egg-rr 74.5%

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

   if 1.9999999999999998e250 < (*.f64 x x)

   1. Initial program 12.7%

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

   3. Taylor expanded in x around inf 78.6%

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

      1. associate--l+ 78.6%

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

      2. distribute-rgt-out-- 78.6%

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

      3. metadata-eval 78.6%

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

   5. Simplified 78.6%

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

      1. pow2 78.6%

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

      2. unpow2 78.6%

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

      3. times-frac 87.1%

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

   7. Applied egg-rr 87.1%

      \[\leadsto 1 + \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \cdot -8 \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-315}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+250}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot -4, y, {x}^{2}\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot -8\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-315}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+250}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot -8\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 2e-315)
   -1.0
   (if (<= (* x x) 2e+250)
     (/ (fma x x (* y (* y -4.0))) (+ (* x x) (* y (* y 4.0))))
     (+ 1.0 (* (* (/ y x) (/ y x)) -8.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2e-315) {
		tmp = -1.0;
	} else if ((x * x) <= 2e+250) {
		tmp = fma(x, x, (y * (y * -4.0))) / ((x * x) + (y * (y * 4.0)));
	} else {
		tmp = 1.0 + (((y / x) * (y / x)) * -8.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 2e-315)
		tmp = -1.0;
	elseif (Float64(x * x) <= 2e+250)
		tmp = Float64(fma(x, x, Float64(y * Float64(y * -4.0))) / Float64(Float64(x * x) + Float64(y * Float64(y * 4.0))));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(y / x) * Float64(y / x)) * -8.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-315], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 2e+250], N[(N[(x * x + N[(y * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 2.0000000019e-315

   1. Initial program 52.5%

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

   3. Taylor expanded in x around 0 93.3%

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

   if 2.0000000019e-315 < (*.f64 x x) < 1.9999999999999998e250

   1. Initial program 74.5%

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

      1. fma-neg 74.5%

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

      2. *-commutative 74.5%

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

      3. distribute-rgt-neg-in 74.5%

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

      4. distribute-rgt-neg-in 74.5%

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

      5. metadata-eval 74.5%

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

   4. Applied egg-rr 74.5%

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

   if 1.9999999999999998e250 < (*.f64 x x)

   1. Initial program 12.7%

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

   3. Taylor expanded in x around inf 78.6%

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

      1. associate--l+ 78.6%

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

      2. distribute-rgt-out-- 78.6%

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

      3. metadata-eval 78.6%

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

   5. Simplified 78.6%

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

      1. pow2 78.6%

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

      2. unpow2 78.6%

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

      3. times-frac 87.1%

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

   7. Applied egg-rr 87.1%

      \[\leadsto 1 + \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \cdot -8 \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-315}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+250}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot -8\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-315}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+250}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot -8\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 2e-315)
     -1.0
     (if (<= (* x x) 2e+250)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (+ 1.0 (* (* (/ y x) (/ y x)) -8.0))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 2e-315) {
		tmp = -1.0;
	} else if ((x * x) <= 2e+250) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (((y / x) * (y / x)) * -8.0);
	}
	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 ((x * x) <= 2d-315) then
        tmp = -1.0d0
    else if ((x * x) <= 2d+250) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else
        tmp = 1.0d0 + (((y / x) * (y / x)) * (-8.0d0))
    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 ((x * x) <= 2e-315) {
		tmp = -1.0;
	} else if ((x * x) <= 2e+250) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (((y / x) * (y / x)) * -8.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if (x * x) <= 2e-315:
		tmp = -1.0
	elif (x * x) <= 2e+250:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	else:
		tmp = 1.0 + (((y / x) * (y / x)) * -8.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 2e-315)
		tmp = -1.0;
	elseif (Float64(x * x) <= 2e+250)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(y / x) * Float64(y / x)) * -8.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if ((x * x) <= 2e-315)
		tmp = -1.0;
	elseif ((x * x) <= 2e+250)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	else
		tmp = 1.0 + (((y / x) * (y / x)) * -8.0);
	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], 2e-315], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 2e+250], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 2.0000000019e-315

   1. Initial program 52.5%

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

   3. Taylor expanded in x around 0 93.3%

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

   if 2.0000000019e-315 < (*.f64 x x) < 1.9999999999999998e250

   1. Initial program 74.5%

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

   if 1.9999999999999998e250 < (*.f64 x x)

   1. Initial program 12.7%

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

   3. Taylor expanded in x around inf 78.6%

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

      1. associate--l+ 78.6%

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

      2. distribute-rgt-out-- 78.6%

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

      3. metadata-eval 78.6%

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

   5. Simplified 78.6%

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

      1. pow2 78.6%

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

      2. unpow2 78.6%

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

      3. times-frac 87.1%

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

   7. Applied egg-rr 87.1%

      \[\leadsto 1 + \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \cdot -8 \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-315}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+250}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot -8\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{-32}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+35} \lor \neg \left(x \leq 1.85 \cdot 10^{+45}\right):\\ \;\;\;\;1 + \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot -8\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 7.2e-32)
   -1.0
   (if (or (<= x 1.5e+35) (not (<= x 1.85e+45)))
     (+ 1.0 (* (* (/ y x) (/ y x)) -8.0))
     -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7.2e-32) {
		tmp = -1.0;
	} else if ((x <= 1.5e+35) || !(x <= 1.85e+45)) {
		tmp = 1.0 + (((y / x) * (y / x)) * -8.0);
	} 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 <= 7.2d-32) then
        tmp = -1.0d0
    else if ((x <= 1.5d+35) .or. (.not. (x <= 1.85d+45))) then
        tmp = 1.0d0 + (((y / x) * (y / x)) * (-8.0d0))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 7.2e-32) {
		tmp = -1.0;
	} else if ((x <= 1.5e+35) || !(x <= 1.85e+45)) {
		tmp = 1.0 + (((y / x) * (y / x)) * -8.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 7.2e-32:
		tmp = -1.0
	elif (x <= 1.5e+35) or not (x <= 1.85e+45):
		tmp = 1.0 + (((y / x) * (y / x)) * -8.0)
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7.2e-32)
		tmp = -1.0;
	elseif ((x <= 1.5e+35) || !(x <= 1.85e+45))
		tmp = Float64(1.0 + Float64(Float64(Float64(y / x) * Float64(y / x)) * -8.0));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 7.2e-32)
		tmp = -1.0;
	elseif ((x <= 1.5e+35) || ~((x <= 1.85e+45)))
		tmp = 1.0 + (((y / x) * (y / x)) * -8.0);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7.2e-32], -1.0, If[Or[LessEqual[x, 1.5e+35], N[Not[LessEqual[x, 1.85e+45]], $MachinePrecision]], N[(1.0 + N[(N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if x < 7.19999999999999986e-32 or 1.49999999999999995e35 < x < 1.84999999999999989e45

   1. Initial program 57.7%

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

   3. Taylor expanded in x around 0 60.7%

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

   if 7.19999999999999986e-32 < x < 1.49999999999999995e35 or 1.84999999999999989e45 < x

   1. Initial program 30.4%

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

   3. Taylor expanded in x around inf 73.0%

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

      1. associate--l+ 73.0%

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

      2. distribute-rgt-out-- 73.0%

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

      3. metadata-eval 73.0%

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

   5. Simplified 73.0%

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

      1. pow2 73.0%

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

      2. unpow2 73.0%

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

      3. times-frac 79.6%

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

   7. Applied egg-rr 79.6%

      \[\leadsto 1 + \color{blue}{\left(\frac{y}{x} \cdot \frac{y}{x}\right)} \cdot -8 \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{-32}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{+35} \lor \neg \left(x \leq 1.85 \cdot 10^{+45}\right):\\ \;\;\;\;1 + \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot -8\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{+45}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 1.85e+45) -1.0 1.0))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.85e+45) {
		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 <= 1.85d+45) 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 <= 1.85e+45) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.85e+45:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.85e+45)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.85e+45)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.85e+45], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.84999999999999989e45

   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. Add Preprocessing

   3. Taylor expanded in x around 0 59.6%

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

   if 1.84999999999999989e45 < x

   1. Initial program 25.8%

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

   3. Taylor expanded in x around inf 79.4%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85 \cdot 10^{+45}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.9% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -1.0)

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 50.3%

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

3. Taylor expanded in x around 0 50.3%

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

4. Final simplification 50.3%

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

Developer target: 49.9% 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 2024048 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

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

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