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

Percentage Accurate: 51.4% → 82.1%

Time: 9.6s

Alternatives: 7

Speedup: 2.8×

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: 51.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-240}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x \cdot 0.5}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+210}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y \cdot -4, y, x \cdot x\right)}{\mathsf{fma}\left(y \cdot 4, y, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y \cdot -8}{x}, \frac{y}{x}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 4e-240)
   (fma (/ x y) (/ (* x 0.5) y) -1.0)
   (if (<= (* x x) 2e+210)
     (/ (fma (* y -4.0) y (* x x)) (fma (* y 4.0) y (* x x)))
     (fma (/ (* y -8.0) x) (/ y x) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 4e-240) {
		tmp = fma((x / y), ((x * 0.5) / y), -1.0);
	} else if ((x * x) <= 2e+210) {
		tmp = fma((y * -4.0), y, (x * x)) / fma((y * 4.0), y, (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) <= 4e-240)
		tmp = fma(Float64(x / y), Float64(Float64(x * 0.5) / y), -1.0);
	elseif (Float64(x * x) <= 2e+210)
		tmp = Float64(fma(Float64(y * -4.0), y, Float64(x * x)) / fma(Float64(y * 4.0), y, 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], 4e-240], N[(N[(x / y), $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] / y), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+210], N[(N[(N[(y * -4.0), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(N[(y * 4.0), $MachinePrecision] * y + 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) < 3.9999999999999999e-240

   1. Initial program 60.8%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   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. accelerator-lowering-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. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{1}{2} \cdot \frac{1}{{y}^{2}}, \mathsf{neg}\left(1\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{1}{2} \cdot \frac{1}{{y}^{2}}, \mathsf{neg}\left(1\right)\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{1}{2} \cdot \frac{1}{{y}^{2}}, \mathsf{neg}\left(1\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{\frac{1}{2} \cdot 1}{{y}^{2}}}, \mathsf{neg}\left(1\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{\color{blue}{\frac{1}{2}}}{{y}^{2}}, \mathsf{neg}\left(1\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{\frac{1}{2}}{{y}^{2}}}, \mathsf{neg}\left(1\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{\frac{1}{2}}{\color{blue}{y \cdot y}}, \mathsf{neg}\left(1\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{\frac{1}{2}}{\color{blue}{y \cdot y}}, \mathsf{neg}\left(1\right)\right) \]

      15. metadata-eval 81.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{0.5}{y \cdot y}, \color{blue}{-1}\right) \]

   5. Simplified 81.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{0.5}{y \cdot y}, -1\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l* N/A

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

      4. times-frac N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 88.1

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

   7. Applied egg-rr 88.1%

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

   if 3.9999999999999999e-240 < (*.f64 x x) < 1.99999999999999985e210

   1. Initial program 81.1%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 81.2

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

   4. Applied egg-rr 81.2%

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

      1. sub-neg N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. *-lowering-*.f64 N/A

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

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

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

      7. *-lowering-*.f64 N/A

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

      8. metadata-eval 81.2

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

   6. Applied egg-rr 81.2%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. *-lowering-*.f64 81.2

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

   8. Applied egg-rr 81.2%

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

   if 1.99999999999999985e210 < (*.f64 x x)

   1. Initial program 11.1%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. *-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. accelerator-lowering-fma.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{y}{\color{blue}{{x}^{2} + 0}} \cdot -8, 1\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{y}{\color{blue}{x \cdot x} + 0} \cdot -8, 1\right) \]

      15. accelerator-lowering-fma.f64 85.9

          \[\leadsto \mathsf{fma}\left(y, \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, 0\right)}} \cdot -8, 1\right) \]

   5. Simplified 85.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{y}{\mathsf{fma}\left(x, x, 0\right)} \cdot -8, 1\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{y}{x \cdot x + 0} \cdot -8\right) \cdot y} + 1 \]

      2. +-rgt-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 87.2

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

   7. Applied egg-rr 87.2%

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

Alternative 2: 82.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \cdot x \leq 4 \cdot 10^{-240}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x \cdot 0.5}{y}, -1\right)\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+210}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, x, y \cdot \left(y \cdot -4\right)\right)}{\mathsf{fma}\left(y \cdot 4, y, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y \cdot -8}{x}, \frac{y}{x}, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* x x) 4e-240)
   (fma (/ x y) (/ (* x 0.5) y) -1.0)
   (if (<= (* x x) 2e+210)
     (/ (fma x x (* y (* y -4.0))) (fma (* y 4.0) y (* x x)))
     (fma (/ (* y -8.0) x) (/ y x) 1.0))))

C

double code(double x, double y) {
	double tmp;
	if ((x * x) <= 4e-240) {
		tmp = fma((x / y), ((x * 0.5) / y), -1.0);
	} else if ((x * x) <= 2e+210) {
		tmp = fma(x, x, (y * (y * -4.0))) / fma((y * 4.0), y, (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) <= 4e-240)
		tmp = fma(Float64(x / y), Float64(Float64(x * 0.5) / y), -1.0);
	elseif (Float64(x * x) <= 2e+210)
		tmp = Float64(fma(x, x, Float64(y * Float64(y * -4.0))) / fma(Float64(y * 4.0), y, 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], 4e-240], N[(N[(x / y), $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] / y), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[N[(x * x), $MachinePrecision], 2e+210], N[(N[(x * x + N[(y * N[(y * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y * 4.0), $MachinePrecision] * y + 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) < 3.9999999999999999e-240

   1. Initial program 60.8%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   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. accelerator-lowering-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. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{1}{2} \cdot \frac{1}{{y}^{2}}, \mathsf{neg}\left(1\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{1}{2} \cdot \frac{1}{{y}^{2}}, \mathsf{neg}\left(1\right)\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{1}{2} \cdot \frac{1}{{y}^{2}}, \mathsf{neg}\left(1\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{\frac{1}{2} \cdot 1}{{y}^{2}}}, \mathsf{neg}\left(1\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{\color{blue}{\frac{1}{2}}}{{y}^{2}}, \mathsf{neg}\left(1\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{\frac{1}{2}}{{y}^{2}}}, \mathsf{neg}\left(1\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{\frac{1}{2}}{\color{blue}{y \cdot y}}, \mathsf{neg}\left(1\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{\frac{1}{2}}{\color{blue}{y \cdot y}}, \mathsf{neg}\left(1\right)\right) \]

      15. metadata-eval 81.3

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{0.5}{y \cdot y}, \color{blue}{-1}\right) \]

   5. Simplified 81.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{0.5}{y \cdot y}, -1\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l* N/A

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

      4. times-frac N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 88.1

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

   7. Applied egg-rr 88.1%

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

   if 3.9999999999999999e-240 < (*.f64 x x) < 1.99999999999999985e210

   1. Initial program 81.1%

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. *-lowering-*.f64 81.2

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

   4. Applied egg-rr 81.2%

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

      1. sub-neg N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. *-lowering-*.f64 N/A

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

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

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

      7. *-lowering-*.f64 N/A

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

      8. metadata-eval 81.2

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

   6. Applied egg-rr 81.2%

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

   if 1.99999999999999985e210 < (*.f64 x x)

   1. Initial program 11.1%

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

   3. Taylor expanded in x around inf

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

      1. associate--l+ N/A

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

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

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. +-commutative N/A

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

      6. *-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. accelerator-lowering-fma.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{y}{\color{blue}{{x}^{2} + 0}} \cdot -8, 1\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{y}{\color{blue}{x \cdot x} + 0} \cdot -8, 1\right) \]

      15. accelerator-lowering-fma.f64 85.9

          \[\leadsto \mathsf{fma}\left(y, \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, 0\right)}} \cdot -8, 1\right) \]

   5. Simplified 85.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{y}{\mathsf{fma}\left(x, x, 0\right)} \cdot -8, 1\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{y}{x \cdot x + 0} \cdot -8\right) \cdot y} + 1 \]

      2. +-rgt-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 87.2

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

   7. Applied egg-rr 87.2%

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

Alternative 3: 74.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y \cdot -8}{x}, \frac{y}{x}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x \cdot 0.5}{y}, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* y 4.0)) 5e+48)
   (fma (/ (* y -8.0) x) (/ y x) 1.0)
   (fma (/ x y) (/ (* x 0.5) y) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (y * 4.0)) <= 5e+48) {
		tmp = fma(((y * -8.0) / x), (y / x), 1.0);
	} else {
		tmp = fma((x / y), ((x * 0.5) / y), -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(y * 4.0)) <= 5e+48)
		tmp = fma(Float64(Float64(y * -8.0) / x), Float64(y / x), 1.0);
	else
		tmp = fma(Float64(x / y), Float64(Float64(x * 0.5) / y), -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], 5e+48], N[(N[(N[(y * -8.0), $MachinePrecision] / x), $MachinePrecision] * N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] / y), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4.99999999999999973e48

   1. Initial program 64.2%

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

   3. Taylor expanded in 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. accelerator-lowering-fma.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. /-lowering-/.f64 N/A

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

      13. +-rgt-identity N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{y}{\color{blue}{{x}^{2} + 0}} \cdot -8, 1\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(y, \frac{y}{\color{blue}{x \cdot x} + 0} \cdot -8, 1\right) \]

      15. accelerator-lowering-fma.f64 76.3

          \[\leadsto \mathsf{fma}\left(y, \frac{y}{\color{blue}{\mathsf{fma}\left(x, x, 0\right)}} \cdot -8, 1\right) \]

   5. Simplified 76.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{y}{\mathsf{fma}\left(x, x, 0\right)} \cdot -8, 1\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{y}{x \cdot x + 0} \cdot -8\right) \cdot y} + 1 \]

      2. +-rgt-identity N/A

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

      3. associate-*l/ N/A

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

      4. associate-*l/ N/A

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

      5. times-frac N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. /-lowering-/.f64 80.2

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

   7. Applied egg-rr 80.2%

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

   if 4.99999999999999973e48 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 41.0%

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

   3. Taylor expanded in x around 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. accelerator-lowering-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. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{1}{2} \cdot \frac{1}{{y}^{2}}, \mathsf{neg}\left(1\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{1}{2} \cdot \frac{1}{{y}^{2}}, \mathsf{neg}\left(1\right)\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{1}{2} \cdot \frac{1}{{y}^{2}}, \mathsf{neg}\left(1\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{\frac{1}{2} \cdot 1}{{y}^{2}}}, \mathsf{neg}\left(1\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{\color{blue}{\frac{1}{2}}}{{y}^{2}}, \mathsf{neg}\left(1\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{\frac{1}{2}}{{y}^{2}}}, \mathsf{neg}\left(1\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{\frac{1}{2}}{\color{blue}{y \cdot y}}, \mathsf{neg}\left(1\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{\frac{1}{2}}{\color{blue}{y \cdot y}}, \mathsf{neg}\left(1\right)\right) \]

      15. metadata-eval 75.1

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{0.5}{y \cdot y}, \color{blue}{-1}\right) \]

   5. Simplified 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{0.5}{y \cdot y}, -1\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l* N/A

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

      4. times-frac N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 79.4

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

   7. Applied egg-rr 79.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \frac{x \cdot 0.5}{y}, -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y \cdot -8}{x}, \frac{y}{x}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x \cdot 0.5}{y}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+48}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x \cdot 0.5}{y}, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* y 4.0)) 5e+48) 1.0 (fma (/ x y) (/ (* x 0.5) y) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (y * 4.0)) <= 5e+48) {
		tmp = 1.0;
	} else {
		tmp = fma((x / y), ((x * 0.5) / y), -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(y * 4.0)) <= 5e+48)
		tmp = 1.0;
	else
		tmp = fma(Float64(x / y), Float64(Float64(x * 0.5) / y), -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], 5e+48], 1.0, N[(N[(x / y), $MachinePrecision] * N[(N[(x * 0.5), $MachinePrecision] / y), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4.99999999999999973e48

   1. Initial program 64.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 79.4%

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

   if 4.99999999999999973e48 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 41.0%

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

   3. Taylor expanded in x around 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. accelerator-lowering-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. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{1}{2} \cdot \frac{1}{{y}^{2}}, \mathsf{neg}\left(1\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{1}{2} \cdot \frac{1}{{y}^{2}}, \mathsf{neg}\left(1\right)\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{1}{2} \cdot \frac{1}{{y}^{2}}, \mathsf{neg}\left(1\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{\frac{1}{2} \cdot 1}{{y}^{2}}}, \mathsf{neg}\left(1\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{\color{blue}{\frac{1}{2}}}{{y}^{2}}, \mathsf{neg}\left(1\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{\frac{1}{2}}{{y}^{2}}}, \mathsf{neg}\left(1\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{\frac{1}{2}}{\color{blue}{y \cdot y}}, \mathsf{neg}\left(1\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{\frac{1}{2}}{\color{blue}{y \cdot y}}, \mathsf{neg}\left(1\right)\right) \]

      15. metadata-eval 75.1

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{0.5}{y \cdot y}, \color{blue}{-1}\right) \]

   5. Simplified 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{0.5}{y \cdot y}, -1\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. associate-*r/ N/A

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

      3. associate-*l* N/A

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

      4. times-frac N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. /-lowering-/.f64 N/A

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

      8. *-lowering-*.f64 79.4

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

   7. Applied egg-rr 79.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \frac{x \cdot 0.5}{y}, -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+48}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{x \cdot 0.5}{y}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+48}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \frac{0.5}{y \cdot y}, x, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (* y (* y 4.0)) 5e+48) 1.0 (fma (* x (/ 0.5 (* y y))) x -1.0)))

C

double code(double x, double y) {
	double tmp;
	if ((y * (y * 4.0)) <= 5e+48) {
		tmp = 1.0;
	} else {
		tmp = fma((x * (0.5 / (y * y))), x, -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(y * 4.0)) <= 5e+48)
		tmp = 1.0;
	else
		tmp = fma(Float64(x * Float64(0.5 / Float64(y * y))), x, -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], 5e+48], 1.0, N[(N[(x * N[(0.5 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4.99999999999999973e48

   1. Initial program 64.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 79.4%

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

   if 4.99999999999999973e48 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 41.0%

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

   3. Taylor expanded in x around 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. accelerator-lowering-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. +-rgt-identity N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{1}{2} \cdot \frac{1}{{y}^{2}}, \mathsf{neg}\left(1\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{1}{2} \cdot \frac{1}{{y}^{2}}, \mathsf{neg}\left(1\right)\right) \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{1}{2} \cdot \frac{1}{{y}^{2}}, \mathsf{neg}\left(1\right)\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{\frac{1}{2} \cdot 1}{{y}^{2}}}, \mathsf{neg}\left(1\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{\color{blue}{\frac{1}{2}}}{{y}^{2}}, \mathsf{neg}\left(1\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{\frac{1}{2}}{{y}^{2}}}, \mathsf{neg}\left(1\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{\frac{1}{2}}{\color{blue}{y \cdot y}}, \mathsf{neg}\left(1\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{\frac{1}{2}}{\color{blue}{y \cdot y}}, \mathsf{neg}\left(1\right)\right) \]

      15. metadata-eval 75.1

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{0.5}{y \cdot y}, \color{blue}{-1}\right) \]

   5. Simplified 75.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{0.5}{y \cdot y}, -1\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. *-lowering-*.f64 78.9

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

   7. Applied egg-rr 78.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{y \cdot y} \cdot x, x, -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+48}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot \frac{0.5}{y \cdot y}, x, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+48}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= (* y (* y 4.0)) 5e+48) 1.0 -1.0))

C

double code(double x, double y) {
	double tmp;
	if ((y * (y * 4.0)) <= 5e+48) {
		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 ((y * (y * 4.0d0)) <= 5d+48) 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 ((y * (y * 4.0)) <= 5e+48) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (y * (y * 4.0)) <= 5e+48:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(y * Float64(y * 4.0)) <= 5e+48)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((y * (y * 4.0)) <= 5e+48)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(y * N[(y * 4.0), $MachinePrecision]), $MachinePrecision], 5e+48], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4.99999999999999973e48

   1. Initial program 64.2%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 79.4%

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

   if 4.99999999999999973e48 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 41.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 78.5%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(y \cdot 4\right) \leq 5 \cdot 10^{+48}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.1% 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 53.1%

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

3. Taylor expanded in x around 0

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

5. Simplified 48.2%

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

Developer Target 1: 51.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot 4\\ t_1 := x \cdot x + t\_0\\ t_2 := \frac{t\_0}{t\_1}\\ t_3 := \left(y \cdot 4\right) \cdot y\\ \mathbf{if}\;\frac{x \cdot x - t\_3}{x \cdot x + t\_3} < 0.9743233849626781:\\ \;\;\;\;\frac{x \cdot x}{t\_1} - t\_2\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{\sqrt{t\_1}}\right)}^{2} - t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* y y) 4.0))
        (t_1 (+ (* x x) t_0))
        (t_2 (/ t_0 t_1))
        (t_3 (* (* y 4.0) y)))
   (if (< (/ (- (* x x) t_3) (+ (* x x) t_3)) 0.9743233849626781)
     (- (/ (* x x) t_1) t_2)
     (- (pow (/ x (sqrt t_1)) 2.0) t_2))))

C

double code(double x, double y) {
	double t_0 = (y * y) * 4.0;
	double t_1 = (x * x) + t_0;
	double t_2 = t_0 / t_1;
	double t_3 = (y * 4.0) * y;
	double tmp;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781) {
		tmp = ((x * x) / t_1) - t_2;
	} else {
		tmp = pow((x / sqrt(t_1)), 2.0) - t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = (y * y) * 4.0d0
    t_1 = (x * x) + t_0
    t_2 = t_0 / t_1
    t_3 = (y * 4.0d0) * y
    if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781d0) then
        tmp = ((x * x) / t_1) - t_2
    else
        tmp = ((x / sqrt(t_1)) ** 2.0d0) - t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * y) * 4.0;
	double t_1 = (x * x) + t_0;
	double t_2 = t_0 / t_1;
	double t_3 = (y * 4.0) * y;
	double tmp;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781) {
		tmp = ((x * x) / t_1) - t_2;
	} else {
		tmp = Math.pow((x / Math.sqrt(t_1)), 2.0) - t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) * 4.0
	t_1 = (x * x) + t_0
	t_2 = t_0 / t_1
	t_3 = (y * 4.0) * y
	tmp = 0
	if (((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781:
		tmp = ((x * x) / t_1) - t_2
	else:
		tmp = math.pow((x / math.sqrt(t_1)), 2.0) - t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * 4.0)
	t_1 = Float64(Float64(x * x) + t_0)
	t_2 = Float64(t_0 / t_1)
	t_3 = Float64(Float64(y * 4.0) * y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * x) - t_3) / Float64(Float64(x * x) + t_3)) < 0.9743233849626781)
		tmp = Float64(Float64(Float64(x * x) / t_1) - t_2);
	else
		tmp = Float64((Float64(x / sqrt(t_1)) ^ 2.0) - t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) * 4.0;
	t_1 = (x * x) + t_0;
	t_2 = t_0 / t_1;
	t_3 = (y * 4.0) * y;
	tmp = 0.0;
	if ((((x * x) - t_3) / ((x * x) + t_3)) < 0.9743233849626781)
		tmp = ((x * x) / t_1) - t_2;
	else
		tmp = ((x / sqrt(t_1)) ^ 2.0) - t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 / t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y * 4.0), $MachinePrecision] * y), $MachinePrecision]}, If[Less[N[(N[(N[(x * x), $MachinePrecision] - t$95$3), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision], 0.9743233849626781], N[(N[(N[(x * x), $MachinePrecision] / t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision], N[(N[Power[N[(x / N[Sqrt[t$95$1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]]]]]]

Reproduce

herbie shell --seed 2024196 
(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))))