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

Percentage Accurate: 50.4% → 81.1%

Time: 4.2s

Alternatives: 8

Speedup: 0.9×

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: 81.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.1 \cdot 10^{-157}:\\ \;\;\;\;\mathsf{fma}\left(-8 \cdot \frac{y\_m}{x}, \frac{y\_m}{x}, 1\right)\\ \mathbf{elif}\;y\_m \leq 2.55 \cdot 10^{+100}:\\ \;\;\;\;\frac{x \cdot x - \left(y\_m \cdot 4\right) \cdot y\_m}{\mathsf{fma}\left(x, x, \left(4 \cdot y\_m\right) \cdot y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{x}{y\_m}\right) \cdot \frac{x}{y\_m} - 1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.1e-157)
   (fma (* -8.0 (/ y_m x)) (/ y_m x) 1.0)
   (if (<= y_m 2.55e+100)
     (/ (- (* x x) (* (* y_m 4.0) y_m)) (fma x x (* (* 4.0 y_m) y_m)))
     (- (* (* 0.5 (/ x y_m)) (/ x y_m)) 1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.1e-157) {
		tmp = fma((-8.0 * (y_m / x)), (y_m / x), 1.0);
	} else if (y_m <= 2.55e+100) {
		tmp = ((x * x) - ((y_m * 4.0) * y_m)) / fma(x, x, ((4.0 * y_m) * y_m));
	} else {
		tmp = ((0.5 * (x / y_m)) * (x / y_m)) - 1.0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.1e-157)
		tmp = fma(Float64(-8.0 * Float64(y_m / x)), Float64(y_m / x), 1.0);
	elseif (y_m <= 2.55e+100)
		tmp = Float64(Float64(Float64(x * x) - Float64(Float64(y_m * 4.0) * y_m)) / fma(x, x, Float64(Float64(4.0 * y_m) * y_m)));
	else
		tmp = Float64(Float64(Float64(0.5 * Float64(x / y_m)) * Float64(x / y_m)) - 1.0);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.1e-157], N[(N[(-8.0 * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[y$95$m, 2.55e+100], N[(N[(N[(x * x), $MachinePrecision] - N[(N[(y$95$m * 4.0), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision] / N[(x * x + N[(N[(4.0 * y$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.10000000000000005e-157

   1. Initial program 46.1%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. distribute-rgt-out-- N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 57.4

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

   5. Applied rewrites 57.4%

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

   if 1.10000000000000005e-157 < y < 2.55000000000000005e100

   1. Initial program 85.9%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 85.9

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 85.9

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

   4. Applied rewrites 85.9%

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

   if 2.55000000000000005e100 < y

   1. Initial program 19.4%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 92.2

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

   5. Applied rewrites 92.2%

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

Alternative 2: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left(y\_m \cdot 4\right) \cdot y\_m\\ t_1 := \frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{y\_m}, x \cdot \frac{x}{y\_m}, -1\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-8 \cdot \frac{y\_m}{x}, \frac{y\_m}{x}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{x}{y\_m}\right) \cdot \frac{x}{y\_m} - 1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (* (* y_m 4.0) y_m)) (t_1 (/ (- (* x x) t_0) (+ (* x x) t_0))))
   (if (<= t_1 -0.5)
     (fma (/ 0.5 y_m) (* x (/ x y_m)) -1.0)
     (if (<= t_1 2.0)
       (fma (* -8.0 (/ y_m x)) (/ y_m x) 1.0)
       (- (* (* 0.5 (/ x y_m)) (/ x y_m)) 1.0)))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = (y_m * 4.0) * y_m;
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double tmp;
	if (t_1 <= -0.5) {
		tmp = fma((0.5 / y_m), (x * (x / y_m)), -1.0);
	} else if (t_1 <= 2.0) {
		tmp = fma((-8.0 * (y_m / x)), (y_m / x), 1.0);
	} else {
		tmp = ((0.5 * (x / y_m)) * (x / y_m)) - 1.0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(Float64(y_m * 4.0) * y_m)
	t_1 = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
	tmp = 0.0
	if (t_1 <= -0.5)
		tmp = fma(Float64(0.5 / y_m), Float64(x * Float64(x / y_m)), -1.0);
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(-8.0 * Float64(y_m / x)), Float64(y_m / x), 1.0);
	else
		tmp = Float64(Float64(Float64(0.5 * Float64(x / y_m)) * Float64(x / y_m)) - 1.0);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(y$95$m * 4.0), $MachinePrecision] * y$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.5], N[(N[(0.5 / y$95$m), $MachinePrecision] * N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(-8.0 * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(N[(0.5 * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y))) < -0.5

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-inverses N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. times-frac N/A

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

      15. *-inverses N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   8. Applied rewrites 99.7%

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

   if -0.5 < (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. distribute-rgt-out-- N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 2 < (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 59.2

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

   5. Applied rewrites 59.2%

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

Alternative 3: 76.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left(y\_m \cdot 4\right) \cdot y\_m\\ t_1 := \frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ \mathbf{if}\;t\_1 \leq -0.5 \lor \neg \left(t\_1 \leq 2\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{y\_m}, x \cdot \frac{x}{y\_m}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-8 \cdot \frac{y\_m}{x}, \frac{y\_m}{x}, 1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (* (* y_m 4.0) y_m)) (t_1 (/ (- (* x x) t_0) (+ (* x x) t_0))))
   (if (or (<= t_1 -0.5) (not (<= t_1 2.0)))
     (fma (/ 0.5 y_m) (* x (/ x y_m)) -1.0)
     (fma (* -8.0 (/ y_m x)) (/ y_m x) 1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = (y_m * 4.0) * y_m;
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double tmp;
	if ((t_1 <= -0.5) || !(t_1 <= 2.0)) {
		tmp = fma((0.5 / y_m), (x * (x / y_m)), -1.0);
	} else {
		tmp = fma((-8.0 * (y_m / x)), (y_m / x), 1.0);
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(Float64(y_m * 4.0) * y_m)
	t_1 = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
	tmp = 0.0
	if ((t_1 <= -0.5) || !(t_1 <= 2.0))
		tmp = fma(Float64(0.5 / y_m), Float64(x * Float64(x / y_m)), -1.0);
	else
		tmp = fma(Float64(-8.0 * Float64(y_m / x)), Float64(y_m / x), 1.0);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(y$95$m * 4.0), $MachinePrecision] * y$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -0.5], N[Not[LessEqual[t$95$1, 2.0]], $MachinePrecision]], N[(N[(0.5 / y$95$m), $MachinePrecision] * N[(x * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(-8.0 * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y))) < -0.5 or 2 < (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)))

   1. Initial program 34.7%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 73.3

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

   5. Applied rewrites 73.3%

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

   6. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-inverses N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. times-frac N/A

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

      15. *-inverses N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   8. Applied rewrites 73.3%

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

   if -0.5 < (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. distribute-rgt-out-- N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 98.7

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

   5. Applied rewrites 98.7%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \leq -0.5 \lor \neg \left(\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \leq 2\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{y}, x \cdot \frac{x}{y}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-8 \cdot \frac{y}{x}, \frac{y}{x}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left(y\_m \cdot 4\right) \cdot y\_m\\ t_1 := \frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x \cdot 0.5}{y\_m \cdot y\_m}, -1\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(-8 \cdot \frac{y\_m}{x}, \frac{y\_m}{x}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (* (* y_m 4.0) y_m)) (t_1 (/ (- (* x x) t_0) (+ (* x x) t_0))))
   (if (<= t_1 -0.5)
     (fma x (/ (* x 0.5) (* y_m y_m)) -1.0)
     (if (<= t_1 2.0) (fma (* -8.0 (/ y_m x)) (/ y_m x) 1.0) -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = (y_m * 4.0) * y_m;
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double tmp;
	if (t_1 <= -0.5) {
		tmp = fma(x, ((x * 0.5) / (y_m * y_m)), -1.0);
	} else if (t_1 <= 2.0) {
		tmp = fma((-8.0 * (y_m / x)), (y_m / x), 1.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(Float64(y_m * 4.0) * y_m)
	t_1 = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
	tmp = 0.0
	if (t_1 <= -0.5)
		tmp = fma(x, Float64(Float64(x * 0.5) / Float64(y_m * y_m)), -1.0);
	elseif (t_1 <= 2.0)
		tmp = fma(Float64(-8.0 * Float64(y_m / x)), Float64(y_m / x), 1.0);
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(y$95$m * 4.0), $MachinePrecision] * y$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.5], N[(x * N[(N[(x * 0.5), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[(-8.0 * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision], -1.0]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y))) < -0.5

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-inverses N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. times-frac N/A

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

      15. *-inverses N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   8. Applied rewrites 99.7%

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

   10. Applied rewrites 99.6%

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

   if -0.5 < (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. distribute-rgt-out-- N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if 2 < (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Applied rewrites 57.7%

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

Alternative 5: 75.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left(y\_m \cdot 4\right) \cdot y\_m\\ t_1 := \frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x \cdot 0.5}{y\_m \cdot y\_m}, -1\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (* (* y_m 4.0) y_m)) (t_1 (/ (- (* x x) t_0) (+ (* x x) t_0))))
   (if (<= t_1 -0.5)
     (fma x (/ (* x 0.5) (* y_m y_m)) -1.0)
     (if (<= t_1 2.0) 1.0 -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = (y_m * 4.0) * y_m;
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double tmp;
	if (t_1 <= -0.5) {
		tmp = fma(x, ((x * 0.5) / (y_m * y_m)), -1.0);
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(Float64(y_m * 4.0) * y_m)
	t_1 = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
	tmp = 0.0
	if (t_1 <= -0.5)
		tmp = fma(x, Float64(Float64(x * 0.5) / Float64(y_m * y_m)), -1.0);
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(y$95$m * 4.0), $MachinePrecision] * y$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.5], N[(x * N[(N[(x * 0.5), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t$95$1, 2.0], 1.0, -1.0]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y))) < -0.5

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. *-inverses N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. *-commutative N/A

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

      13. unpow2 N/A

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

      14. times-frac N/A

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

      15. *-inverses N/A

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

      16. metadata-eval N/A

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

      17. lower-fma.f64 N/A

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

   8. Applied rewrites 99.7%

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

   10. Applied rewrites 99.6%

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

   if -0.5 < (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.6%

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

   if 2 < (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Applied rewrites 57.7%

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

Alternative 6: 75.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left(y\_m \cdot 4\right) \cdot y\_m\\ t_1 := \frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (* (* y_m 4.0) y_m)) (t_1 (/ (- (* x x) t_0) (+ (* x x) t_0))))
   (if (<= t_1 -1e-310) -1.0 (if (<= t_1 INFINITY) 1.0 -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = (y_m * 4.0) * y_m;
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double tmp;
	if (t_1 <= -1e-310) {
		tmp = -1.0;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double t_0 = (y_m * 4.0) * y_m;
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double tmp;
	if (t_1 <= -1e-310) {
		tmp = -1.0;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	t_0 = (y_m * 4.0) * y_m
	t_1 = ((x * x) - t_0) / ((x * x) + t_0)
	tmp = 0
	if t_1 <= -1e-310:
		tmp = -1.0
	elif t_1 <= math.inf:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(Float64(y_m * 4.0) * y_m)
	t_1 = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
	tmp = 0.0
	if (t_1 <= -1e-310)
		tmp = -1.0;
	elseif (t_1 <= Inf)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	t_0 = (y_m * 4.0) * y_m;
	t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	tmp = 0.0;
	if (t_1 <= -1e-310)
		tmp = -1.0;
	elseif (t_1 <= Inf)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(y$95$m * 4.0), $MachinePrecision] * y$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-310], -1.0, If[LessEqual[t$95$1, Infinity], 1.0, -1.0]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y))) < -9.999999999999969e-311 or +inf.0 < (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)))

   1. Initial program 34.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1} \]
   4. Step-by-step derivation

   5. Applied rewrites 71.9%

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

   if -9.999999999999969e-311 < (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y))) < +inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{1} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.6%

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

Alternative 7: 81.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.1 \cdot 10^{-157}:\\ \;\;\;\;\mathsf{fma}\left(-8 \cdot \frac{y\_m}{x}, \frac{y\_m}{x}, 1\right)\\ \mathbf{elif}\;y\_m \leq 2.55 \cdot 10^{+100}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, y\_m \cdot y\_m, x \cdot x\right)}{\mathsf{fma}\left(x, x, \left(4 \cdot y\_m\right) \cdot y\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot \frac{x}{y\_m}\right) \cdot \frac{x}{y\_m} - 1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.1e-157)
   (fma (* -8.0 (/ y_m x)) (/ y_m x) 1.0)
   (if (<= y_m 2.55e+100)
     (/ (fma -4.0 (* y_m y_m) (* x x)) (fma x x (* (* 4.0 y_m) y_m)))
     (- (* (* 0.5 (/ x y_m)) (/ x y_m)) 1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.1e-157) {
		tmp = fma((-8.0 * (y_m / x)), (y_m / x), 1.0);
	} else if (y_m <= 2.55e+100) {
		tmp = fma(-4.0, (y_m * y_m), (x * x)) / fma(x, x, ((4.0 * y_m) * y_m));
	} else {
		tmp = ((0.5 * (x / y_m)) * (x / y_m)) - 1.0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.1e-157)
		tmp = fma(Float64(-8.0 * Float64(y_m / x)), Float64(y_m / x), 1.0);
	elseif (y_m <= 2.55e+100)
		tmp = Float64(fma(-4.0, Float64(y_m * y_m), Float64(x * x)) / fma(x, x, Float64(Float64(4.0 * y_m) * y_m)));
	else
		tmp = Float64(Float64(Float64(0.5 * Float64(x / y_m)) * Float64(x / y_m)) - 1.0);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.1e-157], N[(N[(-8.0 * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[y$95$m, 2.55e+100], N[(N[(-4.0 * N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(x * x + N[(N[(4.0 * y$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.10000000000000005e-157

   1. Initial program 46.1%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

      2. distribute-rgt-out-- N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 57.4

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

   5. Applied rewrites 57.4%

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

   if 1.10000000000000005e-157 < y < 2.55000000000000005e100

   1. Initial program 85.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-lft-neg-in N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 85.9

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 85.9

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 85.9

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

   4. Applied rewrites 85.9%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 85.9

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

   6. Applied rewrites 85.9%

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

   if 2.55000000000000005e100 < y

   1. Initial program 19.4%

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

   3. Taylor expanded in x around 0

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lower--.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. associate-*l/ N/A

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

      9. *-lft-identity N/A

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

      10. unpow2 N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. associate-*r* N/A

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

      14. lower-*.f64 N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lower-/.f64 92.2

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

   5. Applied rewrites 92.2%

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

Alternative 8: 50.0% accurate, 48.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ -1 \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

y_m = math.fabs(y)
def code(x, y_m):
	return -1.0

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := -1.0

Derivation

1. Initial program 52.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-1} \]
4. Step-by-step derivation

5. Applied rewrites 52.9%

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

Developer Target 1: 50.9% accurate, 0.2× 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 2024332 
(FPCore (x y)
  :name "Diagrams.TwoD.Arc:arcBetween from diagrams-lib-1.3.0.3"
  :precision binary64

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

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