Kahan p9 Example

Percentage Accurate: 67.9% → 92.6%

Time: 7.6s

Alternatives: 8

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: 67.9% 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: 92.6% 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 100.0%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   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. associate-+r+ N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. associate--l+ N/A

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

      6. +-lft-identity N/A

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

      7. +-commutative N/A

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

      8. associate--r+ N/A

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

   5. Applied rewrites 79.9%

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

4. Final simplification 93.8%

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

Alternative 2: 92.1% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ y x) (- x y)) (+ (* y y) (* x x)))))
   (if (<= t_0 -0.5)
     (fma 2.0 (/ (* x x) (* y y)) -1.0)
     (if (<= t_0 2.0)
       (fma (/ (* -2.0 y) x) (/ y x) 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(2.0, ((x * x) / (y * y)), -1.0);
	} else if (t_0 <= 2.0) {
		tmp = fma(((-2.0 * y) / x), (y / x), 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(2.0, Float64(Float64(x * x) / Float64(y * y)), -1.0);
	elseif (t_0 <= 2.0)
		tmp = fma(Float64(Float64(-2.0 * y) / x), Float64(y / x), 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[(2.0 * N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(N[(-2.0 * y), $MachinePrecision] / x), $MachinePrecision] * N[(y / x), $MachinePrecision] + 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 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 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. associate-+r+ N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. associate--l+ N/A

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

      6. +-lft-identity N/A

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

      7. +-commutative N/A

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

      8. associate--r+ N/A

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. associate-*r/ N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 99.3

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

   5. Applied rewrites 99.3%

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

   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. associate-+r+ N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. associate--l+ N/A

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

      6. +-lft-identity N/A

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

      7. +-commutative N/A

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

      8. associate--r+ N/A

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

   5. Applied rewrites 79.9%

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

4. Final simplification 93.6%

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

Alternative 3: 92.1% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ y x) (- x y)) (+ (* y y) (* x x)))))
   (if (<= t_0 -0.5)
     (fma 2.0 (/ (* x x) (* y y)) -1.0)
     (if (<= t_0 2.0)
       (fma (* -2.0 y) (/ y (* x x)) 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(2.0, ((x * x) / (y * y)), -1.0);
	} else if (t_0 <= 2.0) {
		tmp = fma((-2.0 * y), (y / (x * x)), 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(2.0, Float64(Float64(x * x) / Float64(y * y)), -1.0);
	elseif (t_0 <= 2.0)
		tmp = fma(Float64(-2.0 * y), Float64(y / Float64(x * x)), 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[(2.0 * N[(N[(x * x), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[t$95$0, 2.0], N[(N[(-2.0 * y), $MachinePrecision] * N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision] + 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 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 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. associate-+r+ N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. associate--l+ N/A

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

      6. +-lft-identity N/A

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

      7. +-commutative N/A

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

      8. associate--r+ N/A

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. associate-*r/ N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 99.3

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

   5. Applied rewrites 99.3%

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

   7. Applied rewrites 99.3%

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

   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. associate-+r+ N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. associate--l+ N/A

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

      6. +-lft-identity N/A

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

      7. +-commutative N/A

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

      8. associate--r+ N/A

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

   5. Applied rewrites 79.9%

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

4. Final simplification 93.6%

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

Alternative 4: 91.5% accurate, 0.3× speedup

Math

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

FPCore

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. associate-+r+ N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. associate--l+ N/A

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

      6. +-lft-identity N/A

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

      7. +-commutative N/A

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

      8. associate--r+ N/A

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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. associate-*r/ N/A

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

      3. unpow2 N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 99.3

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

   5. Applied rewrites 99.3%

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

   7. Applied rewrites 99.3%

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

   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

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

   5. Applied rewrites 76.9%

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

4. Final simplification 92.7%

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

Alternative 5: 91.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 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 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. associate-+r+ N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. associate--l+ N/A

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

      6. +-lft-identity N/A

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

      7. +-commutative N/A

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

      8. associate--r+ N/A

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in 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

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

   5. Applied rewrites 76.9%

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

4. Final simplification 92.6%

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

Alternative 6: 91.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(y + x\right) \cdot \left(x - y\right)}{y \cdot y + x \cdot x}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-311}:\\ \;\;\;\;-1\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (+ y x) (- x y)) (+ (* y y) (* x x)))))
   (if (<= t_0 -2e-311) -1.0 (if (<= t_0 INFINITY) 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 <= -2e-311) {
		tmp = -1.0;
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = ((y + x) * (x - y)) / ((y * y) + (x * x));
	double tmp;
	if (t_0 <= -2e-311) {
		tmp = -1.0;
	} else if (t_0 <= Double.POSITIVE_INFINITY) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = ((y + x) * (x - y)) / ((y * y) + (x * x))
	tmp = 0
	if t_0 <= -2e-311:
		tmp = -1.0
	elif t_0 <= math.inf:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(Float64(y + x) * Float64(x - y)) / Float64(Float64(y * y) + Float64(x * x)))
	tmp = 0.0
	if (t_0 <= -2e-311)
		tmp = -1.0;
	elseif (t_0 <= Inf)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = ((y + x) * (x - y)) / ((y * y) + (x * x));
	tmp = 0.0;
	if (t_0 <= -2e-311)
		tmp = -1.0;
	elseif (t_0 <= Inf)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(y + x), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision] / N[(N[(y * y), $MachinePrecision] + N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e-311], -1.0, If[LessEqual[t$95$0, Infinity], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < -1.9999999999999e-311 or +inf.0 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y)))

   1. Initial program 57.5%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 90.0%

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

   if -1.9999999999999e-311 < (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < +inf.0

   1. Initial program 100.0%

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

   3. Taylor expanded in 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 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y + x\right) \cdot \left(x - y\right)}{y \cdot y + x \cdot x} \leq -2 \cdot 10^{-311}:\\ \;\;\;\;-1\\ \mathbf{elif}\;\frac{\left(y + x\right) \cdot \left(x - y\right)}{y \cdot y + x \cdot x} \leq \infty:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 98.8

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 98.8

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 98.8

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

   4. Applied rewrites 98.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in 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. associate-+r+ N/A

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

      2. distribute-lft1-in N/A

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

      3. metadata-eval N/A

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

      4. mul0-lft N/A

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

      5. associate--l+ N/A

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

      6. +-lft-identity N/A

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

      7. +-commutative N/A

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

      8. associate--r+ N/A

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

   5. Applied rewrites 79.9%

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

4. Final simplification 93.0%

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

Alternative 8: 66.6% 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 69.1%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 65.8%

   \[\leadsto \color{blue}{-1} \]
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 2024276 
(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))))