Kahan p9 Example

Percentage Accurate: 67.7% → 92.5%

Time: 6.9s

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: 67.7% 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: 92.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

   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 y around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

      7. associate--r+ N/A

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

      8. associate-*r/ N/A

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

      9. div-sub N/A

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

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

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

      11. metadata-eval N/A

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

      12. distribute-rgt1-in N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

   5. Applied rewrites 84.4%

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

   7. Applied rewrites 84.4%

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

4. Final simplification 94.5%

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

Alternative 2: 91.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(y + x), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision] / N[(N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(2.0 * x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(2.0 / y), $MachinePrecision] * N[(N[(x / y), $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 99.6%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

      7. associate--r+ N/A

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

      8. associate-*r/ N/A

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

      9. div-sub N/A

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

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

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

      11. metadata-eval N/A

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

      12. distribute-rgt1-in N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 98.7%

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

   9. Applied rewrites 98.7%

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

   11. Applied rewrites 98.7%

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

   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 y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. distribute-neg-frac N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

      7. associate--r+ N/A

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

      8. associate-*r/ N/A

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

      9. div-sub N/A

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

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

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

      11. metadata-eval N/A

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

      12. distribute-rgt1-in N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

   5. Applied rewrites 84.4%

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

4. Final simplification 94.0%

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

Alternative 3: 90.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(y + x), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision] / N[(N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], N[(N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision] * N[(2.0 * x), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x - y), $MachinePrecision]), $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 99.6%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

      7. associate--r+ N/A

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

      8. associate-*r/ N/A

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

      9. div-sub N/A

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

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

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

      11. metadata-eval N/A

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

      12. distribute-rgt1-in N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

   5. Applied rewrites 98.7%

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

   7. Applied rewrites 98.7%

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

   9. Applied rewrites 98.7%

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

   11. Applied rewrites 98.7%

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

   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 y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. distribute-neg-frac N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{{y}^{2}}} \]
   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 0.0

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

   5. Applied rewrites 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 3.1

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

   7. Applied rewrites 3.1%

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

   8. Taylor expanded in y around inf

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

      1. lower-/.f64 83.1

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

   10. Applied rewrites 83.1%

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

4. Final simplification 93.6%

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

Alternative 4: 90.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(y + x), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision] / N[(N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], -1.0, If[LessEqual[t$95$0, 2.0], N[(N[(N[(-2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision] * y + 1.0), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(x - y), $MachinePrecision]), $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 99.6%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 97.7%

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

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

   1. Initial program 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 y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

      8. metadata-eval N/A

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

      9. associate-*r/ N/A

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

      10. unpow2 N/A

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

      11. associate-*r* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. distribute-neg-frac N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{{y}^{2}}} \]
   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 0.0

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

   5. Applied rewrites 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 3.1

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

   7. Applied rewrites 3.1%

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

   8. Taylor expanded in y around inf

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

      1. lower-/.f64 83.1

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

   10. Applied rewrites 83.1%

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

4. Final simplification 93.1%

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

Alternative 5: 91.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(y + x), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision] / N[(N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.5], -1.0, If[LessEqual[t$95$0, 2.0], 1.0, N[(N[(1.0 / y), $MachinePrecision] * N[(x - y), $MachinePrecision]), $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 99.6%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 97.7%

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

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

   1. Initial program 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 y around 0

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

   5. Applied rewrites 99.0%

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

   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 y around inf

      \[\leadsto \frac{\left(x - y\right) \cdot \left(x + y\right)}{\color{blue}{{y}^{2}}} \]
   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 0.0

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

   5. Applied rewrites 0.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 3.1

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

   7. Applied rewrites 3.1%

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

   8. Taylor expanded in y around inf

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

      1. lower-/.f64 83.1

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

   10. Applied rewrites 83.1%

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

4. Final simplification 93.0%

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

Alternative 6: 91.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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 = ((y + x) * (x - y)) / ((y * y) + (x * x))
    if (t_0 <= (-0.5d0)) then
        tmp = -1.0d0
    else if (t_0 <= 2.0d0) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 56.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 91.4%

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

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

   1. Initial program 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 y around 0

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

   5. Applied rewrites 99.0%

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

4. Final simplification 93.0%

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

Alternative 7: 92.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

   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 y around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

      7. associate--r+ N/A

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

      8. associate-*r/ N/A

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

      9. div-sub N/A

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

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

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

      11. metadata-eval N/A

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

      12. distribute-rgt1-in N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

   5. Applied rewrites 84.4%

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

4. Final simplification 94.5%

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

Alternative 8: 92.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lower-fma.f64 99.7

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

   4. Applied rewrites 99.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate-+r+ N/A

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

      3. distribute-lft1-in N/A

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

      4. metadata-eval N/A

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

      5. mul0-lft N/A

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

      6. +-lft-identity N/A

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

      7. associate--r+ N/A

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

      8. associate-*r/ N/A

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

      9. div-sub N/A

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

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

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

      11. metadata-eval N/A

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

      12. distribute-rgt1-in N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

   5. Applied rewrites 84.4%

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

4. Final simplification 94.5%

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

Alternative 9: 66.9% 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 65.8%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 72.8%

   \[\leadsto \color{blue}{-1} \]
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 2024271 
(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))))