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

Percentage Accurate: 49.5% → 81.3%

Time: 7.5s

Alternatives: 11

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: 49.5% 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: 81.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(4 \cdot y\_m\right) \cdot y\_m\\ t_1 := \mathsf{fma}\left(4 \cdot y\_m, y\_m, x\_m \cdot x\_m\right)\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-190}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y\_m}{x\_m} \cdot -8, \frac{y\_m}{x\_m}, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+225}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x\_m}{t\_1}, x\_m, \frac{y\_m}{t\_1} \cdot \left(-4 \cdot y\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{0.5}{{\left(e^{-\log y\_m} \cdot e^{\log x\_m}\right)}^{-2}} - 1}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (* (* 4.0 y_m) y_m)) (t_1 (fma (* 4.0 y_m) y_m (* x_m x_m))))
   (if (<= t_0 5e-190)
     (fma (* (/ y_m x_m) -8.0) (/ y_m x_m) 1.0)
     (if (<= t_0 5e+225)
       (fma (/ x_m t_1) x_m (* (/ y_m t_1) (* -4.0 y_m)))
       (/
        1.0
        (/
         1.0
         (-
          (/ 0.5 (pow (* (exp (- (log y_m))) (exp (log x_m))) -2.0))
          1.0)))))))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double t_0 = (4.0 * y_m) * y_m;
	double t_1 = fma((4.0 * y_m), y_m, (x_m * x_m));
	double tmp;
	if (t_0 <= 5e-190) {
		tmp = fma(((y_m / x_m) * -8.0), (y_m / x_m), 1.0);
	} else if (t_0 <= 5e+225) {
		tmp = fma((x_m / t_1), x_m, ((y_m / t_1) * (-4.0 * y_m)));
	} else {
		tmp = 1.0 / (1.0 / ((0.5 / pow((exp(-log(y_m)) * exp(log(x_m))), -2.0)) - 1.0));
	}
	return tmp;
}

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	t_0 = Float64(Float64(4.0 * y_m) * y_m)
	t_1 = fma(Float64(4.0 * y_m), y_m, Float64(x_m * x_m))
	tmp = 0.0
	if (t_0 <= 5e-190)
		tmp = fma(Float64(Float64(y_m / x_m) * -8.0), Float64(y_m / x_m), 1.0);
	elseif (t_0 <= 5e+225)
		tmp = fma(Float64(x_m / t_1), x_m, Float64(Float64(y_m / t_1) * Float64(-4.0 * y_m)));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(Float64(0.5 / (Float64(exp(Float64(-log(y_m))) * exp(log(x_m))) ^ -2.0)) - 1.0)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := Block[{t$95$0 = N[(N[(4.0 * y$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * y$95$m), $MachinePrecision] * y$95$m + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-190], N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * -8.0), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 5e+225], N[(N[(x$95$m / t$95$1), $MachinePrecision] * x$95$m + N[(N[(y$95$m / t$95$1), $MachinePrecision] * N[(-4.0 * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(N[(0.5 / N[Power[N[(N[Exp[(-N[Log[y$95$m], $MachinePrecision])], $MachinePrecision] * N[Exp[N[Log[x$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 5.00000000000000034e-190

   1. Initial program 59.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 85.9

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

   5. Applied rewrites 85.9%

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

   if 5.00000000000000034e-190 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4.99999999999999981e225

   1. Initial program 79.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 79.7%

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

   if 4.99999999999999981e225 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 14.3%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval 65.6

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

   5. Applied rewrites 65.6%

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

   7. Applied rewrites 80.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{0.5}{{\left(\frac{x}{y}\right)}^{-2}} - 1}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 15.8%

      \[\leadsto \frac{1}{\frac{1}{\frac{0.5}{{\left(e^{\log x} \cdot e^{-\log y}\right)}^{-2}} - 1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(4 \cdot y\right) \cdot y \leq 5 \cdot 10^{-190}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \mathbf{elif}\;\left(4 \cdot y\right) \cdot y \leq 5 \cdot 10^{+225}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)}, x, \frac{y}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} \cdot \left(-4 \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{0.5}{{\left(e^{-\log y} \cdot e^{\log x}\right)}^{-2}} - 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(4 \cdot y\_m\right) \cdot y\_m\\ t_1 := \mathsf{fma}\left(4 \cdot y\_m, y\_m, x\_m \cdot x\_m\right)\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-190}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y\_m}{x\_m} \cdot -8, \frac{y\_m}{x\_m}, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+225}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x\_m}{t\_1}, x\_m, \frac{y\_m}{t\_1} \cdot \left(-4 \cdot y\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(e^{-\log y\_m} \cdot e^{\log x\_m}, \frac{0.5 \cdot x\_m}{y\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (* (* 4.0 y_m) y_m)) (t_1 (fma (* 4.0 y_m) y_m (* x_m x_m))))
   (if (<= t_0 5e-190)
     (fma (* (/ y_m x_m) -8.0) (/ y_m x_m) 1.0)
     (if (<= t_0 5e+225)
       (fma (/ x_m t_1) x_m (* (/ y_m t_1) (* -4.0 y_m)))
       (fma
        (* (exp (- (log y_m))) (exp (log x_m)))
        (/ (* 0.5 x_m) y_m)
        -1.0)))))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double t_0 = (4.0 * y_m) * y_m;
	double t_1 = fma((4.0 * y_m), y_m, (x_m * x_m));
	double tmp;
	if (t_0 <= 5e-190) {
		tmp = fma(((y_m / x_m) * -8.0), (y_m / x_m), 1.0);
	} else if (t_0 <= 5e+225) {
		tmp = fma((x_m / t_1), x_m, ((y_m / t_1) * (-4.0 * y_m)));
	} else {
		tmp = fma((exp(-log(y_m)) * exp(log(x_m))), ((0.5 * x_m) / y_m), -1.0);
	}
	return tmp;
}

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	t_0 = Float64(Float64(4.0 * y_m) * y_m)
	t_1 = fma(Float64(4.0 * y_m), y_m, Float64(x_m * x_m))
	tmp = 0.0
	if (t_0 <= 5e-190)
		tmp = fma(Float64(Float64(y_m / x_m) * -8.0), Float64(y_m / x_m), 1.0);
	elseif (t_0 <= 5e+225)
		tmp = fma(Float64(x_m / t_1), x_m, Float64(Float64(y_m / t_1) * Float64(-4.0 * y_m)));
	else
		tmp = fma(Float64(exp(Float64(-log(y_m))) * exp(log(x_m))), Float64(Float64(0.5 * x_m) / y_m), -1.0);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := Block[{t$95$0 = N[(N[(4.0 * y$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * y$95$m), $MachinePrecision] * y$95$m + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-190], N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * -8.0), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 5e+225], N[(N[(x$95$m / t$95$1), $MachinePrecision] * x$95$m + N[(N[(y$95$m / t$95$1), $MachinePrecision] * N[(-4.0 * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[(-N[Log[y$95$m], $MachinePrecision])], $MachinePrecision] * N[Exp[N[Log[x$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 * x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 5.00000000000000034e-190

   1. Initial program 59.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 85.9

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

   5. Applied rewrites 85.9%

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

   if 5.00000000000000034e-190 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4.99999999999999981e225

   1. Initial program 79.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 79.7%

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

   if 4.99999999999999981e225 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 14.3%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval 65.6

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

   5. Applied rewrites 65.6%

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

   7. Applied rewrites 80.7%

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

   9. Applied rewrites 15.8%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(4 \cdot y\right) \cdot y \leq 5 \cdot 10^{-190}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \mathbf{elif}\;\left(4 \cdot y\right) \cdot y \leq 5 \cdot 10^{+225}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)}, x, \frac{y}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} \cdot \left(-4 \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(e^{-\log y} \cdot e^{\log x}, \frac{0.5 \cdot x}{y}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(4 \cdot y\_m\right) \cdot y\_m\\ t_1 := \mathsf{fma}\left(4 \cdot y\_m, y\_m, x\_m \cdot x\_m\right)\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-190}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y\_m}{x\_m} \cdot -8, \frac{y\_m}{x\_m}, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+225}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x\_m}{t\_1}, x\_m, \frac{y\_m}{t\_1} \cdot \left(-4 \cdot y\_m\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{x\_m}{y\_m}, \frac{x\_m}{y\_m} \cdot 0.5, -1\right)}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (* (* 4.0 y_m) y_m)) (t_1 (fma (* 4.0 y_m) y_m (* x_m x_m))))
   (if (<= t_0 5e-190)
     (fma (* (/ y_m x_m) -8.0) (/ y_m x_m) 1.0)
     (if (<= t_0 5e+225)
       (fma (/ x_m t_1) x_m (* (/ y_m t_1) (* -4.0 y_m)))
       (/ 1.0 (/ 1.0 (fma (/ x_m y_m) (* (/ x_m y_m) 0.5) -1.0)))))))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double t_0 = (4.0 * y_m) * y_m;
	double t_1 = fma((4.0 * y_m), y_m, (x_m * x_m));
	double tmp;
	if (t_0 <= 5e-190) {
		tmp = fma(((y_m / x_m) * -8.0), (y_m / x_m), 1.0);
	} else if (t_0 <= 5e+225) {
		tmp = fma((x_m / t_1), x_m, ((y_m / t_1) * (-4.0 * y_m)));
	} else {
		tmp = 1.0 / (1.0 / fma((x_m / y_m), ((x_m / y_m) * 0.5), -1.0));
	}
	return tmp;
}

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	t_0 = Float64(Float64(4.0 * y_m) * y_m)
	t_1 = fma(Float64(4.0 * y_m), y_m, Float64(x_m * x_m))
	tmp = 0.0
	if (t_0 <= 5e-190)
		tmp = fma(Float64(Float64(y_m / x_m) * -8.0), Float64(y_m / x_m), 1.0);
	elseif (t_0 <= 5e+225)
		tmp = fma(Float64(x_m / t_1), x_m, Float64(Float64(y_m / t_1) * Float64(-4.0 * y_m)));
	else
		tmp = Float64(1.0 / Float64(1.0 / fma(Float64(x_m / y_m), Float64(Float64(x_m / y_m) * 0.5), -1.0)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := Block[{t$95$0 = N[(N[(4.0 * y$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * y$95$m), $MachinePrecision] * y$95$m + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-190], N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * -8.0), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 5e+225], N[(N[(x$95$m / t$95$1), $MachinePrecision] * x$95$m + N[(N[(y$95$m / t$95$1), $MachinePrecision] * N[(-4.0 * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(N[(x$95$m / y$95$m), $MachinePrecision] * N[(N[(x$95$m / y$95$m), $MachinePrecision] * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 5.00000000000000034e-190

   1. Initial program 59.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 85.9

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

   5. Applied rewrites 85.9%

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

   if 5.00000000000000034e-190 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4.99999999999999981e225

   1. Initial program 79.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 79.7%

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

   if 4.99999999999999981e225 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 14.3%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval 65.6

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

   5. Applied rewrites 65.6%

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

   7. Applied rewrites 80.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{0.5}{{\left(\frac{x}{y}\right)}^{-2}} - 1}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 80.7%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(4 \cdot y\right) \cdot y \leq 5 \cdot 10^{-190}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \mathbf{elif}\;\left(4 \cdot y\right) \cdot y \leq 5 \cdot 10^{+225}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)}, x, \frac{y}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)} \cdot \left(-4 \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y} \cdot 0.5, -1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(4 \cdot y\_m\right) \cdot y\_m\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-190}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y\_m}{x\_m} \cdot -8, \frac{y\_m}{x\_m}, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+225}:\\ \;\;\;\;\frac{x\_m \cdot x\_m - t\_0}{x\_m \cdot x\_m + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{x\_m}{y\_m}, \frac{x\_m}{y\_m} \cdot 0.5, -1\right)}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (* (* 4.0 y_m) y_m)))
   (if (<= t_0 5e-190)
     (fma (* (/ y_m x_m) -8.0) (/ y_m x_m) 1.0)
     (if (<= t_0 5e+225)
       (/ (- (* x_m x_m) t_0) (+ (* x_m x_m) t_0))
       (/ 1.0 (/ 1.0 (fma (/ x_m y_m) (* (/ x_m y_m) 0.5) -1.0)))))))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double t_0 = (4.0 * y_m) * y_m;
	double tmp;
	if (t_0 <= 5e-190) {
		tmp = fma(((y_m / x_m) * -8.0), (y_m / x_m), 1.0);
	} else if (t_0 <= 5e+225) {
		tmp = ((x_m * x_m) - t_0) / ((x_m * x_m) + t_0);
	} else {
		tmp = 1.0 / (1.0 / fma((x_m / y_m), ((x_m / y_m) * 0.5), -1.0));
	}
	return tmp;
}

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	t_0 = Float64(Float64(4.0 * y_m) * y_m)
	tmp = 0.0
	if (t_0 <= 5e-190)
		tmp = fma(Float64(Float64(y_m / x_m) * -8.0), Float64(y_m / x_m), 1.0);
	elseif (t_0 <= 5e+225)
		tmp = Float64(Float64(Float64(x_m * x_m) - t_0) / Float64(Float64(x_m * x_m) + t_0));
	else
		tmp = Float64(1.0 / Float64(1.0 / fma(Float64(x_m / y_m), Float64(Float64(x_m / y_m) * 0.5), -1.0)));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := Block[{t$95$0 = N[(N[(4.0 * y$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-190], N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * -8.0), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 5e+225], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x$95$m * x$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(N[(x$95$m / y$95$m), $MachinePrecision] * N[(N[(x$95$m / y$95$m), $MachinePrecision] * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 5.00000000000000034e-190

   1. Initial program 59.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 85.9

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

   5. Applied rewrites 85.9%

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

   if 5.00000000000000034e-190 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4.99999999999999981e225

   1. Initial program 79.4%

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

   if 4.99999999999999981e225 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 14.3%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval 65.6

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

   5. Applied rewrites 65.6%

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

   7. Applied rewrites 80.7%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{0.5}{{\left(\frac{x}{y}\right)}^{-2}} - 1}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 80.7%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(4 \cdot y\right) \cdot y \leq 5 \cdot 10^{-190}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \mathbf{elif}\;\left(4 \cdot y\right) \cdot y \leq 5 \cdot 10^{+225}:\\ \;\;\;\;\frac{x \cdot x - \left(4 \cdot y\right) \cdot y}{x \cdot x + \left(4 \cdot y\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y} \cdot 0.5, -1\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(4 \cdot y\_m\right) \cdot y\_m\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-190}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y\_m}{x\_m} \cdot -8, \frac{y\_m}{x\_m}, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+225}:\\ \;\;\;\;\frac{x\_m \cdot x\_m - t\_0}{x\_m \cdot x\_m + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x\_m}{y\_m}, \frac{0.5 \cdot x\_m}{y\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (* (* 4.0 y_m) y_m)))
   (if (<= t_0 5e-190)
     (fma (* (/ y_m x_m) -8.0) (/ y_m x_m) 1.0)
     (if (<= t_0 5e+225)
       (/ (- (* x_m x_m) t_0) (+ (* x_m x_m) t_0))
       (fma (/ x_m y_m) (/ (* 0.5 x_m) y_m) -1.0)))))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double t_0 = (4.0 * y_m) * y_m;
	double tmp;
	if (t_0 <= 5e-190) {
		tmp = fma(((y_m / x_m) * -8.0), (y_m / x_m), 1.0);
	} else if (t_0 <= 5e+225) {
		tmp = ((x_m * x_m) - t_0) / ((x_m * x_m) + t_0);
	} else {
		tmp = fma((x_m / y_m), ((0.5 * x_m) / y_m), -1.0);
	}
	return tmp;
}

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	t_0 = Float64(Float64(4.0 * y_m) * y_m)
	tmp = 0.0
	if (t_0 <= 5e-190)
		tmp = fma(Float64(Float64(y_m / x_m) * -8.0), Float64(y_m / x_m), 1.0);
	elseif (t_0 <= 5e+225)
		tmp = Float64(Float64(Float64(x_m * x_m) - t_0) / Float64(Float64(x_m * x_m) + t_0));
	else
		tmp = fma(Float64(x_m / y_m), Float64(Float64(0.5 * x_m) / y_m), -1.0);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := Block[{t$95$0 = N[(N[(4.0 * y$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-190], N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * -8.0), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 5e+225], N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[(x$95$m * x$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / y$95$m), $MachinePrecision] * N[(N[(0.5 * x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 5.00000000000000034e-190

   1. Initial program 59.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 85.9

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

   5. Applied rewrites 85.9%

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

   if 5.00000000000000034e-190 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4.99999999999999981e225

   1. Initial program 79.4%

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

   if 4.99999999999999981e225 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 14.3%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval 65.6

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

   5. Applied rewrites 65.6%

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

   7. Applied rewrites 80.7%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(4 \cdot y\right) \cdot y \leq 5 \cdot 10^{-190}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \mathbf{elif}\;\left(4 \cdot y\right) \cdot y \leq 5 \cdot 10^{+225}:\\ \;\;\;\;\frac{x \cdot x - \left(4 \cdot y\right) \cdot y}{x \cdot x + \left(4 \cdot y\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{0.5 \cdot x}{y}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} t_0 := \left(4 \cdot y\_m\right) \cdot y\_m\\ \mathbf{if}\;t\_0 \leq 5 \cdot 10^{-190}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y\_m}{x\_m} \cdot -8, \frac{y\_m}{x\_m}, 1\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+225}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, y\_m \cdot y\_m, x\_m \cdot x\_m\right)}{\mathsf{fma}\left(4 \cdot y\_m, y\_m, x\_m \cdot x\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x\_m}{y\_m}, \frac{0.5 \cdot x\_m}{y\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (let* ((t_0 (* (* 4.0 y_m) y_m)))
   (if (<= t_0 5e-190)
     (fma (* (/ y_m x_m) -8.0) (/ y_m x_m) 1.0)
     (if (<= t_0 5e+225)
       (/ (fma -4.0 (* y_m y_m) (* x_m x_m)) (fma (* 4.0 y_m) y_m (* x_m x_m)))
       (fma (/ x_m y_m) (/ (* 0.5 x_m) y_m) -1.0)))))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double t_0 = (4.0 * y_m) * y_m;
	double tmp;
	if (t_0 <= 5e-190) {
		tmp = fma(((y_m / x_m) * -8.0), (y_m / x_m), 1.0);
	} else if (t_0 <= 5e+225) {
		tmp = fma(-4.0, (y_m * y_m), (x_m * x_m)) / fma((4.0 * y_m), y_m, (x_m * x_m));
	} else {
		tmp = fma((x_m / y_m), ((0.5 * x_m) / y_m), -1.0);
	}
	return tmp;
}

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	t_0 = Float64(Float64(4.0 * y_m) * y_m)
	tmp = 0.0
	if (t_0 <= 5e-190)
		tmp = fma(Float64(Float64(y_m / x_m) * -8.0), Float64(y_m / x_m), 1.0);
	elseif (t_0 <= 5e+225)
		tmp = Float64(fma(-4.0, Float64(y_m * y_m), Float64(x_m * x_m)) / fma(Float64(4.0 * y_m), y_m, Float64(x_m * x_m)));
	else
		tmp = fma(Float64(x_m / y_m), Float64(Float64(0.5 * x_m) / y_m), -1.0);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := Block[{t$95$0 = N[(N[(4.0 * y$95$m), $MachinePrecision] * y$95$m), $MachinePrecision]}, If[LessEqual[t$95$0, 5e-190], N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * -8.0), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[t$95$0, 5e+225], N[(N[(-4.0 * N[(y$95$m * y$95$m), $MachinePrecision] + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(4.0 * y$95$m), $MachinePrecision] * y$95$m + N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / y$95$m), $MachinePrecision] * N[(N[(0.5 * x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 5.00000000000000034e-190

   1. Initial program 59.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 85.9

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

   5. Applied rewrites 85.9%

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

   if 5.00000000000000034e-190 < (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 4.99999999999999981e225

   1. Initial program 79.4%

      \[\frac{x \cdot x - \left(y \cdot 4\right) \cdot y}{x \cdot x + \left(y \cdot 4\right) \cdot y} \]
   2. Add Preprocessing
   3. 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 79.4

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

      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 79.4

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

      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 79.4

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

   4. Applied rewrites 79.4%

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

   if 4.99999999999999981e225 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 14.3%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval 65.6

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

   5. Applied rewrites 65.6%

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

   7. Applied rewrites 80.7%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(4 \cdot y\right) \cdot y \leq 5 \cdot 10^{-190}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \mathbf{elif}\;\left(4 \cdot y\right) \cdot y \leq 5 \cdot 10^{+225}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-4, y \cdot y, x \cdot x\right)}{\mathsf{fma}\left(4 \cdot y, y, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{0.5 \cdot x}{y}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left(4 \cdot y\_m\right) \cdot y\_m \leq 1.5 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y\_m}{x\_m} \cdot -8, \frac{y\_m}{x\_m}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x\_m}{y\_m}, \frac{0.5 \cdot x\_m}{y\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (* (* 4.0 y_m) y_m) 1.5e-27)
   (fma (* (/ y_m x_m) -8.0) (/ y_m x_m) 1.0)
   (fma (/ x_m y_m) (/ (* 0.5 x_m) y_m) -1.0)))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if (((4.0 * y_m) * y_m) <= 1.5e-27) {
		tmp = fma(((y_m / x_m) * -8.0), (y_m / x_m), 1.0);
	} else {
		tmp = fma((x_m / y_m), ((0.5 * x_m) / y_m), -1.0);
	}
	return tmp;
}

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(Float64(4.0 * y_m) * y_m) <= 1.5e-27)
		tmp = fma(Float64(Float64(y_m / x_m) * -8.0), Float64(y_m / x_m), 1.0);
	else
		tmp = fma(Float64(x_m / y_m), Float64(Float64(0.5 * x_m) / y_m), -1.0);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(N[(4.0 * y$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], 1.5e-27], N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * -8.0), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x$95$m / y$95$m), $MachinePrecision] * N[(N[(0.5 * x$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.5000000000000001e-27

   1. Initial program 63.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 79.0

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

   5. Applied rewrites 79.0%

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

   if 1.5000000000000001e-27 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 36.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval 67.6

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

   5. Applied rewrites 67.6%

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

   7. Applied rewrites 77.9%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(4 \cdot y\right) \cdot y \leq 1.5 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, \frac{0.5 \cdot x}{y}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left(4 \cdot y\_m\right) \cdot y\_m \leq 1.5 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y\_m}{x\_m} \cdot -8, \frac{y\_m}{x\_m}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, \frac{0.5 \cdot x\_m}{y\_m \cdot y\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (* (* 4.0 y_m) y_m) 1.5e-27)
   (fma (* (/ y_m x_m) -8.0) (/ y_m x_m) 1.0)
   (fma x_m (/ (* 0.5 x_m) (* y_m y_m)) -1.0)))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if (((4.0 * y_m) * y_m) <= 1.5e-27) {
		tmp = fma(((y_m / x_m) * -8.0), (y_m / x_m), 1.0);
	} else {
		tmp = fma(x_m, ((0.5 * x_m) / (y_m * y_m)), -1.0);
	}
	return tmp;
}

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(Float64(4.0 * y_m) * y_m) <= 1.5e-27)
		tmp = fma(Float64(Float64(y_m / x_m) * -8.0), Float64(y_m / x_m), 1.0);
	else
		tmp = fma(x_m, Float64(Float64(0.5 * x_m) / Float64(y_m * y_m)), -1.0);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(N[(4.0 * y$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], 1.5e-27], N[(N[(N[(y$95$m / x$95$m), $MachinePrecision] * -8.0), $MachinePrecision] * N[(y$95$m / x$95$m), $MachinePrecision] + 1.0), $MachinePrecision], N[(x$95$m * N[(N[(0.5 * x$95$m), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.5000000000000001e-27

   1. Initial program 63.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 79.0

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

   5. Applied rewrites 79.0%

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

   if 1.5000000000000001e-27 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 36.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval 67.6

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

   5. Applied rewrites 67.6%

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

   7. Applied rewrites 77.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{0.5}{{\left(\frac{x}{y}\right)}^{-2}} - 1}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 77.4%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(4 \cdot y\right) \cdot y \leq 1.5 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x} \cdot -8, \frac{y}{x}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{0.5 \cdot x}{y \cdot y}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left(4 \cdot y\_m\right) \cdot y\_m \leq 1.5 \cdot 10^{-27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, \frac{0.5 \cdot x\_m}{y\_m \cdot y\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (* (* 4.0 y_m) y_m) 1.5e-27)
   1.0
   (fma x_m (/ (* 0.5 x_m) (* y_m y_m)) -1.0)))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if (((4.0 * y_m) * y_m) <= 1.5e-27) {
		tmp = 1.0;
	} else {
		tmp = fma(x_m, ((0.5 * x_m) / (y_m * y_m)), -1.0);
	}
	return tmp;
}

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(Float64(4.0 * y_m) * y_m) <= 1.5e-27)
		tmp = 1.0;
	else
		tmp = fma(x_m, Float64(Float64(0.5 * x_m) / Float64(y_m * y_m)), -1.0);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(N[(4.0 * y$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], 1.5e-27], 1.0, N[(x$95$m * N[(N[(0.5 * x$95$m), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.5000000000000001e-27

   1. Initial program 63.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 78.1%

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

   if 1.5000000000000001e-27 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 36.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate--r+ N/A

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

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

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

      4. metadata-eval N/A

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

      5. *-commutative N/A

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

      6. sub-neg N/A

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

      7. *-commutative N/A

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

      8. associate-*l/ N/A

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

      9. unpow2 N/A

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

      10. times-frac N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. metadata-eval 67.6

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

   5. Applied rewrites 67.6%

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

   7. Applied rewrites 77.9%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{0.5}{{\left(\frac{x}{y}\right)}^{-2}} - 1}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 77.4%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(4 \cdot y\right) \cdot y \leq 1.5 \cdot 10^{-27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{0.5 \cdot x}{y \cdot y}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left(4 \cdot y\_m\right) \cdot y\_m \leq 1.5 \cdot 10^{-27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m)
 :precision binary64
 (if (<= (* (* 4.0 y_m) y_m) 1.5e-27) 1.0 -1.0))

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	double tmp;
	if (((4.0 * y_m) * y_m) <= 1.5e-27) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
x_m = abs(x)
real(8) function code(x_m, y_m)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (((4.0d0 * y_m) * y_m) <= 1.5d-27) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	double tmp;
	if (((4.0 * y_m) * y_m) <= 1.5e-27) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	tmp = 0
	if ((4.0 * y_m) * y_m) <= 1.5e-27:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	tmp = 0.0
	if (Float64(Float64(4.0 * y_m) * y_m) <= 1.5e-27)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
x_m = abs(x);
function tmp_2 = code(x_m, y_m)
	tmp = 0.0;
	if (((4.0 * y_m) * y_m) <= 1.5e-27)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := If[LessEqual[N[(N[(4.0 * y$95$m), $MachinePrecision] * y$95$m), $MachinePrecision], 1.5e-27], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 y #s(literal 4 binary64)) y) < 1.5000000000000001e-27

   1. Initial program 63.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 78.1%

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

   if 1.5000000000000001e-27 < (*.f64 (*.f64 y #s(literal 4 binary64)) y)

   1. Initial program 36.5%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 76.4%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(4 \cdot y\right) \cdot y \leq 1.5 \cdot 10^{-27}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.2% accurate, 48.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ x_m = \left|x\right| \\ -1 \end{array} \]

FPCore

y_m = (fabs.f64 y)
x_m = (fabs.f64 x)
(FPCore (x_m y_m) :precision binary64 -1.0)

C

y_m = fabs(y);
x_m = fabs(x);
double code(double x_m, double y_m) {
	return -1.0;
}

Fortran

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

Java

y_m = Math.abs(y);
x_m = Math.abs(x);
public static double code(double x_m, double y_m) {
	return -1.0;
}

Python

y_m = math.fabs(y)
x_m = math.fabs(x)
def code(x_m, y_m):
	return -1.0

Julia

y_m = abs(y)
x_m = abs(x)
function code(x_m, y_m)
	return -1.0
end

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y$95$m_] := -1.0

Derivation

1. Initial program 49.6%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 50.8%

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

Developer Target 1: 50.0% 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 2024254 
(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))))