Kahan p9 Example

Percentage Accurate: 67.5% → 92.0%

Time: 12.7s

Alternatives: 7

Speedup: 5.1×

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: 67.5% 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: 92.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{y\_m + x}{x}}{\frac{x}{x - y\_m}}\\ \mathbf{elif}\;y\_m \leq 1.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\left(y\_m + x\right) \cdot \left(x - y\_m\right)}{\mathsf{fma}\left(y\_m, y\_m, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.6e-162)
   (/ (/ (+ y_m x) x) (/ x (- x y_m)))
   (if (<= y_m 1.5e-25)
     (/ (* (+ y_m x) (- x y_m)) (fma 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 <= 1.6e-162) {
		tmp = ((y_m + x) / x) / (x / (x - y_m));
	} else if (y_m <= 1.5e-25) {
		tmp = ((y_m + x) * (x - y_m)) / fma(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 <= 1.6e-162)
		tmp = Float64(Float64(Float64(y_m + x) / x) / Float64(x / Float64(x - y_m)));
	elseif (y_m <= 1.5e-25)
		tmp = Float64(Float64(Float64(y_m + x) * Float64(x - y_m)) / fma(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, 1.6e-162], N[(N[(N[(y$95$m + x), $MachinePrecision] / x), $MachinePrecision] / N[(x / N[(x - y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1.5e-25], N[(N[(N[(y$95$m + x), $MachinePrecision] * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * y$95$m + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 1.59999999999999988e-162

   1. Initial program 65.5%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 26.8

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

   5. Applied rewrites 26.8%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. *-commutative N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

      7. lower-/.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 39.1

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

   7. Applied rewrites 39.1%

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

   if 1.59999999999999988e-162 < y < 1.4999999999999999e-25

   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. 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 100.0

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

   4. Applied rewrites 100.0%

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

   if 1.4999999999999999e-25 < 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 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{y + x}{x}}{\frac{x}{x - y}}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot \left(x - y\right)}{\mathsf{fma}\left(y, y, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\mathsf{fma}\left(-\frac{y\_m}{x}, \frac{y\_m}{x}, 1\right)\\ \mathbf{elif}\;y\_m \leq 1.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\left(y\_m + x\right) \cdot \left(x - y\_m\right)}{\mathsf{fma}\left(y\_m, y\_m, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.6e-162)
   (fma (- (/ y_m x)) (/ y_m x) 1.0)
   (if (<= y_m 1.5e-25)
     (/ (* (+ y_m x) (- x y_m)) (fma 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 <= 1.6e-162) {
		tmp = fma(-(y_m / x), (y_m / x), 1.0);
	} else if (y_m <= 1.5e-25) {
		tmp = ((y_m + x) * (x - y_m)) / fma(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 <= 1.6e-162)
		tmp = fma(Float64(-Float64(y_m / x)), Float64(y_m / x), 1.0);
	elseif (y_m <= 1.5e-25)
		tmp = Float64(Float64(Float64(y_m + x) * Float64(x - y_m)) / fma(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, 1.6e-162], N[((-N[(y$95$m / x), $MachinePrecision]) * N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[y$95$m, 1.5e-25], N[(N[(N[(y$95$m + x), $MachinePrecision] * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision] / N[(y$95$m * y$95$m + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 1.59999999999999988e-162

   1. Initial program 65.5%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 26.8

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

   5. Applied rewrites 26.8%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto 1 + \left(\color{blue}{0} \cdot \frac{y}{x} + -1 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]

      5. mul0-lft N/A

         \[\leadsto 1 + \left(\color{blue}{0} + -1 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]

      6. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(1 + 0\right) + -1 \cdot \frac{{y}^{2}}{{x}^{2}}} \]

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 27.0

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

   8. Applied rewrites 27.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lift-neg.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 39.1

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

   10. Applied rewrites 39.1%

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

   if 1.59999999999999988e-162 < y < 1.4999999999999999e-25

   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. 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 100.0

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

   4. Applied rewrites 100.0%

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

   if 1.4999999999999999e-25 < 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 51.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\mathsf{fma}\left(-\frac{y}{x}, \frac{y}{x}, 1\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{\left(y + x\right) \cdot \left(x - y\right)}{\mathsf{fma}\left(y, y, x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 5.2 \cdot 10^{-127}:\\ \;\;\;\;\mathsf{fma}\left(-\frac{y\_m}{x}, \frac{y\_m}{x}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{2}{y\_m \cdot y\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 5.2e-127)
   (fma (- (/ y_m x)) (/ y_m x) 1.0)
   (fma (* x x) (/ 2.0 (* y_m y_m)) -1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 5.2e-127) {
		tmp = fma(-(y_m / x), (y_m / x), 1.0);
	} else {
		tmp = fma((x * x), (2.0 / (y_m * y_m)), -1.0);
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 5.2e-127)
		tmp = fma(Float64(-Float64(y_m / x)), Float64(y_m / x), 1.0);
	else
		tmp = fma(Float64(x * x), Float64(2.0 / Float64(y_m * y_m)), -1.0);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 5.2e-127], N[((-N[(y$95$m / x), $MachinePrecision]) * N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * N[(2.0 / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.19999999999999982e-127

   1. Initial program 66.8%

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

   3. Taylor expanded in x around inf

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

      1. unpow2 N/A

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

      2. lower-*.f64 28.6

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

   5. Applied rewrites 28.6%

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

   6. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

         \[\leadsto 1 + \left(\color{blue}{0} \cdot \frac{y}{x} + -1 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]

      5. mul0-lft N/A

         \[\leadsto 1 + \left(\color{blue}{0} + -1 \cdot \frac{{y}^{2}}{{x}^{2}}\right) \]

      6. associate-+r+ N/A

         \[\leadsto \color{blue}{\left(1 + 0\right) + -1 \cdot \frac{{y}^{2}}{{x}^{2}}} \]

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. mul-1-neg N/A

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

      10. unpow2 N/A

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

      11. associate-/l* N/A

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

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

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

      13. mul-1-neg N/A

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

      14. lower-fma.f64 N/A

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

      15. mul-1-neg N/A

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

      16. lower-neg.f64 N/A

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

      17. lower-/.f64 N/A

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

      18. unpow2 N/A

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

      19. lower-*.f64 28.9

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

   8. Applied rewrites 28.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. times-frac N/A

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

      8. lift-neg.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. lift-/.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. lift-/.f64 N/A

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

      14. distribute-neg-frac2 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-neg.f64 40.4

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

   10. Applied rewrites 40.4%

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

   if 5.19999999999999982e-127 < 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

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

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

      3. metadata-eval N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 84.1

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

   5. Applied rewrites 84.1%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.2 \cdot 10^{-127}:\\ \;\;\;\;\mathsf{fma}\left(-\frac{y}{x}, \frac{y}{x}, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{2}{y \cdot y}, -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 5.2 \cdot 10^{-127}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \frac{2}{y\_m \cdot y\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 5.2e-127) 1.0 (fma (* x x) (/ 2.0 (* y_m y_m)) -1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 5.2e-127) {
		tmp = 1.0;
	} else {
		tmp = fma((x * x), (2.0 / (y_m * y_m)), -1.0);
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 5.2e-127)
		tmp = 1.0;
	else
		tmp = fma(Float64(x * x), Float64(2.0 / Float64(y_m * y_m)), -1.0);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 5.2e-127], 1.0, N[(N[(x * x), $MachinePrecision] * N[(2.0 / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.19999999999999982e-127

   1. Initial program 66.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 39.0%

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

   if 5.19999999999999982e-127 < 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

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

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

      3. metadata-eval N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. associate-*r/ N/A

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

      12. metadata-eval N/A

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

      13. lower-/.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 84.1

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

   5. Applied rewrites 84.1%

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

Alternative 5: 82.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 5.2 \cdot 10^{-127}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{x}{y\_m \cdot y\_m}, -1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 5.2e-127) 1.0 (fma x (/ x (* y_m y_m)) -1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 5.2e-127) {
		tmp = 1.0;
	} else {
		tmp = fma(x, (x / (y_m * y_m)), -1.0);
	}
	return tmp;
}

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 5.2e-127)
		tmp = 1.0;
	else
		tmp = fma(x, Float64(x / Float64(y_m * y_m)), -1.0);
	end
	return tmp
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 5.2e-127], 1.0, N[(x * N[(x / N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.19999999999999982e-127

   1. Initial program 66.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 39.0%

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

   if 5.19999999999999982e-127 < 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

      \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{{y}^{2}}} \]
   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 83.3

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

   5. Applied rewrites 83.3%

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

   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. unpow2 N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 83.3

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

   8. Applied rewrites 83.3%

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

Alternative 6: 82.2% accurate, 5.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 5.2 \cdot 10^{-127}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (if (<= y_m 5.2e-127) 1.0 -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 5.2e-127) {
		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) :: tmp
    if (y_m <= 5.2d-127) 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 tmp;
	if (y_m <= 5.2e-127) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 5.2e-127:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 5.2e-127)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 5.2e-127)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 5.2e-127], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 5.19999999999999982e-127

   1. Initial program 66.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 39.0%

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

   if 5.19999999999999982e-127 < 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 82.4%

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

Alternative 7: 66.8% 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 72.3%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 65.3%

   \[\leadsto \color{blue}{-1} \]
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 2024219 
(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))))