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

Percentage Accurate: 50.0% → 80.6%

Time: 7.2s

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: 50.0% 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: 80.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(4 \cdot y\right) \cdot y\\ \mathbf{if}\;x \cdot x \leq 10^{-126}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5 \cdot x}{y}, \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{x \cdot x - t\_0}{t\_0 + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 4.0 y) y)))
   (if (<= (* x x) 1e-126)
     (fma (/ (* 0.5 x) y) (/ x y) -1.0)
     (if (<= (* x x) 5e+292)
       (/ (- (* x x) t_0) (+ t_0 (* x x)))
       (fma (* (/ y x) -8.0) (/ y x) 1.0)))))

C

double code(double x, double y) {
	double t_0 = (4.0 * y) * y;
	double tmp;
	if ((x * x) <= 1e-126) {
		tmp = fma(((0.5 * x) / y), (x / y), -1.0);
	} else if ((x * x) <= 5e+292) {
		tmp = ((x * x) - t_0) / (t_0 + (x * x));
	} else {
		tmp = fma(((y / x) * -8.0), (y / x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(4.0 * y) * y)
	tmp = 0.0
	if (Float64(x * x) <= 1e-126)
		tmp = fma(Float64(Float64(0.5 * x) / y), Float64(x / y), -1.0);
	elseif (Float64(x * x) <= 5e+292)
		tmp = Float64(Float64(Float64(x * x) - t_0) / Float64(t_0 + Float64(x * x)));
	else
		tmp = fma(Float64(Float64(y / x) * -8.0), Float64(y / x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(4.0 * y), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[N[(x * x), $MachinePrecision], 1e-126], N[(N[(N[(0.5 * x), $MachinePrecision] / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e+292], N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(t$95$0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] * -8.0), $MachinePrecision] * N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 9.9999999999999995e-127

   1. Initial program 54.6%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   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 83.2

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

   5. Applied rewrites 83.2%

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

   7. Applied rewrites 83.7%

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

   if 9.9999999999999995e-127 < (*.f64 x x) < 4.9999999999999996e292

   1. Initial program 76.7%

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

   if 4.9999999999999996e292 < (*.f64 x x)

   1. Initial program 0.0%

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

   3. Taylor expanded in 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 92.5

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

   5. Applied rewrites 92.5%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-126}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5 \cdot x}{y}, \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{x \cdot x - \left(4 \cdot y\right) \cdot y}{\left(4 \cdot y\right) \cdot y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-126}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5 \cdot x}{y}, \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, y \cdot y, x \cdot x\right)}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 1e-126)
   (fma (/ (* 0.5 x) y) (/ x y) -1.0)
   (if (<= (* x x) 5e+292)
     (/ (fma -4.0 (* y y) (* x x)) (fma (* 4.0 y) y (* x x)))
     (fma (* (/ y x) -8.0) (/ y x) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1e-126) {
		tmp = fma(((0.5 * x) / y), (x / y), -1.0);
	} else if ((x * x) <= 5e+292) {
		tmp = fma(-4.0, (y * y), (x * x)) / fma((4.0 * y), y, (x * x));
	} else {
		tmp = fma(((y / x) * -8.0), (y / x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 1e-126)
		tmp = fma(Float64(Float64(0.5 * x) / y), Float64(x / y), -1.0);
	elseif (Float64(x * x) <= 5e+292)
		tmp = Float64(fma(-4.0, Float64(y * y), Float64(x * x)) / fma(Float64(4.0 * y), y, Float64(x * x)));
	else
		tmp = fma(Float64(Float64(y / x) * -8.0), Float64(y / x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e-126], N[(N[(N[(0.5 * x), $MachinePrecision] / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e+292], N[(N[(-4.0 * N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * y), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] * -8.0), $MachinePrecision] * N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 9.9999999999999995e-127

   1. Initial program 54.6%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   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 83.2

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

   5. Applied rewrites 83.2%

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

   7. Applied rewrites 83.7%

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

   if 9.9999999999999995e-127 < (*.f64 x x) < 4.9999999999999996e292

   1. Initial program 76.7%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. 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 76.7

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

      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 76.7

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

      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 76.7

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

   4. Applied rewrites 76.7%

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

   if 4.9999999999999996e292 < (*.f64 x x)

   1. Initial program 0.0%

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

   3. Taylor expanded in 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 92.5

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

   5. Applied rewrites 92.5%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-126}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5 \cdot x}{y}, \frac{x}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+292}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, y \cdot y, x \cdot x\right)}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5 \cdot x}{y}, \frac{x}{y}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 1e+92)
   (fma (/ (* 0.5 x) y) (/ x y) -1.0)
   (fma (* (/ y x) -8.0) (/ y x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1e+92) {
		tmp = fma(((0.5 * x) / y), (x / y), -1.0);
	} else {
		tmp = fma(((y / x) * -8.0), (y / x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 1e+92)
		tmp = fma(Float64(Float64(0.5 * x) / y), Float64(x / y), -1.0);
	else
		tmp = fma(Float64(Float64(y / x) * -8.0), Float64(y / x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e+92], N[(N[(N[(0.5 * x), $MachinePrecision] / y), $MachinePrecision] * N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] * -8.0), $MachinePrecision] * N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e92

   1. Initial program 58.9%

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

   3. 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 77.8

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

   5. Applied rewrites 77.8%

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

   7. Applied rewrites 78.2%

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

   if 1e92 < (*.f64 x x)

   1. Initial program 34.3%

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

   3. Taylor expanded in 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 86.6

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

   5. Applied rewrites 86.6%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5 \cdot x}{y}, \frac{x}{y}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{y}, \frac{x \cdot x}{y}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 1e+92)
   (fma (/ 0.5 y) (/ (* x x) y) -1.0)
   (fma (* (/ y x) -8.0) (/ y x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1e+92) {
		tmp = fma((0.5 / y), ((x * x) / y), -1.0);
	} else {
		tmp = fma(((y / x) * -8.0), (y / x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 1e+92)
		tmp = fma(Float64(0.5 / y), Float64(Float64(x * x) / y), -1.0);
	else
		tmp = fma(Float64(Float64(y / x) * -8.0), Float64(y / x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e+92], N[(N[(0.5 / y), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(y / x), $MachinePrecision] * -8.0), $MachinePrecision] * N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e92

   1. Initial program 58.9%

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

   3. 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 77.8

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

   5. Applied rewrites 77.8%

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

   if 1e92 < (*.f64 x x)

   1. Initial program 34.3%

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

   3. Taylor expanded in 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 86.6

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

   5. Applied rewrites 86.6%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{y}, \frac{x \cdot x}{y}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+92}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 1e+92) -1.0 (fma (* (/ y x) -8.0) (/ y x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1e+92) {
		tmp = -1.0;
	} else {
		tmp = fma(((y / x) * -8.0), (y / x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 1e+92)
		tmp = -1.0;
	else
		tmp = fma(Float64(Float64(y / x) * -8.0), Float64(y / x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e+92], -1.0, N[(N[(N[(y / x), $MachinePrecision] * -8.0), $MachinePrecision] * N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e92

   1. Initial program 58.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 77.4%

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

   if 1e92 < (*.f64 x x)

   1. Initial program 34.3%

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

   3. Taylor expanded in 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 86.6

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

   5. Applied rewrites 86.6%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+92}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+92}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-8 \cdot y}{x \cdot x}, y, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 1e+92) -1.0 (fma (/ (* -8.0 y) (* x x)) y 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1e+92) {
		tmp = -1.0;
	} else {
		tmp = fma(((-8.0 * y) / (x * x)), y, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 1e+92)
		tmp = -1.0;
	else
		tmp = fma(Float64(Float64(-8.0 * y) / Float64(x * x)), y, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e+92], -1.0, N[(N[(N[(-8.0 * y), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * y + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1e92

   1. Initial program 58.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 77.4%

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

   if 1e92 < (*.f64 x x)

   1. Initial program 34.3%

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

   3. Taylor expanded in 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 86.6

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

   5. Applied rewrites 86.6%

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

   7. Applied rewrites 86.3%

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

   9. Applied rewrites 86.3%

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

Alternative 7: 74.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+102}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* x x) 1e+102) -1.0 1.0))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1e+102) {
		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) <= 1d+102) 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) <= 1e+102) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x * x) <= 1e+102:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 1e+102)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x * x) <= 1e+102)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 1e+102], -1.0, 1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 9.99999999999999977e101

   1. Initial program 58.8%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 77.0%

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

   if 9.99999999999999977e101 < (*.f64 x x)

   1. Initial program 34.0%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 86.5%

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

Alternative 8: 50.4% 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 48.8%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 51.8%

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

Developer Target 1: 50.4% 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 2024273 
(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))))