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

Percentage Accurate: 50.4% → 99.9%

Time: 8.3s

Alternatives: 12

Speedup: 6.2×

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: 50.4% 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: 99.9% accurate, 0.1× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (hypot x (* y 2.0))))
   (/ (fma y 2.0 x) (/ t_0 (/ (+ x (* y -2.0)) t_0)))))

C

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

Julia

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

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[Sqrt[x ^ 2 + N[(y * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]}, N[(N[(y * 2.0 + x), $MachinePrecision] / N[(t$95$0 / N[(N[(x + N[(y * -2.0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 52.9%

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

   1. add-sqr-sqrt 52.9%

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

   2. difference-of-squares 52.9%

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

   3. *-commutative 52.9%

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

   4. associate-*r* 52.7%

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

   5. *-commutative 52.7%

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

   6. sqrt-prod 52.7%

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

   7. sqrt-prod 27.0%

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

   8. add-sqr-sqrt 39.1%

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

   9. metadata-eval 39.1%

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

   10. *-commutative 39.1%

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

   11. associate-*r* 39.0%

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

   12. *-commutative 39.0%

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

   13. sqrt-prod 39.0%

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

   14. sqrt-prod 27.0%

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

   15. add-sqr-sqrt 52.9%

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

   16. metadata-eval 52.9%

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

3. Applied egg-rr 52.9%

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

   1. add-sqr-sqrt 52.9%

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

   2. times-frac 54.2%

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

   3. +-commutative 54.2%

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

   4. fma-def 54.2%

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

   5. add-sqr-sqrt 54.2%

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

   6. hypot-def 54.2%

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

   7. sqrt-prod 27.7%

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

   8. *-commutative 27.7%

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

   9. sqrt-prod 27.7%

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

   10. metadata-eval 27.7%

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

   11. associate-*r* 27.7%

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

   12. add-sqr-sqrt 54.2%

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

   13. *-commutative 54.2%

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

5. Applied egg-rr 99.9%

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

   1. *-commutative 99.9%

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

   2. clear-num 99.9%

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

   3. frac-times 100.0%

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

   4. *-un-lft-identity 100.0%

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

7. Applied egg-rr 100.0%

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

   1. *-commutative 100.0%

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

   2. clear-num 99.9%

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

   3. un-div-inv 99.9%

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

   4. sub-neg 99.9%

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

   5. distribute-rgt-neg-in 99.9%

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

   6. metadata-eval 99.9%

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

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (hypot x (* y 2.0))))
   (* (/ (fma y 2.0 x) t_0) (/ (- x (* y 2.0)) t_0))))

C

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

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	t_0 = hypot(x, 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

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[Sqrt[x ^ 2 + N[(y * 2.0), $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 52.9%

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

   1. add-sqr-sqrt 52.9%

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

   2. difference-of-squares 52.9%

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

   3. *-commutative 52.9%

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

   4. associate-*r* 52.7%

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

   5. *-commutative 52.7%

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

   6. sqrt-prod 52.7%

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

   7. sqrt-prod 27.0%

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

   8. add-sqr-sqrt 39.1%

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

   9. metadata-eval 39.1%

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

   10. *-commutative 39.1%

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

   11. associate-*r* 39.0%

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

   12. *-commutative 39.0%

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

   13. sqrt-prod 39.0%

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

   14. sqrt-prod 27.0%

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

   15. add-sqr-sqrt 52.9%

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

   16. metadata-eval 52.9%

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

3. Applied egg-rr 52.9%

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

   1. add-sqr-sqrt 52.9%

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

   2. times-frac 54.2%

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

   3. +-commutative 54.2%

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

   4. fma-def 54.2%

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

   5. add-sqr-sqrt 54.2%

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

   6. hypot-def 54.2%

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

   7. sqrt-prod 27.7%

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

   8. *-commutative 27.7%

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

   9. sqrt-prod 27.7%

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

   10. metadata-eval 27.7%

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

   11. associate-*r* 27.7%

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

   12. add-sqr-sqrt 54.2%

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

   13. *-commutative 54.2%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 0.1× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (hypot x (* y 2.0))))
   (/ (fma y 2.0 x) (* t_0 (/ t_0 (- x (* y 2.0)))))))

C

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

Julia

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

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[Sqrt[x ^ 2 + N[(y * 2.0), $MachinePrecision] ^ 2], $MachinePrecision]}, N[(N[(y * 2.0 + x), $MachinePrecision] / N[(t$95$0 * N[(t$95$0 / N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 52.9%

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

   1. add-sqr-sqrt 52.9%

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

   2. difference-of-squares 52.9%

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

   3. *-commutative 52.9%

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

   4. associate-*r* 52.7%

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

   5. *-commutative 52.7%

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

   6. sqrt-prod 52.7%

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

   7. sqrt-prod 27.0%

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

   8. add-sqr-sqrt 39.1%

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

   9. metadata-eval 39.1%

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

   10. *-commutative 39.1%

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

   11. associate-*r* 39.0%

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

   12. *-commutative 39.0%

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

   13. sqrt-prod 39.0%

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

   14. sqrt-prod 27.0%

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

   15. add-sqr-sqrt 52.9%

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

   16. metadata-eval 52.9%

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

3. Applied egg-rr 52.9%

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

   1. add-sqr-sqrt 52.9%

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

   2. times-frac 54.2%

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

   3. +-commutative 54.2%

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

   4. fma-def 54.2%

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

   5. add-sqr-sqrt 54.2%

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

   6. hypot-def 54.2%

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

   7. sqrt-prod 27.7%

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

   8. *-commutative 27.7%

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

   9. sqrt-prod 27.7%

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

   10. metadata-eval 27.7%

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

   11. associate-*r* 27.7%

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

   12. add-sqr-sqrt 54.2%

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

   13. *-commutative 54.2%

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

5. Applied egg-rr 99.9%

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

   1. *-commutative 99.9%

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

   2. clear-num 99.9%

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

   3. frac-times 100.0%

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

   4. *-un-lft-identity 100.0%

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

7. Applied egg-rr 100.0%

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

8. Final simplification 100.0%

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

Alternative 4: 80.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 10^{-218}:\\ \;\;\;\;1 + \left(e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)} + -1\right) \cdot -8\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(x - y \cdot 2\right) \cdot \left(x + y \cdot 2\right)}{t_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 1e-218)
     (+ 1.0 (* (+ (exp (log1p (pow (/ y x) 2.0))) -1.0) -8.0))
     (if (<= t_0 5e+110)
       (/ (* (- x (* y 2.0)) (+ x (* y 2.0))) (+ t_0 (* x x)))
       (fma 0.5 (* (/ x y) (/ x y)) -1.0)))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 1e-218) {
		tmp = 1.0 + ((exp(log1p(pow((y / x), 2.0))) + -1.0) * -8.0);
	} else if (t_0 <= 5e+110) {
		tmp = ((x - (y * 2.0)) * (x + (y * 2.0))) / (t_0 + (x * x));
	} else {
		tmp = fma(0.5, ((x / y) * (x / y)), -1.0);
	}
	return tmp;
}

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 1e-218)
		tmp = Float64(1.0 + Float64(Float64(exp(log1p((Float64(y / x) ^ 2.0))) + -1.0) * -8.0));
	elseif (t_0 <= 5e+110)
		tmp = Float64(Float64(Float64(x - Float64(y * 2.0)) * Float64(x + Float64(y * 2.0))) / Float64(t_0 + Float64(x * x)));
	else
		tmp = fma(0.5, Float64(Float64(x / y) * Float64(x / y)), -1.0);
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-218], N[(1.0 + N[(N[(N[Exp[N[Log[1 + N[Power[N[(y / x), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+110], N[(N[(N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision] * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 1e-218

   1. Initial program 55.3%

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

   2. Taylor expanded in x around inf 76.8%

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

      1. associate--l+ 76.8%

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

      2. distribute-rgt-out-- 76.8%

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

      3. metadata-eval 76.8%

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

      4. *-commutative 76.8%

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

      5. +-commutative 76.8%

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

      6. *-commutative 76.8%

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

      7. fma-def 76.8%

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

      8. unpow2 76.8%

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

      9. unpow2 76.8%

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

      10. times-frac 84.3%

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

   4. Simplified 84.3%

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

      1. fma-udef 84.3%

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

      2. pow2 84.3%

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

   6. Applied egg-rr 84.3%

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

      1. expm1-log1p-u 84.3%

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

      2. expm1-udef 84.3%

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

   8. Applied egg-rr 84.3%

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

   if 1e-218 < (*.f64 (*.f64 y 4) y) < 4.99999999999999978e110

   1. Initial program 85.7%

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

      1. add-sqr-sqrt 85.7%

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

      2. difference-of-squares 85.7%

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

      3. *-commutative 85.7%

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

      4. associate-*r* 85.7%

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

      5. *-commutative 85.7%

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

      6. sqrt-prod 85.7%

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

      7. sqrt-prod 49.0%

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

      8. add-sqr-sqrt 63.8%

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

      9. metadata-eval 63.8%

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

      10. *-commutative 63.8%

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

      11. associate-*r* 63.8%

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

      12. *-commutative 63.8%

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

      13. sqrt-prod 63.8%

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

      14. sqrt-prod 49.0%

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

      15. add-sqr-sqrt 85.7%

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

      16. metadata-eval 85.7%

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

   3. Applied egg-rr 85.7%

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

   if 4.99999999999999978e110 < (*.f64 (*.f64 y 4) y)

   1. Initial program 30.0%

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

   2. Taylor expanded in x around 0 74.2%

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

      1. fma-neg 74.2%

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

      2. unpow2 74.2%

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

      3. unpow2 74.2%

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

      4. metadata-eval 74.2%

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

   4. Simplified 74.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x \cdot x}{y \cdot y}, -1\right)} \]
   5. Step-by-step derivation

      1. times-frac 84.5%

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

   6. Applied egg-rr 84.5%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 10^{-218}:\\ \;\;\;\;1 + \left(e^{\mathsf{log1p}\left({\left(\frac{y}{x}\right)}^{2}\right)} + -1\right) \cdot -8\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(x - y \cdot 2\right) \cdot \left(x + y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \]

Alternative 5: 80.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 10^{-218}:\\ \;\;\;\;1 + {\left(\frac{y}{x}\right)}^{2} \cdot -8\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(x - y \cdot 2\right) \cdot \left(x + y \cdot 2\right)}{t_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 1e-218)
     (+ 1.0 (* (pow (/ y x) 2.0) -8.0))
     (if (<= t_0 5e+110)
       (/ (* (- x (* y 2.0)) (+ x (* y 2.0))) (+ t_0 (* x x)))
       (fma 0.5 (* (/ x y) (/ x y)) -1.0)))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 1e-218) {
		tmp = 1.0 + (pow((y / x), 2.0) * -8.0);
	} else if (t_0 <= 5e+110) {
		tmp = ((x - (y * 2.0)) * (x + (y * 2.0))) / (t_0 + (x * x));
	} else {
		tmp = fma(0.5, ((x / y) * (x / y)), -1.0);
	}
	return tmp;
}

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 1e-218)
		tmp = Float64(1.0 + Float64((Float64(y / x) ^ 2.0) * -8.0));
	elseif (t_0 <= 5e+110)
		tmp = Float64(Float64(Float64(x - Float64(y * 2.0)) * Float64(x + Float64(y * 2.0))) / Float64(t_0 + Float64(x * x)));
	else
		tmp = fma(0.5, Float64(Float64(x / y) * Float64(x / y)), -1.0);
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-218], N[(1.0 + N[(N[Power[N[(y / x), $MachinePrecision], 2.0], $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+110], N[(N[(N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision] * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 1e-218

   1. Initial program 55.3%

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

   2. Taylor expanded in x around inf 76.8%

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

      1. associate--l+ 76.8%

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

      2. distribute-rgt-out-- 76.8%

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

      3. metadata-eval 76.8%

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

      4. *-commutative 76.8%

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

      5. +-commutative 76.8%

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

      6. *-commutative 76.8%

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

      7. fma-def 76.8%

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

      8. unpow2 76.8%

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

      9. unpow2 76.8%

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

      10. times-frac 84.3%

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

   4. Simplified 84.3%

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

      1. fma-udef 84.3%

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

      2. pow2 84.3%

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

   6. Applied egg-rr 84.3%

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

   if 1e-218 < (*.f64 (*.f64 y 4) y) < 4.99999999999999978e110

   1. Initial program 85.7%

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

      1. add-sqr-sqrt 85.7%

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

      2. difference-of-squares 85.7%

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

      3. *-commutative 85.7%

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

      4. associate-*r* 85.7%

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

      5. *-commutative 85.7%

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

      6. sqrt-prod 85.7%

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

      7. sqrt-prod 49.0%

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

      8. add-sqr-sqrt 63.8%

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

      9. metadata-eval 63.8%

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

      10. *-commutative 63.8%

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

      11. associate-*r* 63.8%

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

      12. *-commutative 63.8%

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

      13. sqrt-prod 63.8%

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

      14. sqrt-prod 49.0%

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

      15. add-sqr-sqrt 85.7%

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

      16. metadata-eval 85.7%

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

   3. Applied egg-rr 85.7%

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

   if 4.99999999999999978e110 < (*.f64 (*.f64 y 4) y)

   1. Initial program 30.0%

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

   2. Taylor expanded in x around 0 74.2%

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

      1. fma-neg 74.2%

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

      2. unpow2 74.2%

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

      3. unpow2 74.2%

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

      4. metadata-eval 74.2%

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

   4. Simplified 74.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{x \cdot x}{y \cdot y}, -1\right)} \]
   5. Step-by-step derivation

      1. times-frac 84.5%

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

   6. Applied egg-rr 84.5%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 10^{-218}:\\ \;\;\;\;1 + {\left(\frac{y}{x}\right)}^{2} \cdot -8\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+110}:\\ \;\;\;\;\frac{\left(x - y \cdot 2\right) \cdot \left(x + y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{x}{y} \cdot \frac{x}{y}, -1\right)\\ \end{array} \]

Alternative 6: 80.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;t_0 \leq 10^{-218}:\\ \;\;\;\;1 + {\left(\frac{y}{x}\right)}^{2} \cdot -8\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+208}:\\ \;\;\;\;\frac{\left(x - y \cdot 2\right) \cdot \left(x + y \cdot 2\right)}{t_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.25 \cdot \frac{x}{y \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= t_0 1e-218)
     (+ 1.0 (* (pow (/ y x) 2.0) -8.0))
     (if (<= t_0 5e+208)
       (/ (* (- x (* y 2.0)) (+ x (* y 2.0))) (+ t_0 (* x x)))
       (+ -1.0 (* 0.25 (/ x (* y (/ y x)))))))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 1e-218) {
		tmp = 1.0 + (pow((y / x), 2.0) * -8.0);
	} else if (t_0 <= 5e+208) {
		tmp = ((x - (y * 2.0)) * (x + (y * 2.0))) / (t_0 + (x * x));
	} else {
		tmp = -1.0 + (0.25 * (x / (y * (y / x))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    if (t_0 <= 1d-218) then
        tmp = 1.0d0 + (((y / x) ** 2.0d0) * (-8.0d0))
    else if (t_0 <= 5d+208) then
        tmp = ((x - (y * 2.0d0)) * (x + (y * 2.0d0))) / (t_0 + (x * x))
    else
        tmp = (-1.0d0) + (0.25d0 * (x / (y * (y / x))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (t_0 <= 1e-218) {
		tmp = 1.0 + (Math.pow((y / x), 2.0) * -8.0);
	} else if (t_0 <= 5e+208) {
		tmp = ((x - (y * 2.0)) * (x + (y * 2.0))) / (t_0 + (x * x));
	} else {
		tmp = -1.0 + (0.25 * (x / (y * (y / x))));
	}
	return tmp;
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if t_0 <= 1e-218:
		tmp = 1.0 + (math.pow((y / x), 2.0) * -8.0)
	elif t_0 <= 5e+208:
		tmp = ((x - (y * 2.0)) * (x + (y * 2.0))) / (t_0 + (x * x))
	else:
		tmp = -1.0 + (0.25 * (x / (y * (y / x))))
	return tmp

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (t_0 <= 1e-218)
		tmp = Float64(1.0 + Float64((Float64(y / x) ^ 2.0) * -8.0));
	elseif (t_0 <= 5e+208)
		tmp = Float64(Float64(Float64(x - Float64(y * 2.0)) * Float64(x + Float64(y * 2.0))) / Float64(t_0 + Float64(x * x)));
	else
		tmp = Float64(-1.0 + Float64(0.25 * Float64(x / Float64(y * Float64(y / x)))));
	end
	return tmp
end

MATLAB

x = abs(x)
y = abs(y)
function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if (t_0 <= 1e-218)
		tmp = 1.0 + (((y / x) ^ 2.0) * -8.0);
	elseif (t_0 <= 5e+208)
		tmp = ((x - (y * 2.0)) * (x + (y * 2.0))) / (t_0 + (x * x));
	else
		tmp = -1.0 + (0.25 * (x / (y * (y / x))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1e-218], N[(1.0 + N[(N[Power[N[(y / x), $MachinePrecision], 2.0], $MachinePrecision] * -8.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+208], N[(N[(N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision] * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(0.25 * N[(x / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y 4) y) < 1e-218

   1. Initial program 55.3%

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

   2. Taylor expanded in x around inf 76.8%

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

      1. associate--l+ 76.8%

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

      2. distribute-rgt-out-- 76.8%

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

      3. metadata-eval 76.8%

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

      4. *-commutative 76.8%

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

      5. +-commutative 76.8%

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

      6. *-commutative 76.8%

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

      7. fma-def 76.8%

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

      8. unpow2 76.8%

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

      9. unpow2 76.8%

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

      10. times-frac 84.3%

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

   4. Simplified 84.3%

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

      1. fma-udef 84.3%

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

      2. pow2 84.3%

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

   6. Applied egg-rr 84.3%

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

   if 1e-218 < (*.f64 (*.f64 y 4) y) < 5.0000000000000004e208

   1. Initial program 82.0%

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

      1. add-sqr-sqrt 82.0%

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

      2. difference-of-squares 82.0%

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

      3. *-commutative 82.0%

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

      4. associate-*r* 82.0%

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

      5. *-commutative 82.0%

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

      6. sqrt-prod 82.0%

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

      7. sqrt-prod 44.7%

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

      8. add-sqr-sqrt 56.8%

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

      9. metadata-eval 56.8%

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

      10. *-commutative 56.8%

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

      11. associate-*r* 56.8%

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

      12. *-commutative 56.8%

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

      13. sqrt-prod 56.8%

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

      14. sqrt-prod 44.7%

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

      15. add-sqr-sqrt 82.0%

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

      16. metadata-eval 82.0%

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

   3. Applied egg-rr 82.0%

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

   if 5.0000000000000004e208 < (*.f64 (*.f64 y 4) y)

   1. Initial program 23.5%

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

   2. Taylor expanded in x around 0 23.6%

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

      1. unpow2 23.6%

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

   4. Simplified 23.6%

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

      1. div-sub 23.6%

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

      2. *-un-lft-identity 23.6%

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

      3. times-frac 23.6%

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

      4. metadata-eval 23.6%

         \[\leadsto \color{blue}{0.25} \cdot \frac{x \cdot x}{y \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{4 \cdot \left(y \cdot y\right)} \]

      5. times-frac 23.6%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} - \frac{\left(y \cdot 4\right) \cdot y}{4 \cdot \left(y \cdot y\right)} \]

      6. pow2 23.6%

         \[\leadsto 0.25 \cdot \color{blue}{{\left(\frac{x}{y}\right)}^{2}} - \frac{\left(y \cdot 4\right) \cdot y}{4 \cdot \left(y \cdot y\right)} \]

      7. associate-*r* 23.6%

         \[\leadsto 0.25 \cdot {\left(\frac{x}{y}\right)}^{2} - \frac{\left(y \cdot 4\right) \cdot y}{\color{blue}{\left(4 \cdot y\right) \cdot y}} \]

      8. *-commutative 23.6%

         \[\leadsto 0.25 \cdot {\left(\frac{x}{y}\right)}^{2} - \frac{\left(y \cdot 4\right) \cdot y}{\color{blue}{\left(y \cdot 4\right)} \cdot y} \]

      9. *-inverses 87.0%

         \[\leadsto 0.25 \cdot {\left(\frac{x}{y}\right)}^{2} - \color{blue}{1} \]

   6. Applied egg-rr 87.0%

      \[\leadsto \color{blue}{0.25 \cdot {\left(\frac{x}{y}\right)}^{2} - 1} \]
   7. Step-by-step derivation

      1. unpow2 87.0%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} - 1 \]

      2. clear-num 87.0%

         \[\leadsto 0.25 \cdot \left(\color{blue}{\frac{1}{\frac{y}{x}}} \cdot \frac{x}{y}\right) - 1 \]

      3. frac-times 87.0%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{1 \cdot x}{\frac{y}{x} \cdot y}} - 1 \]

      4. *-un-lft-identity 87.0%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{x}}{\frac{y}{x} \cdot y} - 1 \]

   8. Applied egg-rr 87.0%

      \[\leadsto 0.25 \cdot \color{blue}{\frac{x}{\frac{y}{x} \cdot y}} - 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 10^{-218}:\\ \;\;\;\;1 + {\left(\frac{y}{x}\right)}^{2} \cdot -8\\ \mathbf{elif}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+208}:\\ \;\;\;\;\frac{\left(x - y \cdot 2\right) \cdot \left(x + y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.25 \cdot \frac{x}{y \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 7: 80.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-110}:\\ \;\;\;\;1 + -8 \cdot \frac{y}{x \cdot \frac{x}{y}}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+104}:\\ \;\;\;\;\frac{\left(x - y \cdot 2\right) \cdot \left(x + y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.25 \cdot \frac{x}{y \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 7.2e-110)
   (+ 1.0 (* -8.0 (/ y (* x (/ x y)))))
   (if (<= y 1.25e+104)
     (/ (* (- x (* y 2.0)) (+ x (* y 2.0))) (+ (* y (* y 4.0)) (* x x)))
     (+ -1.0 (* 0.25 (/ x (* y (/ y x))))))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 7.2e-110) {
		tmp = 1.0 + (-8.0 * (y / (x * (x / y))));
	} else if (y <= 1.25e+104) {
		tmp = ((x - (y * 2.0)) * (x + (y * 2.0))) / ((y * (y * 4.0)) + (x * x));
	} else {
		tmp = -1.0 + (0.25 * (x / (y * (y / x))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 7.2d-110) then
        tmp = 1.0d0 + ((-8.0d0) * (y / (x * (x / y))))
    else if (y <= 1.25d+104) then
        tmp = ((x - (y * 2.0d0)) * (x + (y * 2.0d0))) / ((y * (y * 4.0d0)) + (x * x))
    else
        tmp = (-1.0d0) + (0.25d0 * (x / (y * (y / x))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 7.2e-110) {
		tmp = 1.0 + (-8.0 * (y / (x * (x / y))));
	} else if (y <= 1.25e+104) {
		tmp = ((x - (y * 2.0)) * (x + (y * 2.0))) / ((y * (y * 4.0)) + (x * x));
	} else {
		tmp = -1.0 + (0.25 * (x / (y * (y / x))));
	}
	return tmp;
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 7.2e-110:
		tmp = 1.0 + (-8.0 * (y / (x * (x / y))))
	elif y <= 1.25e+104:
		tmp = ((x - (y * 2.0)) * (x + (y * 2.0))) / ((y * (y * 4.0)) + (x * x))
	else:
		tmp = -1.0 + (0.25 * (x / (y * (y / x))))
	return tmp

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 7.2e-110)
		tmp = Float64(1.0 + Float64(-8.0 * Float64(y / Float64(x * Float64(x / y)))));
	elseif (y <= 1.25e+104)
		tmp = Float64(Float64(Float64(x - Float64(y * 2.0)) * Float64(x + Float64(y * 2.0))) / Float64(Float64(y * Float64(y * 4.0)) + Float64(x * x)));
	else
		tmp = Float64(-1.0 + Float64(0.25 * Float64(x / Float64(y * Float64(y / x)))));
	end
	return tmp
end

MATLAB

x = abs(x)
y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7.2e-110)
		tmp = 1.0 + (-8.0 * (y / (x * (x / y))));
	elseif (y <= 1.25e+104)
		tmp = ((x - (y * 2.0)) * (x + (y * 2.0))) / ((y * (y * 4.0)) + (x * x));
	else
		tmp = -1.0 + (0.25 * (x / (y * (y / x))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 7.2e-110], N[(1.0 + N[(-8.0 * N[(y / N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e+104], N[(N[(N[(x - N[(y * 2.0), $MachinePrecision]), $MachinePrecision] * N[(x + N[(y * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(0.25 * N[(x / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 7.1999999999999999e-110

   1. Initial program 53.1%

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

   2. Taylor expanded in x around inf 50.9%

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

      1. associate--l+ 50.9%

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

      2. distribute-rgt-out-- 50.9%

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

      3. metadata-eval 50.9%

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

      4. *-commutative 50.9%

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

      5. +-commutative 50.9%

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

      6. *-commutative 50.9%

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

      7. fma-def 50.9%

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

      8. unpow2 50.9%

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

      9. unpow2 50.9%

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

      10. times-frac 58.4%

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

   4. Simplified 58.4%

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

      1. fma-udef 58.4%

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

      2. pow2 58.4%

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

   6. Applied egg-rr 58.4%

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

      1. unpow2 58.4%

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

      2. clear-num 58.4%

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

      3. frac-times 58.4%

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

      4. *-un-lft-identity 58.4%

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

   8. Applied egg-rr 58.4%

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

   if 7.1999999999999999e-110 < y < 1.2499999999999999e104

   1. Initial program 81.3%

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

      1. add-sqr-sqrt 81.3%

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

      2. difference-of-squares 81.3%

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

      3. *-commutative 81.3%

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

      4. associate-*r* 81.3%

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

      5. *-commutative 81.3%

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

      6. sqrt-prod 81.3%

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

      7. sqrt-prod 81.1%

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

      8. add-sqr-sqrt 81.3%

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

      9. metadata-eval 81.3%

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

      10. *-commutative 81.3%

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

      11. associate-*r* 81.3%

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

      12. *-commutative 81.3%

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

      13. sqrt-prod 81.3%

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

      14. sqrt-prod 81.1%

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

      15. add-sqr-sqrt 81.3%

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

      16. metadata-eval 81.3%

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

   3. Applied egg-rr 81.3%

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

   if 1.2499999999999999e104 < y

   1. Initial program 18.9%

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

   2. Taylor expanded in x around 0 18.9%

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

      1. unpow2 18.9%

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

   4. Simplified 18.9%

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

      1. div-sub 18.9%

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

      2. *-un-lft-identity 18.9%

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

      3. times-frac 18.9%

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

      4. metadata-eval 18.9%

         \[\leadsto \color{blue}{0.25} \cdot \frac{x \cdot x}{y \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{4 \cdot \left(y \cdot y\right)} \]

      5. times-frac 18.9%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} - \frac{\left(y \cdot 4\right) \cdot y}{4 \cdot \left(y \cdot y\right)} \]

      6. pow2 18.9%

         \[\leadsto 0.25 \cdot \color{blue}{{\left(\frac{x}{y}\right)}^{2}} - \frac{\left(y \cdot 4\right) \cdot y}{4 \cdot \left(y \cdot y\right)} \]

      7. associate-*r* 18.9%

         \[\leadsto 0.25 \cdot {\left(\frac{x}{y}\right)}^{2} - \frac{\left(y \cdot 4\right) \cdot y}{\color{blue}{\left(4 \cdot y\right) \cdot y}} \]

      8. *-commutative 18.9%

         \[\leadsto 0.25 \cdot {\left(\frac{x}{y}\right)}^{2} - \frac{\left(y \cdot 4\right) \cdot y}{\color{blue}{\left(y \cdot 4\right)} \cdot y} \]

      9. *-inverses 87.7%

         \[\leadsto 0.25 \cdot {\left(\frac{x}{y}\right)}^{2} - \color{blue}{1} \]

   6. Applied egg-rr 87.7%

      \[\leadsto \color{blue}{0.25 \cdot {\left(\frac{x}{y}\right)}^{2} - 1} \]
   7. Step-by-step derivation

      1. unpow2 87.7%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} - 1 \]

      2. clear-num 87.7%

         \[\leadsto 0.25 \cdot \left(\color{blue}{\frac{1}{\frac{y}{x}}} \cdot \frac{x}{y}\right) - 1 \]

      3. frac-times 87.7%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{1 \cdot x}{\frac{y}{x} \cdot y}} - 1 \]

      4. *-un-lft-identity 87.7%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{x}}{\frac{y}{x} \cdot y} - 1 \]

   8. Applied egg-rr 87.7%

      \[\leadsto 0.25 \cdot \color{blue}{\frac{x}{\frac{y}{x} \cdot y}} - 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{-110}:\\ \;\;\;\;1 + -8 \cdot \frac{y}{x \cdot \frac{x}{y}}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+104}:\\ \;\;\;\;\frac{\left(x - y \cdot 2\right) \cdot \left(x + y \cdot 2\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.25 \cdot \frac{x}{y \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 8: 80.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;y \leq 4.3 \cdot 10^{-110}:\\ \;\;\;\;1 + -8 \cdot \frac{y}{x \cdot \frac{x}{y}}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+104}:\\ \;\;\;\;\frac{x \cdot x - t_0}{t_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.25 \cdot \frac{x}{y \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= y 4.3e-110)
     (+ 1.0 (* -8.0 (/ y (* x (/ x y)))))
     (if (<= y 1.35e+104)
       (/ (- (* x x) t_0) (+ t_0 (* x x)))
       (+ -1.0 (* 0.25 (/ x (* y (/ y x)))))))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (y <= 4.3e-110) {
		tmp = 1.0 + (-8.0 * (y / (x * (x / y))));
	} else if (y <= 1.35e+104) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = -1.0 + (0.25 * (x / (y * (y / x))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    if (y <= 4.3d-110) then
        tmp = 1.0d0 + ((-8.0d0) * (y / (x * (x / y))))
    else if (y <= 1.35d+104) then
        tmp = ((x * x) - t_0) / (t_0 + (x * x))
    else
        tmp = (-1.0d0) + (0.25d0 * (x / (y * (y / x))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if (y <= 4.3e-110) {
		tmp = 1.0 + (-8.0 * (y / (x * (x / y))));
	} else if (y <= 1.35e+104) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = -1.0 + (0.25 * (x / (y * (y / x))));
	}
	return tmp;
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if y <= 4.3e-110:
		tmp = 1.0 + (-8.0 * (y / (x * (x / y))))
	elif y <= 1.35e+104:
		tmp = ((x * x) - t_0) / (t_0 + (x * x))
	else:
		tmp = -1.0 + (0.25 * (x / (y * (y / x))))
	return tmp

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (y <= 4.3e-110)
		tmp = Float64(1.0 + Float64(-8.0 * Float64(y / Float64(x * Float64(x / y)))));
	elseif (y <= 1.35e+104)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(t_0 + Float64(x * x)));
	else
		tmp = Float64(-1.0 + Float64(0.25 * Float64(x / Float64(y * Float64(y / x)))));
	end
	return tmp
end

MATLAB

x = abs(x)
y = abs(y)
function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if (y <= 4.3e-110)
		tmp = 1.0 + (-8.0 * (y / (x * (x / y))));
	elseif (y <= 1.35e+104)
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	else
		tmp = -1.0 + (0.25 * (x / (y * (y / x))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 4.3e-110], N[(1.0 + N[(-8.0 * N[(y / N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+104], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(0.25 * N[(x / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.30000000000000025e-110

   1. Initial program 53.1%

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

   2. Taylor expanded in x around inf 50.9%

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

      1. associate--l+ 50.9%

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

      2. distribute-rgt-out-- 50.9%

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

      3. metadata-eval 50.9%

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

      4. *-commutative 50.9%

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

      5. +-commutative 50.9%

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

      6. *-commutative 50.9%

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

      7. fma-def 50.9%

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

      8. unpow2 50.9%

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

      9. unpow2 50.9%

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

      10. times-frac 58.4%

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

   4. Simplified 58.4%

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

      1. fma-udef 58.4%

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

      2. pow2 58.4%

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

   6. Applied egg-rr 58.4%

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

      1. unpow2 58.4%

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

      2. clear-num 58.4%

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

      3. frac-times 58.4%

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

      4. *-un-lft-identity 58.4%

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

   8. Applied egg-rr 58.4%

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

   if 4.30000000000000025e-110 < y < 1.34999999999999992e104

   1. Initial program 81.3%

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

   if 1.34999999999999992e104 < y

   1. Initial program 18.9%

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

   2. Taylor expanded in x around 0 18.9%

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

      1. unpow2 18.9%

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

   4. Simplified 18.9%

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

      1. div-sub 18.9%

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

      2. *-un-lft-identity 18.9%

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

      3. times-frac 18.9%

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

      4. metadata-eval 18.9%

         \[\leadsto \color{blue}{0.25} \cdot \frac{x \cdot x}{y \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{4 \cdot \left(y \cdot y\right)} \]

      5. times-frac 18.9%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} - \frac{\left(y \cdot 4\right) \cdot y}{4 \cdot \left(y \cdot y\right)} \]

      6. pow2 18.9%

         \[\leadsto 0.25 \cdot \color{blue}{{\left(\frac{x}{y}\right)}^{2}} - \frac{\left(y \cdot 4\right) \cdot y}{4 \cdot \left(y \cdot y\right)} \]

      7. associate-*r* 18.9%

         \[\leadsto 0.25 \cdot {\left(\frac{x}{y}\right)}^{2} - \frac{\left(y \cdot 4\right) \cdot y}{\color{blue}{\left(4 \cdot y\right) \cdot y}} \]

      8. *-commutative 18.9%

         \[\leadsto 0.25 \cdot {\left(\frac{x}{y}\right)}^{2} - \frac{\left(y \cdot 4\right) \cdot y}{\color{blue}{\left(y \cdot 4\right)} \cdot y} \]

      9. *-inverses 87.7%

         \[\leadsto 0.25 \cdot {\left(\frac{x}{y}\right)}^{2} - \color{blue}{1} \]

   6. Applied egg-rr 87.7%

      \[\leadsto \color{blue}{0.25 \cdot {\left(\frac{x}{y}\right)}^{2} - 1} \]
   7. Step-by-step derivation

      1. unpow2 87.7%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} - 1 \]

      2. clear-num 87.7%

         \[\leadsto 0.25 \cdot \left(\color{blue}{\frac{1}{\frac{y}{x}}} \cdot \frac{x}{y}\right) - 1 \]

      3. frac-times 87.7%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{1 \cdot x}{\frac{y}{x} \cdot y}} - 1 \]

      4. *-un-lft-identity 87.7%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{x}}{\frac{y}{x} \cdot y} - 1 \]

   8. Applied egg-rr 87.7%

      \[\leadsto 0.25 \cdot \color{blue}{\frac{x}{\frac{y}{x} \cdot y}} - 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-110}:\\ \;\;\;\;1 + -8 \cdot \frac{y}{x \cdot \frac{x}{y}}\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+104}:\\ \;\;\;\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{y \cdot \left(y \cdot 4\right) + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.25 \cdot \frac{x}{y \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 9: 74.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;1 + -8 \cdot \frac{y}{x \cdot \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 6.2e-6) (+ 1.0 (* -8.0 (/ y (* x (/ x y))))) -1.0))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 6.2e-6) {
		tmp = 1.0 + (-8.0 * (y / (x * (x / y))));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 6.2d-6) then
        tmp = 1.0d0 + ((-8.0d0) * (y / (x * (x / y))))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 6.2e-6) {
		tmp = 1.0 + (-8.0 * (y / (x * (x / y))));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 6.2e-6:
		tmp = 1.0 + (-8.0 * (y / (x * (x / y))))
	else:
		tmp = -1.0
	return tmp

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 6.2e-6)
		tmp = Float64(1.0 + Float64(-8.0 * Float64(y / Float64(x * Float64(x / y)))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

x = abs(x)
y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6.2e-6)
		tmp = 1.0 + (-8.0 * (y / (x * (x / y))));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 6.2e-6], N[(1.0 + N[(-8.0 * N[(y / N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 6.1999999999999999e-6

   1. Initial program 57.2%

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

   2. Taylor expanded in x around inf 52.5%

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

      1. associate--l+ 52.5%

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

      2. distribute-rgt-out-- 52.5%

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

      3. metadata-eval 52.5%

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

      4. *-commutative 52.5%

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

      5. +-commutative 52.5%

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

      6. *-commutative 52.5%

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

      7. fma-def 52.5%

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

      8. unpow2 52.5%

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

      9. unpow2 52.5%

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

      10. times-frac 59.2%

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

   4. Simplified 59.2%

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

      1. fma-udef 59.2%

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

      2. pow2 59.2%

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

   6. Applied egg-rr 59.2%

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

      1. unpow2 59.2%

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

      2. clear-num 59.2%

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

      3. frac-times 59.1%

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

      4. *-un-lft-identity 59.1%

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

   8. Applied egg-rr 59.1%

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

   if 6.1999999999999999e-6 < y

   1. Initial program 37.4%

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

   2. Taylor expanded in x around 0 77.4%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{-6}:\\ \;\;\;\;1 + -8 \cdot \frac{y}{x \cdot \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 10: 74.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{-7}:\\ \;\;\;\;1 + -8 \cdot \frac{y}{x \cdot \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.25 \cdot \frac{x}{y \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 4.7e-7)
   (+ 1.0 (* -8.0 (/ y (* x (/ x y)))))
   (+ -1.0 (* 0.25 (/ x (* y (/ y x)))))))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.7e-7) {
		tmp = 1.0 + (-8.0 * (y / (x * (x / y))));
	} else {
		tmp = -1.0 + (0.25 * (x / (y * (y / x))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 4.7d-7) then
        tmp = 1.0d0 + ((-8.0d0) * (y / (x * (x / y))))
    else
        tmp = (-1.0d0) + (0.25d0 * (x / (y * (y / x))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 4.7e-7) {
		tmp = 1.0 + (-8.0 * (y / (x * (x / y))));
	} else {
		tmp = -1.0 + (0.25 * (x / (y * (y / x))));
	}
	return tmp;
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 4.7e-7:
		tmp = 1.0 + (-8.0 * (y / (x * (x / y))))
	else:
		tmp = -1.0 + (0.25 * (x / (y * (y / x))))
	return tmp

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 4.7e-7)
		tmp = Float64(1.0 + Float64(-8.0 * Float64(y / Float64(x * Float64(x / y)))));
	else
		tmp = Float64(-1.0 + Float64(0.25 * Float64(x / Float64(y * Float64(y / x)))));
	end
	return tmp
end

MATLAB

x = abs(x)
y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.7e-7)
		tmp = 1.0 + (-8.0 * (y / (x * (x / y))));
	else
		tmp = -1.0 + (0.25 * (x / (y * (y / x))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 4.7e-7], N[(1.0 + N[(-8.0 * N[(y / N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(0.25 * N[(x / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.7e-7

   1. Initial program 57.2%

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

   2. Taylor expanded in x around inf 52.5%

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

      1. associate--l+ 52.5%

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

      2. distribute-rgt-out-- 52.5%

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

      3. metadata-eval 52.5%

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

      4. *-commutative 52.5%

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

      5. +-commutative 52.5%

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

      6. *-commutative 52.5%

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

      7. fma-def 52.5%

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

      8. unpow2 52.5%

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

      9. unpow2 52.5%

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

      10. times-frac 59.2%

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

   4. Simplified 59.2%

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

      1. fma-udef 59.2%

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

      2. pow2 59.2%

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

   6. Applied egg-rr 59.2%

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

      1. unpow2 59.2%

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

      2. clear-num 59.2%

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

      3. frac-times 59.1%

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

      4. *-un-lft-identity 59.1%

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

   8. Applied egg-rr 59.1%

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

   if 4.7e-7 < y

   1. Initial program 37.4%

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

   2. Taylor expanded in x around 0 32.9%

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

      1. unpow2 32.9%

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

   4. Simplified 32.9%

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

      1. div-sub 32.9%

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

      2. *-un-lft-identity 32.9%

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

      3. times-frac 32.9%

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

      4. metadata-eval 32.9%

         \[\leadsto \color{blue}{0.25} \cdot \frac{x \cdot x}{y \cdot y} - \frac{\left(y \cdot 4\right) \cdot y}{4 \cdot \left(y \cdot y\right)} \]

      5. times-frac 33.0%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} - \frac{\left(y \cdot 4\right) \cdot y}{4 \cdot \left(y \cdot y\right)} \]

      6. pow2 33.0%

         \[\leadsto 0.25 \cdot \color{blue}{{\left(\frac{x}{y}\right)}^{2}} - \frac{\left(y \cdot 4\right) \cdot y}{4 \cdot \left(y \cdot y\right)} \]

      7. associate-*r* 33.0%

         \[\leadsto 0.25 \cdot {\left(\frac{x}{y}\right)}^{2} - \frac{\left(y \cdot 4\right) \cdot y}{\color{blue}{\left(4 \cdot y\right) \cdot y}} \]

      8. *-commutative 33.0%

         \[\leadsto 0.25 \cdot {\left(\frac{x}{y}\right)}^{2} - \frac{\left(y \cdot 4\right) \cdot y}{\color{blue}{\left(y \cdot 4\right)} \cdot y} \]

      9. *-inverses 78.4%

         \[\leadsto 0.25 \cdot {\left(\frac{x}{y}\right)}^{2} - \color{blue}{1} \]

   6. Applied egg-rr 78.4%

      \[\leadsto \color{blue}{0.25 \cdot {\left(\frac{x}{y}\right)}^{2} - 1} \]
   7. Step-by-step derivation

      1. unpow2 78.4%

         \[\leadsto 0.25 \cdot \color{blue}{\left(\frac{x}{y} \cdot \frac{x}{y}\right)} - 1 \]

      2. clear-num 78.4%

         \[\leadsto 0.25 \cdot \left(\color{blue}{\frac{1}{\frac{y}{x}}} \cdot \frac{x}{y}\right) - 1 \]

      3. frac-times 78.4%

         \[\leadsto 0.25 \cdot \color{blue}{\frac{1 \cdot x}{\frac{y}{x} \cdot y}} - 1 \]

      4. *-un-lft-identity 78.4%

         \[\leadsto 0.25 \cdot \frac{\color{blue}{x}}{\frac{y}{x} \cdot y} - 1 \]

   8. Applied egg-rr 78.4%

      \[\leadsto 0.25 \cdot \color{blue}{\frac{x}{\frac{y}{x} \cdot y}} - 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.7 \cdot 10^{-7}:\\ \;\;\;\;1 + -8 \cdot \frac{y}{x \cdot \frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;-1 + 0.25 \cdot \frac{x}{y \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 11: 73.9% accurate, 6.2× speedup

Math

\[\begin{array}{l} x = |x|\\ y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
(FPCore (x y) :precision binary64 (if (<= y 5e-7) 1.0 -1.0))

C

x = abs(x);
y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 5e-7) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5d-7) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 5e-7) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

x = abs(x)
y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 5e-7:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

x = abs(x)
y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 5e-7)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

x = abs(x)
y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5e-7)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 5e-7], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 4.99999999999999977e-7

   1. Initial program 57.2%

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

   2. Taylor expanded in x around inf 57.6%

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

   if 4.99999999999999977e-7 < y

   1. Initial program 37.4%

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

   2. Taylor expanded in x around 0 77.4%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 12: 50.1% accurate, 19.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.9%

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

2. Taylor expanded in x around 0 50.2%

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

3. Final simplification 50.2%

   \[\leadsto -1 \]

Developer target: 50.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023257 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

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

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