Kahan p9 Example

Percentage Accurate: 68.0% → 92.7%

Time: 7.6s

Alternatives: 10

Speedup: 0.5×

Specification

\[\left(0 < x \land x < 1\right) \land y < 1\]

Math

\[\begin{array}{l} \\ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 68.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 92.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left(y\_m + x\right) \cdot \left(x - y\_m\right)\\ \mathbf{if}\;\frac{t\_0}{y\_m \cdot y\_m + x \cdot x} \leq 2:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(y\_m, y\_m, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{y\_m}, x, -1\right), x, -y\_m\right)}{y\_m + x}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (* (+ y_m x) (- x y_m))))
   (if (<= (/ t_0 (+ (* y_m y_m) (* x x))) 2.0)
     (/ t_0 (fma y_m y_m (* x x)))
     (/ (fma (fma (/ 2.0 y_m) x -1.0) x (- y_m)) (+ y_m x)))))

C

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

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(Float64(y_m + x) * Float64(x - y_m))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(y_m * y_m) + Float64(x * x))) <= 2.0)
		tmp = Float64(t_0 / fma(y_m, y_m, Float64(x * x)));
	else
		tmp = Float64(fma(fma(Float64(2.0 / y_m), x, -1.0), x, Float64(-y_m)) / Float64(y_m + x));
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(y$95$m + x), $MachinePrecision] * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(t$95$0 / N[(y$95$m * y$95$m + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(2.0 / y$95$m), $MachinePrecision] * x + -1.0), $MachinePrecision] * x + (-y$95$m)), $MachinePrecision] / N[(y$95$m + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. associate-*l/ N/A

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

      10. div-inv N/A

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

      11. lower-*.f64 N/A

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. sub-neg N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. mul-1-neg N/A

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

      15. lower-neg.f64 76.8

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

   7. Applied rewrites 76.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow-1 N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 76.9

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

      6. lift-+.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 76.9

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

   9. Applied rewrites 76.9%

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

4. Final simplification 92.2%

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

Alternative 2: 92.3% accurate, 0.3× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (+ y_m x) (- x y_m)) (+ (* y_m y_m) (* x x)))))
   (if (<= t_0 -0.5)
     (fma x (/ (* 2.0 x) (* y_m y_m)) -1.0)
     (if (<= t_0 2.0)
       (fma (/ (/ (* -2.0 y_m) x) x) y_m 1.0)
       (fma (/ 2.0 y_m) (* (/ x y_m) x) -1.0)))))

C

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

Julia

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[(y$95$m + x), $MachinePrecision] * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(x * N[(N[(2.0 * x), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(N[(N[(-2.0 * y$95$m), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision] * y$95$m + 1.0), $MachinePrecision], N[(N[(2.0 / y$95$m), $MachinePrecision] * N[(N[(x / y$95$m), $MachinePrecision] * x), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < -0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. associate-*r/ N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{y}, \frac{x}{y} \cdot x, -1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.1%

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

   9. Applied rewrites 100.0%

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

   if -0.5 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-2}{x \cdot x} \cdot y, y, 1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.8%

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

   if 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. associate-*r/ N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 76.9

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

   5. Applied rewrites 76.9%

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

4. Final simplification 92.0%

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

Alternative 3: 92.3% accurate, 0.3× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (+ y_m x) (- x y_m)) (+ (* y_m y_m) (* x x)))))
   (if (<= t_0 -0.5)
     (fma x (/ (* 2.0 x) (* y_m y_m)) -1.0)
     (if (<= t_0 2.0)
       (fma (/ (* -2.0 y_m) (* x x)) y_m 1.0)
       (fma (/ 2.0 y_m) (* (/ x y_m) x) -1.0)))))

C

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

Julia

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[(y$95$m + x), $MachinePrecision] * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(x * N[(N[(2.0 * x), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(N[(-2.0 * y$95$m), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * y$95$m + 1.0), $MachinePrecision], N[(N[(2.0 / y$95$m), $MachinePrecision] * N[(N[(x / y$95$m), $MachinePrecision] * x), $MachinePrecision] + -1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < -0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. associate-*r/ N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{y}, \frac{x}{y} \cdot x, -1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.1%

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

   9. Applied rewrites 100.0%

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

   if -0.5 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-2}{x \cdot x} \cdot y, y, 1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.8%

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

   if 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. associate-*r/ N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 76.9

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

   5. Applied rewrites 76.9%

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

4. Final simplification 92.0%

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

Alternative 4: 91.7% accurate, 0.3× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (+ y_m x) (- x y_m)) (+ (* y_m y_m) (* x x)))))
   (if (<= t_0 -0.5)
     (fma x (/ (* 2.0 x) (* y_m y_m)) -1.0)
     (if (<= t_0 2.0) (fma (/ (* -2.0 y_m) (* x x)) y_m 1.0) -1.0))))

C

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

Julia

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[(y$95$m + x), $MachinePrecision] * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(x * N[(N[(2.0 * x), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(N[(-2.0 * y$95$m), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * y$95$m + 1.0), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < -0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. associate-*r/ N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{y}, \frac{x}{y} \cdot x, -1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.1%

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

   9. Applied rewrites 100.0%

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

   if -0.5 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 98.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-2}{x \cdot x} \cdot y, y, 1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.8%

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

   if 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 75.6%

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

4. Final simplification 91.6%

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

Alternative 5: 91.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{\left(y\_m + x\right) \cdot \left(x - y\_m\right)}{y\_m \cdot y\_m + x \cdot x}\\ \mathbf{if}\;t\_0 \leq 0.999:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{2 \cdot x}{y\_m \cdot y\_m}, -1\right)\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{-2}{x \cdot x} \cdot y\_m, y\_m, 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (+ y_m x) (- x y_m)) (+ (* y_m y_m) (* x x)))))
   (if (<= t_0 0.999)
     (fma x (/ (* 2.0 x) (* y_m y_m)) -1.0)
     (if (<= t_0 2.0) (fma (* (/ -2.0 (* x x)) y_m) y_m 1.0) -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = ((y_m + x) * (x - y_m)) / ((y_m * y_m) + (x * x));
	double tmp;
	if (t_0 <= 0.999) {
		tmp = fma(x, ((2.0 * x) / (y_m * y_m)), -1.0);
	} else if (t_0 <= 2.0) {
		tmp = fma(((-2.0 / (x * x)) * y_m), y_m, 1.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(Float64(Float64(y_m + x) * Float64(x - y_m)) / Float64(Float64(y_m * y_m) + Float64(x * x)))
	tmp = 0.0
	if (t_0 <= 0.999)
		tmp = fma(x, Float64(Float64(2.0 * x) / Float64(y_m * y_m)), -1.0);
	elseif (t_0 <= 2.0)
		tmp = fma(Float64(Float64(-2.0 / Float64(x * x)) * y_m), y_m, 1.0);
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[(y$95$m + x), $MachinePrecision] * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.999], N[(x * N[(N[(2.0 * x), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] * y$95$m), $MachinePrecision] * y$95$m + 1.0), $MachinePrecision], -1.0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 0.998999999999999999

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. associate-*r/ N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 99.2

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

   5. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{y}, \frac{x}{y} \cdot x, -1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.2%

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

   9. Applied rewrites 99.2%

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

   if 0.998999999999999999 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 99.6%

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

   if 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 75.6%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y + x\right) \cdot \left(x - y\right)}{y \cdot y + x \cdot x} \leq 0.999:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{2 \cdot x}{y \cdot y}, -1\right)\\ \mathbf{elif}\;\frac{\left(y + x\right) \cdot \left(x - y\right)}{y \cdot y + x \cdot x} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{-2}{x \cdot x} \cdot y, y, 1\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.6% accurate, 0.4× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (+ y_m x) (- x y_m)) (+ (* y_m y_m) (* x x)))))
   (if (<= t_0 -0.5)
     (fma x (/ (* 2.0 x) (* y_m y_m)) -1.0)
     (if (<= t_0 2.0) 1.0 -1.0))))

C

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

Julia

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[(y$95$m + x), $MachinePrecision] * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(x * N[(N[(2.0 * x), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, -1.0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < -0.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. associate-*r/ N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 100.0

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

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{y}, \frac{x}{y} \cdot x, -1\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 98.1%

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

   9. Applied rewrites 100.0%

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

   if -0.5 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.5%

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

   if 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 75.6%

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

4. Final simplification 91.3%

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

Alternative 7: 91.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{\left(y\_m + x\right) \cdot \left(x - y\_m\right)}{y\_m \cdot y\_m + x \cdot x}\\ \mathbf{if}\;t\_0 \leq -0.5:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (+ y_m x) (- x y_m)) (+ (* y_m y_m) (* x x)))))
   (if (<= t_0 -0.5) -1.0 (if (<= t_0 2.0) 1.0 -1.0))))

C

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

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y_m + x) * (x - y_m)) / ((y_m * y_m) + (x * x))
    if (t_0 <= (-0.5d0)) then
        tmp = -1.0d0
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double t_0 = ((y_m + x) * (x - y_m)) / ((y_m * y_m) + (x * x));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = -1.0;
	} else if (t_0 <= 2.0) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	t_0 = ((y_m + x) * (x - y_m)) / ((y_m * y_m) + (x * x))
	tmp = 0
	if t_0 <= -0.5:
		tmp = -1.0
	elif t_0 <= 2.0:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

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

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	t_0 = ((y_m + x) * (x - y_m)) / ((y_m * y_m) + (x * x));
	tmp = 0.0;
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= 2.0)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[(y$95$m + x), $MachinePrecision] * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], -1.0, If[LessEqual[t$95$0, 2.0], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < -0.5 or 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 55.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 89.0%

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

   if -0.5 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 97.5%

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

4. Final simplification 91.2%

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

Alternative 8: 92.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \left(y\_m + x\right) \cdot \left(x - y\_m\right)\\ \mathbf{if}\;\frac{t\_0}{y\_m \cdot y\_m + x \cdot x} \leq 2:\\ \;\;\;\;\frac{t\_0}{\mathsf{fma}\left(y\_m, y\_m, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{y\_m}, \frac{x}{y\_m} \cdot x, -1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (* (+ y_m x) (- x y_m))))
   (if (<= (/ t_0 (+ (* y_m y_m) (* x x))) 2.0)
     (/ t_0 (fma y_m y_m (* x x)))
     (fma (/ 2.0 y_m) (* (/ x y_m) x) -1.0))))

C

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

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(Float64(y_m + x) * Float64(x - y_m))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(y_m * y_m) + Float64(x * x))) <= 2.0)
		tmp = Float64(t_0 / fma(y_m, y_m, Float64(x * x)));
	else
		tmp = fma(Float64(2.0 / y_m), Float64(Float64(x / y_m) * x), -1.0);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(y$95$m + x), $MachinePrecision] * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(t$95$0 / N[(y$95$m * y$95$m + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / y$95$m), $MachinePrecision] * N[(N[(x / y$95$m), $MachinePrecision] * x), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. associate-*r/ N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 76.9

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

   5. Applied rewrites 76.9%

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

4. Final simplification 92.1%

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

Alternative 9: 91.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;\frac{\left(y\_m + x\right) \cdot \left(x - y\_m\right)}{y\_m \cdot y\_m + x \cdot x} \leq 2:\\ \;\;\;\;\frac{x - y\_m}{\mathsf{fma}\left(x, x, y\_m \cdot y\_m\right)} \cdot \left(y\_m + x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{y\_m}, \frac{x}{y\_m} \cdot x, -1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= (/ (* (+ y_m x) (- x y_m)) (+ (* y_m y_m) (* x x))) 2.0)
   (* (/ (- x y_m) (fma x x (* y_m y_m))) (+ y_m x))
   (fma (/ 2.0 y_m) (* (/ x y_m) x) -1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((((y_m + x) * (x - y_m)) / ((y_m * y_m) + (x * x))) <= 2.0) {
		tmp = ((x - y_m) / fma(x, x, (y_m * y_m))) * (y_m + x);
	} else {
		tmp = fma((2.0 / y_m), ((x / y_m) * x), -1.0);
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (Float64(Float64(Float64(y_m + x) * Float64(x - y_m)) / Float64(Float64(y_m * y_m) + Float64(x * x))) <= 2.0)
		tmp = Float64(Float64(Float64(x - y_m) / fma(x, x, Float64(y_m * y_m))) * Float64(y_m + x));
	else
		tmp = fma(Float64(2.0 / y_m), Float64(Float64(x / y_m) * x), -1.0);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[N[(N[(N[(y$95$m + x), $MachinePrecision] * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(N[(x - y$95$m), $MachinePrecision] / N[(x * x + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y$95$m + x), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / y$95$m), $MachinePrecision] * N[(N[(x / y$95$m), $MachinePrecision] * x), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < 2

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. associate-*r/ N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. lift-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-+.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. associate-*l/ N/A

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

   6. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. cancel-sign-sub-inv N/A

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

      11. associate-*r/ N/A

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

      12. associate-*l/ N/A

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

      13. lift-+.f64 N/A

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

      14. +-commutative N/A

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

      15. flip-- N/A

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

      16. lift-/.f64 N/A

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

      17. associate-*r/ N/A

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

   8. Applied rewrites 98.9%

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

   if 2 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. associate-*r/ N/A

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

      3. unpow2 N/A

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

      4. times-frac N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 76.9

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

   5. Applied rewrites 76.9%

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

4. Final simplification 91.6%

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

Alternative 10: 67.1% accurate, 36.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 66.5%

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

Developer Target 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;0.5 < t\_0 \land t\_0 < 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (fabs (/ x y))))
   (if (and (< 0.5 t_0) (< t_0 2.0))
     (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))
     (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))))

C

double code(double x, double y) {
	double t_0 = fabs((x / y));
	double tmp;
	if ((0.5 < t_0) && (t_0 < 2.0)) {
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	} else {
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))));
	}
	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 = abs((x / y))
    if ((0.5d0 < t_0) .and. (t_0 < 2.0d0)) then
        tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
    else
        tmp = 1.0d0 - (2.0d0 / (1.0d0 + ((x / y) * (x / y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.abs((x / y));
	double tmp;
	if ((0.5 < t_0) && (t_0 < 2.0)) {
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	} else {
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.fabs((x / y))
	tmp = 0
	if (0.5 < t_0) and (t_0 < 2.0):
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
	else:
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))))
	return tmp

Julia

function code(x, y)
	t_0 = abs(Float64(x / y))
	tmp = 0.0
	if ((0.5 < t_0) && (t_0 < 2.0))
		tmp = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)));
	else
		tmp = Float64(1.0 - Float64(2.0 / Float64(1.0 + Float64(Float64(x / y) * Float64(x / y)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = abs((x / y));
	tmp = 0.0;
	if ((0.5 < t_0) && (t_0 < 2.0))
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	else
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Reproduce

herbie shell --seed 2024327 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

  :alt
  (! :herbie-platform default (if (< 1/2 (fabs (/ x y)) 2) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1 (/ 2 (+ 1 (* (/ x y) (/ x y)))))))

  (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))