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

Percentage Accurate: 49.7% → 80.5%

Time: 6.3s

Alternatives: 6

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: 49.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(4 \cdot y\right) \cdot y\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-240}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+173}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot y, y, x \cdot x\right)}{\mathsf{fma}\left(y \cdot y, 4, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{y}, \frac{x}{y} \cdot x, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 4.0 y) y)))
   (if (<= t_0 4e-240)
     (fma (* (/ y x) -8.0) (/ y x) 1.0)
     (if (<= t_0 2e+173)
       (/ (fma (* -4.0 y) y (* x x)) (fma (* y y) 4.0 (* x x)))
       (fma (/ 0.5 y) (* (/ x y) x) -1.0)))))

C

double code(double x, double y) {
	double t_0 = (4.0 * y) * y;
	double tmp;
	if (t_0 <= 4e-240) {
		tmp = fma(((y / x) * -8.0), (y / x), 1.0);
	} else if (t_0 <= 2e+173) {
		tmp = fma((-4.0 * y), y, (x * x)) / fma((y * y), 4.0, (x * x));
	} else {
		tmp = fma((0.5 / y), ((x / y) * x), -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(4.0 * y) * y)
	tmp = 0.0
	if (t_0 <= 4e-240)
		tmp = fma(Float64(Float64(y / x) * -8.0), Float64(y / x), 1.0);
	elseif (t_0 <= 2e+173)
		tmp = Float64(fma(Float64(-4.0 * y), y, Float64(x * x)) / fma(Float64(y * y), 4.0, Float64(x * x)));
	else
		tmp = fma(Float64(0.5 / y), Float64(Float64(x / y) * x), -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(4.0 * y), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-240], N[(N[(N[(y / x), $MachinePrecision] * -8.0), $MachinePrecision] * N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2e+173], N[(N[(N[(-4.0 * y), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(N[(y * y), $MachinePrecision] * 4.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / y), $MachinePrecision] * N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 3.9999999999999999e-240

   1. Initial program 59.3%

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

   3. Taylor expanded in 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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 86.0

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

   5. Applied rewrites 86.0%

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

   if 3.9999999999999999e-240 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 2e173

   1. Initial program 79.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

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

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

      10. lower-*.f64 N/A

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

      11. metadata-eval 79.2

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

   4. Applied rewrites 79.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 79.2

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

   6. Applied rewrites 79.2%

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

   if 2e173 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 24.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   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. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 76.8

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 82.5%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(4 \cdot y\right) \cdot y \leq 4 \cdot 10^{-240}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \mathbf{elif}\;\left(4 \cdot y\right) \cdot y \leq 2 \cdot 10^{+173}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4 \cdot y, y, x \cdot x\right)}{\mathsf{fma}\left(y \cdot y, 4, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{y}, \frac{x}{y} \cdot x, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(4 \cdot y\right) \cdot y\\ t_1 := \frac{x \cdot x - t\_0}{x \cdot x + t\_0}\\ t_2 := \mathsf{fma}\left(\frac{0.5}{y}, \frac{x}{y} \cdot x, -1\right)\\ \mathbf{if}\;t\_1 \leq -0.5:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 4.0 y) y))
        (t_1 (/ (- (* x x) t_0) (+ (* x x) t_0)))
        (t_2 (fma (/ 0.5 y) (* (/ x y) x) -1.0)))
   (if (<= t_1 -0.5) t_2 (if (<= t_1 2.0) 1.0 t_2))))

C

double code(double x, double y) {
	double t_0 = (4.0 * y) * y;
	double t_1 = ((x * x) - t_0) / ((x * x) + t_0);
	double t_2 = fma((0.5 / y), ((x / y) * x), -1.0);
	double tmp;
	if (t_1 <= -0.5) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(4.0 * y) * y)
	t_1 = Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0))
	t_2 = fma(Float64(0.5 / y), Float64(Float64(x / y) * x), -1.0)
	tmp = 0.0
	if (t_1 <= -0.5)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = 1.0;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(4.0 * y), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.5 / y), $MachinePrecision] * N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[t$95$1, -0.5], t$95$2, If[LessEqual[t$95$1, 2.0], 1.0, t$95$2]]]]]

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))) < -0.5 or 2 < (/.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 35.9%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 62.1

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

   5. Applied rewrites 62.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 69.9%

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

   if -0.5 < (/.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

   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 inf

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

   5. Applied rewrites 100.0%

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

4. Final simplification 77.4%

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

Alternative 3: 76.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y)) (+.f64 (*.f64 x x) (*.f64 (*.f64 y #s(literal 4 binary64)) y))) < -0.5

   1. Initial program 99.9%

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

   3. Taylor expanded in 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. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 96.8

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

   5. Applied rewrites 96.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 99.7%

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

   if -0.5 < (/.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 34.2%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 67.6

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

   5. Applied rewrites 67.6%

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

4. Final simplification 76.2%

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

Alternative 4: 80.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(4 \cdot y\right) \cdot y\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-240}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+173}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, y \cdot y, x \cdot x\right)}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{y}, \frac{x}{y} \cdot x, -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 4.0 y) y)))
   (if (<= t_0 4e-240)
     (fma (* (/ y x) -8.0) (/ y x) 1.0)
     (if (<= t_0 2e+173)
       (/ (fma -4.0 (* y y) (* x x)) (fma (* 4.0 y) y (* x x)))
       (fma (/ 0.5 y) (* (/ x y) x) -1.0)))))

C

double code(double x, double y) {
	double t_0 = (4.0 * y) * y;
	double tmp;
	if (t_0 <= 4e-240) {
		tmp = fma(((y / x) * -8.0), (y / x), 1.0);
	} else if (t_0 <= 2e+173) {
		tmp = fma(-4.0, (y * y), (x * x)) / fma((4.0 * y), y, (x * x));
	} else {
		tmp = fma((0.5 / y), ((x / y) * x), -1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(4.0 * y) * y)
	tmp = 0.0
	if (t_0 <= 4e-240)
		tmp = fma(Float64(Float64(y / x) * -8.0), Float64(y / x), 1.0);
	elseif (t_0 <= 2e+173)
		tmp = Float64(fma(-4.0, Float64(y * y), Float64(x * x)) / fma(Float64(4.0 * y), y, Float64(x * x)));
	else
		tmp = fma(Float64(0.5 / y), Float64(Float64(x / y) * x), -1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(4.0 * y), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$0, 4e-240], N[(N[(N[(y / x), $MachinePrecision] * -8.0), $MachinePrecision] * N[(y / x), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 2e+173], N[(N[(-4.0 * N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * y), $MachinePrecision] * y + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 / y), $MachinePrecision] * N[(N[(x / y), $MachinePrecision] * x), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 3.9999999999999999e-240

   1. Initial program 59.3%

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

   3. Taylor expanded in 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. unpow2 N/A

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

      7. unpow2 N/A

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

      8. times-frac N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower-/.f64 86.0

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

   5. Applied rewrites 86.0%

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

   if 3.9999999999999999e-240 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 2e173

   1. Initial program 79.2%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 79.2

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lower-fma.f64 79.2

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lower-*.f64 79.2

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

   4. Applied rewrites 79.2%

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

   if 2e173 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 24.5%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   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. lower-fma.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 N/A

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

      13. metadata-eval 76.8

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 82.5%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(4 \cdot y\right) \cdot y \leq 4 \cdot 10^{-240}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \mathbf{elif}\;\left(4 \cdot y\right) \cdot y \leq 2 \cdot 10^{+173}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, y \cdot y, x \cdot x\right)}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{0.5}{y}, \frac{x}{y} \cdot x, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(4 \cdot y\right) \cdot y\\ \mathbf{if}\;\frac{x \cdot x - t\_0}{x \cdot x + t\_0} \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (* 4.0 y) y)))
   (if (<= (/ (- (* x x) t_0) (+ (* x x) t_0)) -0.5) -1.0 1.0)))

C

double code(double x, double y) {
	double t_0 = (4.0 * y) * y;
	double tmp;
	if ((((x * x) - t_0) / ((x * x) + t_0)) <= -0.5) {
		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 = (4.0d0 * y) * y
    if ((((x * x) - t_0) / ((x * x) + t_0)) <= (-0.5d0)) 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 = (4.0 * y) * y;
	double tmp;
	if ((((x * x) - t_0) / ((x * x) + t_0)) <= -0.5) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (4.0 * y) * y
	tmp = 0
	if (((x * x) - t_0) / ((x * x) + t_0)) <= -0.5:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(4.0 * y) * y)
	tmp = 0.0
	if (Float64(Float64(Float64(x * x) - t_0) / Float64(Float64(x * x) + t_0)) <= -0.5)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (4.0 * y) * y;
	tmp = 0.0;
	if ((((x * x) - t_0) / ((x * x) + t_0)) <= -0.5)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(4.0 * y), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[N[(N[(N[(x * x), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], -0.5], -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))) < -0.5

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 98.8%

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

   if -0.5 < (/.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 34.2%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 66.8%

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

4. Final simplification 75.4%

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

Alternative 6: 49.9% 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 51.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 51.8%

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

Developer Target 1: 50.3% 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 2024298 
(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))))