Kahan p9 Example

Percentage Accurate: 68.4% → 99.9%

Time: 7.4s

Alternatives: 9

Speedup: 0.5×

Specification

\[\left(0 < x \land x < 1\right) \land y < 1\]

Math

\[\begin{array}{l} \\ \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))

C

double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x - y) * (x + y)) / ((x * x) + (y * y))
end function

Java

public static double code(double x, double y) {
	return ((x - y) * (x + y)) / ((x * x) + (y * y));
}

Python

def code(x, y):
	return ((x - y) * (x + y)) / ((x * x) + (y * y))

Julia

function code(x, y)
	return Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
end

Wolfram

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

Initial Program: 68.4% 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: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (/ (- x y) (hypot x y)) (/ (+ x y) (hypot x y))))

C

double code(double x, double y) {
	return ((x - y) / hypot(x, y)) * ((x + y) / hypot(x, y));
}

Java

public static double code(double x, double y) {
	return ((x - y) / Math.hypot(x, y)) * ((x + y) / Math.hypot(x, y));
}

Python

def code(x, y):
	return ((x - y) / math.hypot(x, y)) * ((x + y) / math.hypot(x, y))

Julia

function code(x, y)
	return Float64(Float64(Float64(x - y) / hypot(x, y)) * Float64(Float64(x + y) / hypot(x, y)))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x - y) / hypot(x, y)) * ((x + y) / hypot(x, y));
end

Wolfram

code[x_, y_] := N[(N[(N[(x - y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 66.0%

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

   1. add-sqr-sqrt 66.0%

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

   2. times-frac 66.5%

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

   3. hypot-def 66.5%

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

   4. hypot-def 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 92.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   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. Taylor expanded in x around 0 51.7%

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

      1. unpow2 51.7%

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

      2. unpow2 51.7%

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

      3. associate-*r/ 51.7%

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

      4. times-frac 76.8%

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

      5. unpow2 76.8%

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

      6. fma-neg 76.8%

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

      7. unpow2 76.8%

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

      8. associate-/l* 78.9%

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

      9. metadata-eval 78.9%

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

   4. Simplified 78.9%

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

      1. fma-udef 78.9%

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

      2. div-inv 78.9%

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

      3. clear-num 78.9%

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

   6. Applied egg-rr 78.9%

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

4. Final simplification 92.8%

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

Alternative 3: 92.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], t$95$0, N[(N[(x - y), $MachinePrecision] / N[(y + N[(N[(2.0 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $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 100.0%

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

   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. Step-by-step derivation

      1. associate-/l* 3.1%

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

      2. +-commutative 3.1%

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

      3. remove-double-neg 3.1%

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

      4. sub-neg 3.1%

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

      5. +-commutative 3.1%

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

      6. fma-def 3.1%

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

      7. sub-neg 3.1%

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

      8. remove-double-neg 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in x around 0 76.8%

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

      1. +-commutative 76.8%

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

      2. mul-1-neg 76.8%

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

      3. unsub-neg 76.8%

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

      4. unpow2 76.8%

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

      5. associate-/l* 78.2%

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

   6. Simplified 78.2%

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

4. Final simplification 92.6%

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

Alternative 4: 43.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-163}:\\ \;\;\;\;1 + \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.8e-163)
   (+ 1.0 (* (* (/ y x) (/ y x)) -2.0))
   (+ -1.0 (* (/ x y) (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.8e-163) {
		tmp = 1.0 + (((y / x) * (y / x)) * -2.0);
	} else {
		tmp = -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) :: tmp
    if (y <= 2.8d-163) then
        tmp = 1.0d0 + (((y / x) * (y / x)) * (-2.0d0))
    else
        tmp = (-1.0d0) + ((x / y) * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.8e-163) {
		tmp = 1.0 + (((y / x) * (y / x)) * -2.0);
	} else {
		tmp = -1.0 + ((x / y) * (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.8e-163:
		tmp = 1.0 + (((y / x) * (y / x)) * -2.0)
	else:
		tmp = -1.0 + ((x / y) * (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.8e-163)
		tmp = Float64(1.0 + Float64(Float64(Float64(y / x) * Float64(y / x)) * -2.0));
	else
		tmp = Float64(-1.0 + Float64(Float64(x / y) * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.8e-163)
		tmp = 1.0 + (((y / x) * (y / x)) * -2.0);
	else
		tmp = -1.0 + ((x / y) * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.8e-163], N[(1.0 + N[(N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.8e-163

   1. Initial program 60.0%

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

      1. add-sqr-sqrt 60.0%

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

      2. times-frac 60.8%

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

      3. hypot-def 60.8%

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

      4. hypot-def 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in y around 0 26.7%

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

      1. *-commutative 26.7%

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

      2. unpow2 26.7%

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

      3. unpow2 26.7%

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

   6. Simplified 26.7%

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

      1. frac-times 36.0%

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

   8. Applied egg-rr 36.0%

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

   if 2.8e-163 < y

   1. Initial program 97.5%

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

   2. Taylor expanded in x around 0 77.3%

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

      1. unpow2 77.3%

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

   4. Simplified 77.3%

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

   5. Taylor expanded in x around 0 77.3%

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

      1. sub-neg 77.3%

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

      2. unpow2 77.3%

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

      3. unpow2 77.3%

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

      4. metadata-eval 77.3%

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

   7. Simplified 77.3%

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

      1. times-frac 79.7%

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

   9. Applied egg-rr 79.7%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-163}:\\ \;\;\;\;1 + \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 5: 43.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-163}:\\ \;\;\;\;1 + \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y} \cdot \left(x \cdot \frac{x}{y}\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 2.4e-163)
   (+ 1.0 (* (* (/ y x) (/ y x)) -2.0))
   (+ (* (/ 2.0 y) (* x (/ x y))) -1.0)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 2.4e-163) {
		tmp = 1.0 + (((y / x) * (y / x)) * -2.0);
	} else {
		tmp = ((2.0 / y) * (x * (x / y))) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 2.4d-163) then
        tmp = 1.0d0 + (((y / x) * (y / x)) * (-2.0d0))
    else
        tmp = ((2.0d0 / y) * (x * (x / y))) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 2.4e-163) {
		tmp = 1.0 + (((y / x) * (y / x)) * -2.0);
	} else {
		tmp = ((2.0 / y) * (x * (x / y))) + -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 2.4e-163:
		tmp = 1.0 + (((y / x) * (y / x)) * -2.0)
	else:
		tmp = ((2.0 / y) * (x * (x / y))) + -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 2.4e-163)
		tmp = Float64(1.0 + Float64(Float64(Float64(y / x) * Float64(y / x)) * -2.0));
	else
		tmp = Float64(Float64(Float64(2.0 / y) * Float64(x * Float64(x / y))) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.4e-163)
		tmp = 1.0 + (((y / x) * (y / x)) * -2.0);
	else
		tmp = ((2.0 / y) * (x * (x / y))) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 2.4e-163], N[(1.0 + N[(N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 / y), $MachinePrecision] * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.4000000000000001e-163

   1. Initial program 60.0%

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

      1. add-sqr-sqrt 60.0%

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

      2. times-frac 60.8%

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

      3. hypot-def 60.8%

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

      4. hypot-def 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in y around 0 26.7%

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

      1. *-commutative 26.7%

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

      2. unpow2 26.7%

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

      3. unpow2 26.7%

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

   6. Simplified 26.7%

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

      1. frac-times 36.0%

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

   8. Applied egg-rr 36.0%

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

   if 2.4000000000000001e-163 < y

   1. Initial program 97.5%

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

   2. Taylor expanded in x around 0 77.2%

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

      1. unpow2 77.2%

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

      2. unpow2 77.2%

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

      3. associate-*r/ 77.2%

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

      4. times-frac 79.7%

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

      5. unpow2 79.7%

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

      6. fma-neg 79.7%

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

      7. unpow2 79.7%

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

      8. associate-/l* 79.7%

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

      9. metadata-eval 79.7%

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

   4. Simplified 79.7%

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

      1. fma-udef 79.7%

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

      2. div-inv 79.7%

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

      3. clear-num 79.7%

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

   6. Applied egg-rr 79.7%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-163}:\\ \;\;\;\;1 + \left(\frac{y}{x} \cdot \frac{y}{x}\right) \cdot -2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{y} \cdot \left(x \cdot \frac{x}{y}\right) + -1\\ \end{array} \]

Alternative 6: 42.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-179}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.4e-179) 1.0 (+ -1.0 (* (/ x y) (/ x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.4e-179) {
		tmp = 1.0;
	} else {
		tmp = -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) :: tmp
    if (y <= 4.4d-179) then
        tmp = 1.0d0
    else
        tmp = (-1.0d0) + ((x / y) * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.4e-179) {
		tmp = 1.0;
	} else {
		tmp = -1.0 + ((x / y) * (x / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.4e-179:
		tmp = 1.0
	else:
		tmp = -1.0 + ((x / y) * (x / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.4e-179)
		tmp = 1.0;
	else
		tmp = Float64(-1.0 + Float64(Float64(x / y) * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.4e-179)
		tmp = 1.0;
	else
		tmp = -1.0 + ((x / y) * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.4e-179], 1.0, N[(-1.0 + N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.40000000000000009e-179

   1. Initial program 60.5%

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

   2. Taylor expanded in x around inf 33.6%

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

   if 4.40000000000000009e-179 < y

   1. Initial program 91.3%

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

   2. Taylor expanded in x around 0 69.0%

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

      1. unpow2 69.0%

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

   4. Simplified 69.0%

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

   5. Taylor expanded in x around 0 69.0%

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

      1. sub-neg 69.0%

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

      2. unpow2 69.0%

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

      3. unpow2 69.0%

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

      4. metadata-eval 69.0%

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

   7. Simplified 69.0%

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

      1. times-frac 77.7%

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

   9. Applied egg-rr 77.7%

      \[\leadsto \color{blue}{\frac{x}{y} \cdot \frac{x}{y}} + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-179}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 7: 41.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-163}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 3.5e-163) (/ (- x y) x) -1.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.5e-163) {
		tmp = (x - y) / x;
	} 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) :: tmp
    if (y <= 3.5d-163) then
        tmp = (x - y) / x
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.5e-163) {
		tmp = (x - y) / x;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.5e-163:
		tmp = (x - y) / x
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.5e-163)
		tmp = Float64(Float64(x - y) / x);
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.5e-163)
		tmp = (x - y) / x;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.5e-163], N[(N[(x - y), $MachinePrecision] / x), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 3.50000000000000027e-163

   1. Initial program 60.0%

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

      1. associate-/l* 60.7%

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

      2. +-commutative 60.7%

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

      3. remove-double-neg 60.7%

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

      4. sub-neg 60.7%

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

      5. +-commutative 60.7%

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

      6. fma-def 60.7%

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

      7. sub-neg 60.7%

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

      8. remove-double-neg 60.7%

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

   3. Simplified 60.7%

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

   4. Taylor expanded in x around inf 33.1%

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

   if 3.50000000000000027e-163 < y

   1. Initial program 97.5%

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

   2. Taylor expanded in x around 0 78.4%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-163}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 8: 41.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-163}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 4.6e-163) 1.0 -1.0))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.6e-163) {
		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) :: tmp
    if (y <= 4.6d-163) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.6e-163) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.6e-163:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.6e-163)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.6e-163)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.6e-163], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 4.5999999999999999e-163

   1. Initial program 60.0%

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

   2. Taylor expanded in x around inf 33.7%

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

   if 4.5999999999999999e-163 < y

   1. Initial program 97.5%

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

   2. Taylor expanded in x around 0 78.4%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.6 \cdot 10^{-163}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 9: 66.3% accurate, 15.0× 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 66.0%

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

2. Taylor expanded in x around 0 67.9%

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

3. Final simplification 67.9%

   \[\leadsto -1 \]

Developer target: 99.9% accurate, 0.1× 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 2023271 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

  :herbie-target
  (if (and (< 0.5 (fabs (/ x y))) (< (fabs (/ x y)) 2.0)) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))

  (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))))