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

Percentage Accurate: 50.2% → 81.5%

Time: 7.7s

Alternatives: 8

Speedup: 48.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 50.2% 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: 81.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.35 \cdot 10^{-101}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y\_m}{x} \cdot -8, \frac{y\_m}{x}, 1\right)\\ \mathbf{elif}\;y\_m \leq 2 \cdot 10^{+140}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -4 \cdot \left(y\_m \cdot y\_m\right)\right)}{\mathsf{fma}\left(4 \cdot y\_m, y\_m, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5 \cdot x}{y\_m}, \frac{x}{y\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.35e-101)
   (fma (* (/ y_m x) -8.0) (/ y_m x) 1.0)
   (if (<= y_m 2e+140)
     (/ (fma x x (* -4.0 (* y_m y_m))) (fma (* 4.0 y_m) y_m (* x x)))
     (fma (/ (* 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 <= 1.35e-101) {
		tmp = fma(((y_m / x) * -8.0), (y_m / x), 1.0);
	} else if (y_m <= 2e+140) {
		tmp = fma(x, x, (-4.0 * (y_m * y_m))) / fma((4.0 * y_m), y_m, (x * x));
	} else {
		tmp = fma(((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 <= 1.35e-101)
		tmp = fma(Float64(Float64(y_m / x) * -8.0), Float64(y_m / x), 1.0);
	elseif (y_m <= 2e+140)
		tmp = Float64(fma(x, x, Float64(-4.0 * Float64(y_m * y_m))) / fma(Float64(4.0 * y_m), y_m, Float64(x * x)));
	else
		tmp = fma(Float64(Float64(0.5 * 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, 1.35e-101], N[(N[(N[(y$95$m / x), $MachinePrecision] * -8.0), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[y$95$m, 2e+140], N[(N[(x * x + N[(-4.0 * N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * y$95$m), $MachinePrecision] * y$95$m + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * x), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.3500000000000001e-101

   1. Initial program 50.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 62.5

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

   5. Applied rewrites 62.5%

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

   if 1.3500000000000001e-101 < y < 2.00000000000000012e140

   1. Initial program 80.4%

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

      1. 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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-neg-in 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 80.4

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

      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 80.4

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

      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 80.4

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

   4. Applied rewrites 80.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 80.4

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

   6. Applied rewrites 80.4%

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

   if 2.00000000000000012e140 < y

   1. Initial program 5.6%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 75.4

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

   5. Applied rewrites 75.4%

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

   7. Applied rewrites 89.4%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-101}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+140}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, -4 \cdot \left(y \cdot y\right)\right)}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5 \cdot x}{y}, \frac{x}{y}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.35 \cdot 10^{-101}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y\_m}{x} \cdot -8, \frac{y\_m}{x}, 1\right)\\ \mathbf{elif}\;y\_m \leq 2 \cdot 10^{+140}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, y\_m \cdot y\_m, x \cdot x\right)}{\mathsf{fma}\left(4 \cdot y\_m, y\_m, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5 \cdot x}{y\_m}, \frac{x}{y\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.35e-101)
   (fma (* (/ y_m x) -8.0) (/ y_m x) 1.0)
   (if (<= y_m 2e+140)
     (/ (fma -4.0 (* y_m y_m) (* x x)) (fma (* 4.0 y_m) y_m (* x x)))
     (fma (/ (* 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 <= 1.35e-101) {
		tmp = fma(((y_m / x) * -8.0), (y_m / x), 1.0);
	} else if (y_m <= 2e+140) {
		tmp = fma(-4.0, (y_m * y_m), (x * x)) / fma((4.0 * y_m), y_m, (x * x));
	} else {
		tmp = fma(((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 <= 1.35e-101)
		tmp = fma(Float64(Float64(y_m / x) * -8.0), Float64(y_m / x), 1.0);
	elseif (y_m <= 2e+140)
		tmp = Float64(fma(-4.0, Float64(y_m * y_m), Float64(x * x)) / fma(Float64(4.0 * y_m), y_m, Float64(x * x)));
	else
		tmp = fma(Float64(Float64(0.5 * 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, 1.35e-101], N[(N[(N[(y$95$m / x), $MachinePrecision] * -8.0), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[y$95$m, 2e+140], N[(N[(-4.0 * N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * y$95$m), $MachinePrecision] * y$95$m + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.5 * x), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.3500000000000001e-101

   1. Initial program 50.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 62.5

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

   5. Applied rewrites 62.5%

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

   if 1.3500000000000001e-101 < y < 2.00000000000000012e140

   1. Initial program 80.4%

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

      1. 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. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-neg-in 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 80.4

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

      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 80.4

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

      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 80.4

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

   4. Applied rewrites 80.4%

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

   if 2.00000000000000012e140 < y

   1. Initial program 5.6%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 75.4

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

   5. Applied rewrites 75.4%

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

   7. Applied rewrites 89.4%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-101}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+140}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, y \cdot y, x \cdot x\right)}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5 \cdot x}{y}, \frac{x}{y}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5 \cdot x}{y\_m}, \frac{x}{y\_m}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y\_m}{x} \cdot -8, \frac{y\_m}{x}, 1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= (* x x) 1e-80)
   (fma (/ (* 0.5 x) y_m) (/ x y_m) -1.0)
   (fma (* (/ y_m x) -8.0) (/ y_m x) 1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((x * x) <= 1e-80) {
		tmp = fma(((0.5 * x) / y_m), (x / y_m), -1.0);
	} else {
		tmp = fma(((y_m / x) * -8.0), (y_m / x), 1.0);
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (Float64(x * x) <= 1e-80)
		tmp = fma(Float64(Float64(0.5 * x) / y_m), Float64(x / y_m), -1.0);
	else
		tmp = fma(Float64(Float64(y_m / x) * -8.0), Float64(y_m / x), 1.0);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-80], N[(N[(N[(0.5 * x), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(x / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(y$95$m / x), $MachinePrecision] * -8.0), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.99999999999999961e-81

   1. Initial program 61.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 78.3

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

   5. Applied rewrites 78.3%

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

   7. Applied rewrites 79.0%

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

   if 9.99999999999999961e-81 < (*.f64 x x)

   1. Initial program 38.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 77.4

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

   5. Applied rewrites 77.4%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5 \cdot x}{y}, \frac{x}{y}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{y\_m}, \frac{x \cdot x}{y\_m}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y\_m}{x} \cdot -8, \frac{y\_m}{x}, 1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= (* x x) 1e-80)
   (fma (/ 0.5 y_m) (/ (* x x) y_m) -1.0)
   (fma (* (/ y_m x) -8.0) (/ y_m x) 1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((x * x) <= 1e-80) {
		tmp = fma((0.5 / y_m), ((x * x) / y_m), -1.0);
	} else {
		tmp = fma(((y_m / x) * -8.0), (y_m / x), 1.0);
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (Float64(x * x) <= 1e-80)
		tmp = fma(Float64(0.5 / y_m), Float64(Float64(x * x) / y_m), -1.0);
	else
		tmp = fma(Float64(Float64(y_m / x) * -8.0), Float64(y_m / x), 1.0);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-80], N[(N[(0.5 / y$95$m), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(y$95$m / x), $MachinePrecision] * -8.0), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.99999999999999961e-81

   1. Initial program 61.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. associate-*r/ N/A

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

      8. unpow2 N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. metadata-eval 78.3

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

   5. Applied rewrites 78.3%

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

   if 9.99999999999999961e-81 < (*.f64 x x)

   1. Initial program 38.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 77.4

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

   5. Applied rewrites 77.4%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{y}, \frac{x \cdot x}{y}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-86}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y\_m}{x} \cdot -8, \frac{y\_m}{x}, 1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= (* x x) 1e-86) -1.0 (fma (* (/ y_m x) -8.0) (/ y_m x) 1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((x * x) <= 1e-86) {
		tmp = -1.0;
	} else {
		tmp = fma(((y_m / x) * -8.0), (y_m / x), 1.0);
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (Float64(x * x) <= 1e-86)
		tmp = -1.0;
	else
		tmp = fma(Float64(Float64(y_m / x) * -8.0), Float64(y_m / x), 1.0);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-86], -1.0, N[(N[(N[(y$95$m / x), $MachinePrecision] * -8.0), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.00000000000000008e-86

   1. Initial program 61.7%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 78.1%

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

   if 1.00000000000000008e-86 < (*.f64 x x)

   1. Initial program 39.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 77.1

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

   5. Applied rewrites 77.1%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-86}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-86}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-8 \cdot y\_m}{x \cdot x}, y\_m, 1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= (* x x) 1e-86) -1.0 (fma (/ (* -8.0 y_m) (* x x)) y_m 1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((x * x) <= 1e-86) {
		tmp = -1.0;
	} else {
		tmp = fma(((-8.0 * y_m) / (x * x)), y_m, 1.0);
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (Float64(x * x) <= 1e-86)
		tmp = -1.0;
	else
		tmp = fma(Float64(Float64(-8.0 * y_m) / Float64(x * x)), y_m, 1.0);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-86], -1.0, N[(N[(N[(-8.0 * y$95$m), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * y$95$m + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.00000000000000008e-86

   1. Initial program 61.7%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 78.1%

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

   if 1.00000000000000008e-86 < (*.f64 x x)

   1. Initial program 39.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 77.1

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

   5. Applied rewrites 77.1%

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

   7. Applied rewrites 76.7%

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

   9. Applied rewrites 76.7%

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

Alternative 7: 74.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-86}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (if (<= (* x x) 1e-86) -1.0 1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((x * x) <= 1e-86) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((x * x) <= 1d-86) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if ((x * x) <= 1e-86) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if (x * x) <= 1e-86:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (Float64(x * x) <= 1e-86)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if ((x * x) <= 1e-86)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-86], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.00000000000000008e-86

   1. Initial program 61.7%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 78.1%

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

   if 1.00000000000000008e-86 < (*.f64 x x)

   1. Initial program 39.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 75.8%

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

Alternative 8: 50.7% 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 49.2%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 48.8%

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

Developer Target 1: 50.7% 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 2024268 
(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))))