Kahan p9 Example

Percentage Accurate: 68.5% → 92.8%

Time: 11.8s

Alternatives: 9

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: 68.5% 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: 92.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(t$95$0 / N[(y * y + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * 2.0), $MachinePrecision] / y), $MachinePrecision] * N[(x / y), $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. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. 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

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

      2. lift-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 3.1

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 3.1

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

   4. Applied rewrites 3.1%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   6. 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. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. *-rgt-identity N/A

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

      7. associate-*r/ N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 52.3

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

   7. Applied rewrites 52.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*r* N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lift-*.f64 N/A

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

      9. times-frac N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 77.5

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

   9. Applied rewrites 77.5%

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

4. Final simplification 93.3%

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

Alternative 2: 91.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(2.0 * N[(x * N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(x * x + N[(N[(y * y), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $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 99.5%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 99.2

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   6. 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. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. *-rgt-identity N/A

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

      7. associate-*r/ N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 98.4

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

   7. Applied rewrites 98.4%

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

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

   1. Initial program 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 93.5%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 98.9%

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

   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 76.1%

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

Alternative 3: 91.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(2.0 * N[(x * N[(x / N[(y * y), $MachinePrecision]), $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 99.5%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 99.2

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   6. 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. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. associate-/l* N/A

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

      6. *-rgt-identity N/A

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

      7. associate-*r/ N/A

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

      8. lower-*.f64 N/A

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

      9. associate-*r/ N/A

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

      10. *-rgt-identity N/A

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

      11. lower-/.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 98.4

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

   7. Applied rewrites 98.4%

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

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

   1. Initial program 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

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

   5. Applied rewrites 97.8%

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

   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 76.1%

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

Alternative 4: 91.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-*r/ N/A

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

      9. *-rgt-identity N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 97.3

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

   8. Applied rewrites 97.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. sub-neg N/A

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

      6. lower--.f64 97.3

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 97.3

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

   10. Applied rewrites 97.3%

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

   if -0.5 < (/.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

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

   5. Applied rewrites 97.8%

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

   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 76.1%

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

4. Final simplification 91.3%

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

Alternative 5: 91.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      1. unpow2 N/A

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

      2. lower-*.f64 97.3

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

   5. Applied rewrites 97.3%

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

   6. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. *-rgt-identity N/A

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

      6. associate-*r/ N/A

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

      7. lower-fma.f64 N/A

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

      8. associate-*r/ N/A

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

      9. *-rgt-identity N/A

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

      10. lower-/.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 97.3

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

   8. Applied rewrites 97.3%

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

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

   1. Initial program 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

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

   5. Applied rewrites 97.8%

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

   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 76.1%

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

Alternative 6: 91.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;t\_0 \leq -1 \cdot 10^{-309}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))))
   (if (<= t_0 -1e-309) -1.0 (if (<= t_0 INFINITY) 1.0 -1.0))))

C

double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (t_0 <= -1e-309) {
		tmp = -1.0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (t_0 <= -1e-309) {
		tmp = -1.0;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y))
	tmp = 0
	if t_0 <= -1e-309:
		tmp = -1.0
	elif t_0 <= math.inf:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)))
	tmp = 0.0
	if (t_0 <= -1e-309)
		tmp = -1.0;
	elseif (t_0 <= Inf)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	tmp = 0.0;
	if (t_0 <= -1e-309)
		tmp = -1.0;
	elseif (t_0 <= Inf)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1e-309], -1.0, If[LessEqual[t$95$0, Infinity], 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))) < -1.000000000000002e-309 or +inf.0 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 59.1%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 88.6%

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

   if -1.000000000000002e-309 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < +inf.0

   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

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

   5. Applied rewrites 97.8%

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

Alternative 7: 92.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(t$95$0 / N[(y * y + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -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))) < 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. lift-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. 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

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

   5. Applied rewrites 76.1%

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

Alternative 8: 91.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(x - y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / N[(x * x + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -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))) < 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

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

      2. lift-+.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 98.5

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 98.5

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

   4. Applied rewrites 98.5%

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

   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

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

   5. Applied rewrites 76.1%

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

4. Final simplification 92.0%

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

Alternative 9: 65.1% accurate, 36.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -1.0)

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

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

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 70.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 63.5%

   \[\leadsto \color{blue}{-1} \]
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 2024214 
(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))))