Kahan p9 Example

Percentage Accurate: 67.3% → 92.1%

Time: 6.6s

Alternatives: 6

Speedup: 0.7×

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: 67.3% 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: 92.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.02 \cdot 10^{-162}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(-2, \frac{\frac{y\_m}{x} \cdot y\_m}{x}, 1\right)}}\\ \mathbf{elif}\;y\_m \leq 10^{-12}:\\ \;\;\;\;\frac{\left(x + y\_m\right) \cdot \left(x - y\_m\right)}{y\_m \cdot y\_m + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.02e-162)
   (/ 1.0 (/ 1.0 (fma -2.0 (/ (* (/ y_m x) y_m) x) 1.0)))
   (if (<= y_m 1e-12)
     (/ (* (+ x y_m) (- x y_m)) (+ (* y_m y_m) (* x x)))
     -1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.02e-162) {
		tmp = 1.0 / (1.0 / fma(-2.0, (((y_m / x) * y_m) / x), 1.0));
	} else if (y_m <= 1e-12) {
		tmp = ((x + y_m) * (x - y_m)) / ((y_m * y_m) + (x * x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.02e-162)
		tmp = Float64(1.0 / Float64(1.0 / fma(-2.0, Float64(Float64(Float64(y_m / x) * y_m) / x), 1.0)));
	elseif (y_m <= 1e-12)
		tmp = Float64(Float64(Float64(x + y_m) * Float64(x - y_m)) / Float64(Float64(y_m * y_m) + Float64(x * x)));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.02e-162], N[(1.0 / N[(1.0 / N[(-2.0 * N[(N[(N[(y$95$m / x), $MachinePrecision] * y$95$m), $MachinePrecision] / x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1e-12], N[(N[(N[(x + y$95$m), $MachinePrecision] * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 1.01999999999999998e-162

   1. Initial program 65.1%

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

   3. Taylor expanded in x around 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 43.1%

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

   7. Applied rewrites 43.1%

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

   9. Applied rewrites 43.1%

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

   if 1.01999999999999998e-162 < y < 9.9999999999999998e-13

   1. Initial program 100.0%

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

   if 9.9999999999999998e-13 < y

   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

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

   5. Applied rewrites 100.0%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{-162}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left(-2, \frac{\frac{y}{x} \cdot y}{x}, 1\right)}}\\ \mathbf{elif}\;y \leq 10^{-12}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{y \cdot y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.0% accurate, 0.3× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (+ x y_m) (- x y_m)) (+ (* y_m y_m) (* x x)))))
   (if (<= t_0 -0.5)
     (fma 2.0 (/ (* x x) (* y_m y_m)) -1.0)
     (if (<= t_0 2.0) 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 = ((x + y_m) * (x - y_m)) / ((y_m * y_m) + (x * x));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = fma(2.0, ((x * x) / (y_m * y_m)), -1.0);
	} else if (t_0 <= 2.0) {
		tmp = 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(x + y_m) * Float64(x - y_m)) / Float64(Float64(y_m * y_m) + Float64(x * x)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = fma(2.0, Float64(Float64(x * x) / Float64(y_m * y_m)), -1.0);
	elseif (t_0 <= 2.0)
		tmp = 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[(x + y$95$m), $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[(2.0 * N[(N[(x * x), $MachinePrecision] / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], 1.0, 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 100.0%

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

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

   1. Initial program 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 100.0%

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

   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

      \[\leadsto \color{blue}{2 \cdot \frac{{x}^{2}}{{y}^{2}} - 1} \]
   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 69.3

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

   5. Applied rewrites 69.3%

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

4. Final simplification 91.0%

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

Alternative 3: 91.5% accurate, 0.4× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (+ x y_m) (- x y_m)) (+ (* y_m y_m) (* x x)))))
   (if (<= t_0 -0.5)
     (fma 2.0 (/ (* x 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 = ((x + y_m) * (x - y_m)) / ((y_m * y_m) + (x * x));
	double tmp;
	if (t_0 <= -0.5) {
		tmp = fma(2.0, ((x * 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(x + y_m) * Float64(x - y_m)) / Float64(Float64(y_m * y_m) + Float64(x * x)))
	tmp = 0.0
	if (t_0 <= -0.5)
		tmp = fma(2.0, Float64(Float64(x * 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[(x + y$95$m), $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[(2.0 * N[(N[(x * 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 100.0%

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

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

   1. Initial program 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 100.0%

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

   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 67.0%

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

4. Final simplification 90.3%

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

Alternative 4: 91.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(Float64(Float64(x + y_m) * 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 <= Inf)
		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 = ((x + y_m) * (x - y_m)) / ((y_m * y_m) + (x * x));
	tmp = 0.0;
	if (t_0 <= -0.5)
		tmp = -1.0;
	elseif (t_0 <= Inf)
		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[(x + y$95$m), $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, 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))) < -0.5 or +inf.0 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 59.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 86.4%

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

   if -0.5 < (/.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 100.0%

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

4. Final simplification 90.2%

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

Alternative 5: 92.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.02 \cdot 10^{-162}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-2 \cdot y\_m}{x}, \frac{y\_m}{x}, 1\right)\\ \mathbf{elif}\;y\_m \leq 10^{-12}:\\ \;\;\;\;\frac{\left(x + y\_m\right) \cdot \left(x - y\_m\right)}{y\_m \cdot y\_m + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.02e-162)
   (fma (/ (* -2.0 y_m) x) (/ y_m x) 1.0)
   (if (<= y_m 1e-12)
     (/ (* (+ x y_m) (- x y_m)) (+ (* y_m y_m) (* x x)))
     -1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.02e-162) {
		tmp = fma(((-2.0 * y_m) / x), (y_m / x), 1.0);
	} else if (y_m <= 1e-12) {
		tmp = ((x + y_m) * (x - y_m)) / ((y_m * y_m) + (x * x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.02e-162)
		tmp = fma(Float64(Float64(-2.0 * y_m) / x), Float64(y_m / x), 1.0);
	elseif (y_m <= 1e-12)
		tmp = Float64(Float64(Float64(x + y_m) * Float64(x - y_m)) / Float64(Float64(y_m * y_m) + Float64(x * x)));
	else
		tmp = -1.0;
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.02e-162], N[(N[(N[(-2.0 * y$95$m), $MachinePrecision] / x), $MachinePrecision] * N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[y$95$m, 1e-12], N[(N[(N[(x + y$95$m), $MachinePrecision] * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(y$95$m * y$95$m), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 1.01999999999999998e-162

   1. Initial program 65.1%

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

   3. Taylor expanded in x around 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 43.1%

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

   if 1.01999999999999998e-162 < y < 9.9999999999999998e-13

   1. Initial program 100.0%

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

   if 9.9999999999999998e-13 < y

   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

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

   5. Applied rewrites 100.0%

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

4. Final simplification 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.02 \cdot 10^{-162}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-2 \cdot y}{x}, \frac{y}{x}, 1\right)\\ \mathbf{elif}\;y \leq 10^{-12}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{y \cdot y + x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.4% 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 70.7%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 62.6%

   \[\leadsto \color{blue}{-1} \]
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 2024288 
(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))))