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

Percentage Accurate: 51.0% → 80.7%

Time: 9.4s

Alternatives: 7

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: 51.0% 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: 80.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-253}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x \cdot 0.5}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+238}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot y, -4, x \cdot x\right)}{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot 4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y \cdot -8}{x}, \frac{y}{x}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 2e-253)
   (fma (/ x y) (/ (* x 0.5) y) -1.0)
   (if (<= (* x x) 5e+238)
     (/ (fma (* y y) -4.0 (* x x)) (fma x x (* y (* y 4.0))))
     (fma (/ (* y -8.0) x) (/ y x) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2e-253) {
		tmp = fma((x / y), ((x * 0.5) / y), -1.0);
	} else if ((x * x) <= 5e+238) {
		tmp = fma((y * y), -4.0, (x * x)) / fma(x, x, (y * (y * 4.0)));
	} else {
		tmp = fma(((y * -8.0) / x), (y / x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 2e-253)
		tmp = fma(Float64(x / y), Float64(Float64(x * 0.5) / y), -1.0);
	elseif (Float64(x * x) <= 5e+238)
		tmp = Float64(fma(Float64(y * y), -4.0, Float64(x * x)) / fma(x, x, Float64(y * Float64(y * 4.0))));
	else
		tmp = fma(Float64(Float64(y * -8.0) / x), Float64(y / x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-253], N[(N[(x / y), $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] / y), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e+238], N[(N[(N[(y * y), $MachinePrecision] * -4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(x * x + N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * -8.0), $MachinePrecision] / x), $MachinePrecision] * N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 2.0000000000000001e-253

   1. Initial program 67.5%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 80.1

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

   5. Applied rewrites 80.1%

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

   7. Applied rewrites 90.1%

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

   if 2.0000000000000001e-253 < (*.f64 x x) < 4.99999999999999995e238

   1. Initial program 81.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 81.7

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 81.7

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

   4. Applied rewrites 81.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lift-*.f64 N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-*l* N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 81.7

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

   6. Applied rewrites 81.7%

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

   if 4.99999999999999995e238 < (*.f64 x x)

   1. Initial program 9.7%

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

   3. Taylor expanded in x around 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 80.6

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

   5. Applied rewrites 80.6%

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

   7. Applied rewrites 82.3%

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

Alternative 2: 80.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-253}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x \cdot 0.5}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+238}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, \left(y \cdot y\right) \cdot -4\right)}{\mathsf{fma}\left(y, y \cdot 4, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y \cdot -8}{x}, \frac{y}{x}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 2e-253)
   (fma (/ x y) (/ (* x 0.5) y) -1.0)
   (if (<= (* x x) 5e+238)
     (/ (fma x x (* (* y y) -4.0)) (fma y (* y 4.0) (* x x)))
     (fma (/ (* y -8.0) x) (/ y x) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 2e-253) {
		tmp = fma((x / y), ((x * 0.5) / y), -1.0);
	} else if ((x * x) <= 5e+238) {
		tmp = fma(x, x, ((y * y) * -4.0)) / fma(y, (y * 4.0), (x * x));
	} else {
		tmp = fma(((y * -8.0) / x), (y / x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 2e-253)
		tmp = fma(Float64(x / y), Float64(Float64(x * 0.5) / y), -1.0);
	elseif (Float64(x * x) <= 5e+238)
		tmp = Float64(fma(x, x, Float64(Float64(y * y) * -4.0)) / fma(y, Float64(y * 4.0), Float64(x * x)));
	else
		tmp = fma(Float64(Float64(y * -8.0) / x), Float64(y / x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 2e-253], N[(N[(x / y), $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] / y), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 5e+238], 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[(N[(y * -8.0), $MachinePrecision] / x), $MachinePrecision] * N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 x x) < 2.0000000000000001e-253

   1. Initial program 67.5%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 80.1

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

   5. Applied rewrites 80.1%

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

   7. Applied rewrites 90.1%

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

   if 2.0000000000000001e-253 < (*.f64 x x) < 4.99999999999999995e238

   1. Initial program 81.7%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   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. 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 81.7

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

      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 81.7

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

   4. Applied rewrites 81.7%

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

   if 4.99999999999999995e238 < (*.f64 x x)

   1. Initial program 9.7%

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

   3. Taylor expanded in x around 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 80.6

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

   5. Applied rewrites 80.6%

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

   7. Applied rewrites 82.3%

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

Alternative 3: 74.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \left(y \cdot 4\right)\\ \mathbf{if}\;\frac{x \cdot x - t\_0}{x \cdot x + t\_0} \leq -2 \cdot 10^{-312}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* y (* y 4.0))))
   (if (<= (/ (- (* x x) t_0) (+ (* x x) t_0)) -2e-312) -1.0 1.0)))

C

double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((((x * x) - t_0) / ((x * x) + t_0)) <= -2e-312) {
		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) :: t_0
    real(8) :: tmp
    t_0 = y * (y * 4.0d0)
    if ((((x * x) - t_0) / ((x * x) + t_0)) <= (-2d-312)) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = y * (y * 4.0);
	double tmp;
	if ((((x * x) - t_0) / ((x * x) + t_0)) <= -2e-312) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = y * (y * 4.0)
	tmp = 0
	if (((x * x) - t_0) / ((x * x) + t_0)) <= -2e-312:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(y * Float64(y * 4.0))
	tmp = 0.0
	if (Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0)) <= -2e-312)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = y * (y * 4.0);
	tmp = 0.0;
	if ((((x * x) - t_0) / ((x * x) + t_0)) <= -2e-312)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], -2e-312], -1.0, 1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y))) < -2.0000000000019e-312

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.7%

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

   if -2.0000000000019e-312 < (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)))

   1. Initial program 34.5%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 68.3%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot x - y \cdot \left(y \cdot 4\right)}{x \cdot x + y \cdot \left(y \cdot 4\right)} \leq -2 \cdot 10^{-312}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x \cdot 0.5}{y}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y \cdot -8}{x}, \frac{y}{x}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 5e-35)
   (fma (/ x y) (/ (* x 0.5) y) -1.0)
   (fma (/ (* y -8.0) x) (/ y x) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 5e-35) {
		tmp = fma((x / y), ((x * 0.5) / y), -1.0);
	} else {
		tmp = fma(((y * -8.0) / x), (y / x), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 5e-35)
		tmp = fma(Float64(x / y), Float64(Float64(x * 0.5) / y), -1.0);
	else
		tmp = fma(Float64(Float64(y * -8.0) / x), Float64(y / x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-35], N[(N[(x / y), $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] / y), $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(N[(y * -8.0), $MachinePrecision] / x), $MachinePrecision] * N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.99999999999999964e-35

   1. Initial program 70.8%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 78.2

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

   5. Applied rewrites 78.2%

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

   7. Applied rewrites 84.6%

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

   if 4.99999999999999964e-35 < (*.f64 x x)

   1. Initial program 43.4%

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

   3. Taylor expanded in x around 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 74.5

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

   5. Applied rewrites 74.5%

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

   7. Applied rewrites 75.4%

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

Alternative 5: 74.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x \cdot 0.5}{y}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -8 \cdot \frac{y}{x \cdot x}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 5e-35)
   (fma (/ x y) (/ (* x 0.5) y) -1.0)
   (fma y (* -8.0 (/ y (* x x))) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 5e-35) {
		tmp = fma((x / y), ((x * 0.5) / y), -1.0);
	} else {
		tmp = fma(y, (-8.0 * (y / (x * x))), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 5e-35)
		tmp = fma(Float64(x / y), Float64(Float64(x * 0.5) / y), -1.0);
	else
		tmp = fma(y, Float64(-8.0 * Float64(y / Float64(x * x))), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 5e-35], N[(N[(x / y), $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] / y), $MachinePrecision] + -1.0), $MachinePrecision], N[(y * N[(-8.0 * N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 4.99999999999999964e-35

   1. Initial program 70.8%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-lft-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval 78.2

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

   5. Applied rewrites 78.2%

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

   7. Applied rewrites 84.6%

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

   if 4.99999999999999964e-35 < (*.f64 x x)

   1. Initial program 43.4%

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

   3. Taylor expanded in x around 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 74.5

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

   5. Applied rewrites 74.5%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 5 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x \cdot 0.5}{y}, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -8 \cdot \frac{y}{x \cdot x}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.7 \cdot 10^{-28}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -8 \cdot \frac{y}{x \cdot x}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 1.7e-28) -1.0 (fma y (* -8.0 (/ y (* x x))) 1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 1.7e-28) {
		tmp = -1.0;
	} else {
		tmp = fma(y, (-8.0 * (y / (x * x))), 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(x * x) <= 1.7e-28)
		tmp = -1.0;
	else
		tmp = fma(y, Float64(-8.0 * Float64(y / Float64(x * x))), 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(x * x), $MachinePrecision], 1.7e-28], -1.0, N[(y * N[(-8.0 * N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 x x) < 1.7e-28

   1. Initial program 70.8%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 83.6%

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

   if 1.7e-28 < (*.f64 x x)

   1. Initial program 43.4%

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

   3. Taylor expanded in x around 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 74.5

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

   5. Applied rewrites 74.5%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot x \leq 1.7 \cdot 10^{-28}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, -8 \cdot \frac{y}{x \cdot x}, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 51.5% 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 57.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 54.4%

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

Developer Target 1: 51.4% 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 2024219 
(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))))