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

Percentage Accurate: 50.9% → 99.9%

Time: 8.8s

Alternatives: 7

Speedup: 3.2×

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: 50.9% 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: 99.9% accurate, 0.0× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (hypot x (fabs (* y 2.0)))))
   (* (/ (fma y 2.0 x) t_0) (/ (+ x (* y -2.0)) t_0))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 47.6%

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

   1. sub-neg 47.6%

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

   2. +-commutative 47.6%

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

   3. distribute-lft-neg-in 47.6%

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

   4. fma-define 47.6%

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

   5. distribute-rgt-neg-in 47.6%

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

   6. metadata-eval 47.6%

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

   7. pow2 47.6%

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

4. Applied egg-rr 47.6%

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

   1. pow2 47.6%

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

6. Applied egg-rr 47.6%

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

   1. fma-undefine 47.6%

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

   2. *-commutative 47.6%

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

   3. +-commutative 47.6%

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

   4. *-commutative 47.6%

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

   5. *-commutative 47.6%

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

   6. associate-*l* 47.6%

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

   7. metadata-eval 47.6%

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

   8. cancel-sign-sub-inv 47.6%

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

   9. associate-*r* 47.6%

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

   10. *-commutative 47.6%

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

   11. add-sqr-sqrt 47.6%

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

   12. difference-of-squares 47.6%

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

   13. *-commutative 47.6%

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

   14. associate-*r* 47.6%

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

   15. sqrt-prod 47.6%

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

   16. sqrt-prod 21.8%

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

   17. add-sqr-sqrt 36.6%

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

   18. metadata-eval 36.6%

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

   19. *-commutative 36.6%

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

   20. associate-*r* 36.6%

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

   21. sqrt-prod 36.6%

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

   22. sqrt-prod 21.8%

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

   23. add-sqr-sqrt 47.6%

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

   24. metadata-eval 47.6%

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

8. Applied egg-rr 47.6%

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

   1. *-un-lft-identity 47.6%

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

   2. add-sqr-sqrt 47.6%

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

   3. times-frac 47.4%

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

   4. add-sqr-sqrt 47.4%

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

   5. hypot-define 47.5%

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

   6. *-commutative 47.5%

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

   7. +-commutative 47.5%

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

   8. fma-define 47.5%

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

   9. add-sqr-sqrt 47.5%

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

   10. hypot-define 48.3%

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

10. Applied egg-rr 48.3%

    \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(x, \sqrt{y \cdot \left(y \cdot 4\right)}\right)} \cdot \frac{\mathsf{fma}\left(y, 2, x\right) \cdot \left(x - y \cdot 2\right)}{\mathsf{hypot}\left(x, \sqrt{y \cdot \left(y \cdot 4\right)}\right)}} \]
11. Step-by-step derivation

    1. associate-*l/ 48.4%

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

    2. *-lft-identity 48.4%

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

    3. associate-/l* 72.2%

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

    4. associate-*l/ 72.2%

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

    5. associate-*r* 72.2%

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

    6. metadata-eval 72.2%

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

    7. swap-sqr 72.2%

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

    8. rem-sqrt-square 72.3%

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

    9. *-commutative 72.3%

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

    10. cancel-sign-sub-inv 72.3%

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

    11. metadata-eval 72.3%

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

    12. *-commutative 72.3%

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

    13. associate-*r* 72.3%

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

    14. metadata-eval 72.3%

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

12. Simplified 99.9%

    \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{hypot}\left(x, \left|y \cdot 2\right|\right)} \cdot \frac{x + y \cdot -2}{\mathsf{hypot}\left(x, \left|y \cdot 2\right|\right)}} \]
13. Add Preprocessing

Alternative 2: 80.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-257}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 10^{+182}:\\ \;\;\;\;\left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{fma}\left(y, y \cdot 4, {x}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 + \frac{y}{x \cdot \frac{x}{y}}\right) \cdot -8 + 8\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 1e-257)
   -1.0
   (if (<= (* x x) 1e+182)
     (* (- x (* y 2.0)) (/ (fma y 2.0 x) (fma y (* y 4.0) (pow x 2.0))))
     (+ 1.0 (+ (* (+ 1.0 (/ y (* x (/ x y)))) -8.0) 8.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1e-257) {
		tmp = -1.0;
	} else if ((x * x) <= 1e+182) {
		tmp = (x - (y * 2.0)) * (fma(y, 2.0, x) / fma(y, (y * 4.0), pow(x, 2.0)));
	} else {
		tmp = 1.0 + (((1.0 + (y / (x * (x / y)))) * -8.0) + 8.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 1e-257)
		tmp = -1.0;
	elseif (Float64(x * x) <= 1e+182)
		tmp = Float64(Float64(x - Float64(y * 2.0)) * Float64(fma(y, 2.0, x) / fma(y, Float64(y * 4.0), (x ^ 2.0))));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(1.0 + Float64(y / Float64(x * Float64(x / y)))) * -8.0) + 8.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-257], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 1e+182], N[(N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision] * N[(N[(y * 2.0 + x), $MachinePrecision] / N[(y * N[(y * 4.0), $MachinePrecision] + N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(1.0 + N[(y / N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -8.0), $MachinePrecision] + 8.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 9.9999999999999998e-258

   1. Initial program 45.6%

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

   3. Taylor expanded in x around 0 86.6%

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

   if 9.9999999999999998e-258 < (*.f64 x x) < 1.0000000000000001e182

   1. Initial program 79.4%

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

      1. sub-neg 79.4%

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

      2. +-commutative 79.4%

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

      3. distribute-lft-neg-in 79.4%

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

      4. fma-define 79.4%

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

      5. distribute-rgt-neg-in 79.4%

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

      6. metadata-eval 79.4%

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

      7. pow2 79.4%

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

   4. Applied egg-rr 79.4%

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

      1. pow2 79.4%

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

   6. Applied egg-rr 79.4%

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

      1. fma-undefine 79.4%

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

      2. *-commutative 79.4%

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

      3. +-commutative 79.4%

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

      4. *-commutative 79.4%

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

      5. *-commutative 79.4%

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

      6. associate-*l* 79.4%

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

      7. metadata-eval 79.4%

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

      8. cancel-sign-sub-inv 79.4%

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

      9. associate-*r* 79.4%

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

      10. *-commutative 79.4%

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

      11. add-sqr-sqrt 79.4%

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

      12. difference-of-squares 79.4%

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

      13. *-commutative 79.4%

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

      14. associate-*r* 79.4%

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

      15. sqrt-prod 79.4%

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

      16. sqrt-prod 34.1%

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

      17. add-sqr-sqrt 63.3%

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

      18. metadata-eval 63.3%

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

      19. *-commutative 63.3%

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

      20. associate-*r* 63.3%

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

      21. sqrt-prod 63.3%

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

      22. sqrt-prod 34.1%

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

      23. add-sqr-sqrt 79.4%

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

      24. metadata-eval 79.4%

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

   8. Applied egg-rr 79.4%

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

      1. *-un-lft-identity 79.4%

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

      2. associate-/l* 79.7%

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

      3. +-commutative 79.7%

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

      4. fma-define 79.7%

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

      5. +-commutative 79.7%

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

      6. *-commutative 79.7%

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

      7. fma-define 79.7%

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

      8. pow2 79.7%

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

   10. Applied egg-rr 79.7%

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

       1. *-lft-identity 79.7%

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

       2. associate-*r/ 79.4%

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

       3. *-commutative 79.4%

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

       4. associate-*r/ 79.7%

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

   12. Simplified 79.7%

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

   if 1.0000000000000001e182 < (*.f64 x x)

   1. Initial program 11.6%

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

   3. Taylor expanded in y around 0 74.8%

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

      1. pow2 74.8%

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

      2. unpow2 74.8%

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

      3. times-frac 82.9%

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

   5. Applied egg-rr 82.9%

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

      1. pow2 82.9%

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

      2. expm1-log1p-u 82.9%

         \[\leadsto 1 + -8 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)\right)} \]

      3. expm1-define 82.9%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)} - 1\right)} \]

      4. sub-neg 82.9%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)} + \left(-1\right)\right)} \]

      5. distribute-rgt-in 82.9%

         \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)} \cdot -8 + \left(-1\right) \cdot -8\right)} \]

      6. log1p-undefine 82.9%

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

      7. rem-exp-log 82.9%

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

      8. metadata-eval 82.9%

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

      9. metadata-eval 82.9%

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

   7. Applied egg-rr 82.9%

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

      1. unpow2 82.9%

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

      2. clear-num 82.9%

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

      3. frac-times 82.9%

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

      4. *-un-lft-identity 82.9%

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

   9. Applied egg-rr 82.9%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-257}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 10^{+182}:\\ \;\;\;\;\left(x - y \cdot 2\right) \cdot \frac{\mathsf{fma}\left(y, 2, x\right)}{\mathsf{fma}\left(y, y \cdot 4, {x}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 + \frac{y}{x \cdot \frac{x}{y}}\right) \cdot -8 + 8\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-257}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 10^{+182}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot -4, y, x \cdot x\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 + \frac{y}{x \cdot \frac{x}{y}}\right) \cdot -8 + 8\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 1e-257)
   -1.0
   (if (<= (* x x) 1e+182)
     (/ (fma (* y -4.0) y (* x x)) (+ (* x x) (* y (* y 4.0))))
     (+ 1.0 (+ (* (+ 1.0 (/ y (* x (/ x y)))) -8.0) 8.0)))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1e-257) {
		tmp = -1.0;
	} else if ((x * x) <= 1e+182) {
		tmp = fma((y * -4.0), y, (x * x)) / ((x * x) + (y * (y * 4.0)));
	} else {
		tmp = 1.0 + (((1.0 + (y / (x * (x / y)))) * -8.0) + 8.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 1e-257)
		tmp = -1.0;
	elseif (Float64(x * x) <= 1e+182)
		tmp = Float64(fma(Float64(y * -4.0), y, Float64(x * x)) / Float64(Float64(x * x) + Float64(y * Float64(y * 4.0))));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(1.0 + Float64(y / Float64(x * Float64(x / y)))) * -8.0) + 8.0));
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-257], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 1e+182], N[(N[(N[(y * -4.0), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[(1.0 + N[(y / N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -8.0), $MachinePrecision] + 8.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 9.9999999999999998e-258

   1. Initial program 45.6%

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

   3. Taylor expanded in x around 0 86.6%

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

   if 9.9999999999999998e-258 < (*.f64 x x) < 1.0000000000000001e182

   1. Initial program 79.4%

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

      1. sub-neg 79.4%

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

      2. +-commutative 79.4%

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

      3. distribute-lft-neg-in 79.4%

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

      4. fma-define 79.4%

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

      5. distribute-rgt-neg-in 79.4%

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

      6. metadata-eval 79.4%

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

      7. pow2 79.4%

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

   4. Applied egg-rr 79.4%

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

      1. pow2 79.4%

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

   6. Applied egg-rr 79.4%

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

   if 1.0000000000000001e182 < (*.f64 x x)

   1. Initial program 11.6%

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

   3. Taylor expanded in y around 0 74.8%

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

      1. pow2 74.8%

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

      2. unpow2 74.8%

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

      3. times-frac 82.9%

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

   5. Applied egg-rr 82.9%

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

      1. pow2 82.9%

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

      2. expm1-log1p-u 82.9%

         \[\leadsto 1 + -8 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)\right)} \]

      3. expm1-define 82.9%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)} - 1\right)} \]

      4. sub-neg 82.9%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)} + \left(-1\right)\right)} \]

      5. distribute-rgt-in 82.9%

         \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)} \cdot -8 + \left(-1\right) \cdot -8\right)} \]

      6. log1p-undefine 82.9%

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

      7. rem-exp-log 82.9%

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

      8. metadata-eval 82.9%

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

      9. metadata-eval 82.9%

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

   7. Applied egg-rr 82.9%

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

      1. unpow2 82.9%

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

      2. clear-num 82.9%

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

      3. frac-times 82.9%

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

      4. *-un-lft-identity 82.9%

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

   9. Applied egg-rr 82.9%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-257}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 10^{+182}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot -4, y, x \cdot x\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 + \frac{y}{x \cdot \frac{x}{y}}\right) \cdot -8 + 8\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;x \cdot x \leq 10^{-257}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 10^{+182}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 + \frac{y}{x \cdot \frac{x}{y}}\right) \cdot -8 + 8\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (* x x) 1e-257)
     -1.0
     (if (<= (* x x) 1e+182)
       (/ (- (* x x) t_0) (+ (* x x) t_0))
       (+ 1.0 (+ (* (+ 1.0 (/ y (* x (/ x y)))) -8.0) 8.0))))))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((x * x) <= 1e-257) {
		tmp = -1.0;
	} else if ((x * x) <= 1e+182) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (((1.0 + (y / (x * (x / y)))) * -8.0) + 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) <= 1d-257) then
        tmp = -1.0d0
    else if ((x * x) <= 1d+182) then
        tmp = ((x * x) - t_0) / ((x * x) + t_0)
    else
        tmp = 1.0d0 + (((1.0d0 + (y / (x * (x / y)))) * (-8.0d0)) + 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) <= 1e-257) {
		tmp = -1.0;
	} else if ((x * x) <= 1e+182) {
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	} else {
		tmp = 1.0 + (((1.0 + (y / (x * (x / y)))) * -8.0) + 8.0);
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if (x * x) <= 1e-257:
		tmp = -1.0
	elif (x * x) <= 1e+182:
		tmp = ((x * x) - t_0) / ((x * x) + t_0)
	else:
		tmp = 1.0 + (((1.0 + (y / (x * (x / y)))) * -8.0) + 8.0)
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(x * x) <= 1e-257)
		tmp = -1.0;
	elseif (Float64(x * x) <= 1e+182)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0));
	else
		tmp = Float64(1.0 + Float64(Float64(Float64(1.0 + Float64(y / Float64(x * Float64(x / y)))) * -8.0) + 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) <= 1e-257)
		tmp = -1.0;
	elseif ((x * x) <= 1e+182)
		tmp = ((x * x) - t_0) / ((x * x) + t_0);
	else
		tmp = 1.0 + (((1.0 + (y / (x * (x / y)))) * -8.0) + 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], 1e-257], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 1e+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[(N[(N[(1.0 + N[(y / N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -8.0), $MachinePrecision] + 8.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 9.9999999999999998e-258

   1. Initial program 45.6%

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

   3. Taylor expanded in x around 0 86.6%

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

   if 9.9999999999999998e-258 < (*.f64 x x) < 1.0000000000000001e182

   1. Initial program 79.4%

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

   if 1.0000000000000001e182 < (*.f64 x x)

   1. Initial program 11.6%

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

   3. Taylor expanded in y around 0 74.8%

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

      1. pow2 74.8%

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

      2. unpow2 74.8%

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

      3. times-frac 82.9%

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

   5. Applied egg-rr 82.9%

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

      1. pow2 82.9%

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

      2. expm1-log1p-u 82.9%

         \[\leadsto 1 + -8 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)\right)} \]

      3. expm1-define 82.9%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)} - 1\right)} \]

      4. sub-neg 82.9%

         \[\leadsto 1 + -8 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)} + \left(-1\right)\right)} \]

      5. distribute-rgt-in 82.9%

         \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)} \cdot -8 + \left(-1\right) \cdot -8\right)} \]

      6. log1p-undefine 82.9%

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

      7. rem-exp-log 82.9%

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

      8. metadata-eval 82.9%

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

      9. metadata-eval 82.9%

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

   7. Applied egg-rr 82.9%

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

      1. unpow2 82.9%

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

      2. clear-num 82.9%

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

      3. frac-times 82.9%

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

      4. *-un-lft-identity 82.9%

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

   9. Applied egg-rr 82.9%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-257}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 10^{+182}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\left(1 + \frac{y}{x \cdot \frac{x}{y}}\right) \cdot -8 + 8\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.8 \cdot 10^{+64}:\\ \;\;\;\;1 + -8 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.8e+64) (+ 1.0 (* -8.0 (* (/ y x) (/ y x)))) -1.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.8e+64) {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 7.8d+64) then
        tmp = 1.0d0 + ((-8.0d0) * ((y / x) * (y / x)))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7.8e+64) {
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.8e+64:
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.8e+64)
		tmp = Float64(1.0 + Float64(-8.0 * Float64(Float64(y / x) * Float64(y / x))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7.8e+64)
		tmp = 1.0 + (-8.0 * ((y / x) * (y / x)));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.8e+64], N[(1.0 + N[(-8.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 7.7999999999999996e64

   1. Initial program 51.4%

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

   3. Taylor expanded in y around 0 57.7%

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

      1. pow2 57.7%

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

      2. unpow2 57.7%

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

      3. times-frac 63.5%

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

   5. Applied egg-rr 63.5%

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

   if 7.7999999999999996e64 < y

   1. Initial program 32.7%

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

   3. Taylor expanded in x around 0 92.4%

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

Alternative 6: 62.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+59}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 2e+59) 1.0 -1.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2e+59) {
		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 <= 2d+59) 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 <= 2e+59) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2e+59:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2e+59)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2e+59)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2e+59], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.99999999999999994e59

   1. Initial program 51.5%

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

   3. Taylor expanded in x around inf 62.0%

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

   if 1.99999999999999994e59 < y

   1. Initial program 33.3%

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

   3. Taylor expanded in x around 0 90.9%

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

Alternative 7: 50.6% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 47.6%

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

3. Taylor expanded in x around 0 48.6%

   \[\leadsto \color{blue}{-1} \]
4. Add Preprocessing

Developer Target 1: 51.4% 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 2024181 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ (- (* x x) (* (* y 4) y)) (+ (* x x) (* (* y 4) y))) 9743233849626781/10000000000000000) (- (/ (* x x) (+ (* x x) (* (* y y) 4))) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4)))) (- (pow (/ x (sqrt (+ (* x x) (* (* y y) 4)))) 2) (/ (* (* y y) 4) (+ (* x x) (* (* y y) 4))))))

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