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

Percentage Accurate: 50.7% → 80.9%

Time: 8.9s

Alternatives: 7

Speedup: 4.0×

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: 50.7% 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: 80.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-242}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 10^{+265}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(y, y \cdot 4, x \cdot x\right)}}{\frac{1}{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot -4\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x \cdot -0.125}, \frac{y}{x}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 5e-242)
   -1.0
   (if (<= (* x x) 1e+265)
     (/ (/ 1.0 (fma y (* y 4.0) (* x x))) (/ 1.0 (fma x x (* (* y y) -4.0))))
     (fma (/ y (* x -0.125)) (/ y x) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 5e-242) {
		tmp = -1.0;
	} else if ((x * x) <= 1e+265) {
		tmp = (1.0 / fma(y, (y * 4.0), (x * x))) / (1.0 / fma(x, x, ((y * y) * -4.0)));
	} else {
		tmp = fma((y / (x * -0.125)), (y / x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 5e-242)
		tmp = -1.0;
	elseif (Float64(x * x) <= 1e+265)
		tmp = Float64(Float64(1.0 / fma(y, Float64(y * 4.0), Float64(x * x))) / Float64(1.0 / fma(x, x, Float64(Float64(y * y) * -4.0))));
	else
		tmp = fma(Float64(y / Float64(x * -0.125)), Float64(y / x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-242], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 1e+265], N[(N[(1.0 / N[(y * N[(y * 4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(x * x + N[(N[(y * y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(x * -0.125), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 4.9999999999999998e-242

   1. Initial program 48.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 86.2%

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

   if 4.9999999999999998e-242 < (*.f64 x x) < 1.00000000000000007e265

   1. Initial program 77.8%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. lift--.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. lift-+.f64 N/A

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

      9. clear-num N/A

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

   4. Applied rewrites 77.9%

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

   if 1.00000000000000007e265 < (*.f64 x x)

   1. Initial program 10.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. lift--.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. lift-+.f64 N/A

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

      9. clear-num N/A

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

   4. Applied rewrites 10.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. distribute-neg-frac N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 90.3

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

   7. Applied rewrites 90.3%

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

   9. Applied rewrites 90.4%

      \[\leadsto \mathsf{fma}\left(\frac{y}{x \cdot -0.125}, \color{blue}{\frac{y}{x}}, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 80.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-242}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 10^{+265}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot -4\right)}{\mathsf{fma}\left(y, y \cdot 4, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x \cdot -0.125}, \frac{y}{x}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 5e-242)
   -1.0
   (if (<= (* x x) 1e+265)
     (/ (fma x x (* (* y y) -4.0)) (fma y (* y 4.0) (* x x)))
     (fma (/ y (* x -0.125)) (/ y x) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 5e-242) {
		tmp = -1.0;
	} else if ((x * x) <= 1e+265) {
		tmp = fma(x, x, ((y * y) * -4.0)) / fma(y, (y * 4.0), (x * x));
	} else {
		tmp = fma((y / (x * -0.125)), (y / x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 5e-242)
		tmp = -1.0;
	elseif (Float64(x * x) <= 1e+265)
		tmp = Float64(fma(x, x, Float64(Float64(y * y) * -4.0)) / fma(y, Float64(y * 4.0), Float64(x * x)));
	else
		tmp = fma(Float64(y / Float64(x * -0.125)), Float64(y / x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-242], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 1e+265], N[(N[(x * x + N[(N[(y * y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] / N[(y * N[(y * 4.0), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(x * -0.125), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 4.9999999999999998e-242

   1. Initial program 48.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 86.2%

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

   if 4.9999999999999998e-242 < (*.f64 x x) < 1.00000000000000007e265

   1. Initial program 77.8%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 77.8

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 77.8

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

   4. Applied rewrites 77.8%

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

   if 1.00000000000000007e265 < (*.f64 x x)

   1. Initial program 10.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. lift--.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. lift-+.f64 N/A

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

      9. clear-num N/A

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

   4. Applied rewrites 10.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. distribute-neg-frac N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 90.3

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

   7. Applied rewrites 90.3%

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

   9. Applied rewrites 90.4%

      \[\leadsto \mathsf{fma}\left(\frac{y}{x \cdot -0.125}, \color{blue}{\frac{y}{x}}, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 80.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-242}:\\ \;\;\;\;-1\\ \mathbf{elif}\;x \cdot x \leq 10^{+265}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot -4\right)}{\mathsf{fma}\left(x, x, 4 \cdot \left(y \cdot y\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x \cdot -0.125}, \frac{y}{x}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 5e-242)
   -1.0
   (if (<= (* x x) 1e+265)
     (/ (fma x x (* (* y y) -4.0)) (fma x x (* 4.0 (* y y))))
     (fma (/ y (* x -0.125)) (/ y x) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 5e-242) {
		tmp = -1.0;
	} else if ((x * x) <= 1e+265) {
		tmp = fma(x, x, ((y * y) * -4.0)) / fma(x, x, (4.0 * (y * y)));
	} else {
		tmp = fma((y / (x * -0.125)), (y / x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 5e-242)
		tmp = -1.0;
	elseif (Float64(x * x) <= 1e+265)
		tmp = Float64(fma(x, x, Float64(Float64(y * y) * -4.0)) / fma(x, x, Float64(4.0 * Float64(y * y))));
	else
		tmp = fma(Float64(y / Float64(x * -0.125)), Float64(y / x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-242], -1.0, If[LessEqual[N[(x * x), $MachinePrecision], 1e+265], N[(N[(x * x + N[(N[(y * y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision] / N[(x * x + N[(4.0 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(x * -0.125), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 4.9999999999999998e-242

   1. Initial program 48.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 86.2%

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

   if 4.9999999999999998e-242 < (*.f64 x x) < 1.00000000000000007e265

   1. Initial program 77.8%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. 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. lift-*.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. metadata-eval 77.8

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 77.8

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

   4. Applied rewrites 77.8%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lift-*.f64 N/A

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

      10. lower-*.f64 77.8

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

   6. Applied rewrites 77.8%

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

   if 1.00000000000000007e265 < (*.f64 x x)

   1. Initial program 10.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. lift--.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. lift-+.f64 N/A

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

      9. clear-num N/A

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

   4. Applied rewrites 10.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. distribute-neg-frac N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 90.3

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

   7. Applied rewrites 90.3%

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

   9. Applied rewrites 90.4%

      \[\leadsto \mathsf{fma}\left(\frac{y}{x \cdot -0.125}, \color{blue}{\frac{y}{x}}, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+83}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x \cdot -0.125}, \frac{y}{x}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 2e+83) -1.0 (fma (/ y (* x -0.125)) (/ y x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2e+83) {
		tmp = -1.0;
	} else {
		tmp = fma((y / (x * -0.125)), (y / x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 2e+83)
		tmp = -1.0;
	else
		tmp = fma(Float64(y / Float64(x * -0.125)), Float64(y / x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+83], -1.0, N[(N[(y / N[(x * -0.125), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.00000000000000006e83

   1. Initial program 56.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 77.5%

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

   if 2.00000000000000006e83 < (*.f64 x x)

   1. Initial program 41.0%

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

      1. lift-/.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. flip-+ N/A

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

      4. lift--.f64 N/A

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

      5. associate-/r/ N/A

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

      6. lift--.f64 N/A

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

      7. flip-- N/A

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

      8. lift-+.f64 N/A

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

      9. clear-num N/A

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

   4. Applied rewrites 41.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. associate-*r/ N/A

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

      9. unpow2 N/A

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

      10. associate-*l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. distribute-neg-frac N/A

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

      16. metadata-eval N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 81.9

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

   7. Applied rewrites 81.9%

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

   9. Applied rewrites 82.0%

      \[\leadsto \mathsf{fma}\left(\frac{y}{x \cdot -0.125}, \color{blue}{\frac{y}{x}}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 74.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 3.2 \cdot 10^{+83}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{y}{x \cdot x} \cdot -8, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 3.2e+83) -1.0 (fma y (* (/ y (* x x)) -8.0) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 3.2e+83) {
		tmp = -1.0;
	} else {
		tmp = fma(y, ((y / (x * x)) * -8.0), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 3.2e+83)
		tmp = -1.0;
	else
		tmp = fma(y, Float64(Float64(y / Float64(x * x)) * -8.0), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 3.2e+83], -1.0, N[(y * N[(N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision] * -8.0), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 3.1999999999999999e83

   1. Initial program 56.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 77.5%

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

   if 3.1999999999999999e83 < (*.f64 x x)

   1. Initial program 41.0%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   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. *-commutative N/A

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

      7. unpow2 N/A

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

      8. associate-/l* N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 81.9

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

   5. Applied rewrites 81.9%

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

Alternative 6: 74.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+83}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* x x) 2e+83) -1.0 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2e+83) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if ((x * x) <= 2d+83) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2e+83) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 2e+83:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 2e+83)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 2e+83)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e+83], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 2.00000000000000006e83

   1. Initial program 56.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 77.5%

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

   if 2.00000000000000006e83 < (*.f64 x x)

   1. Initial program 41.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 81.3%

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

Alternative 7: 49.6% accurate, 48.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 50.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 55.2%

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

Developer Target 1: 51.1% 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 2024225 
(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))))