Kahan p9 Example

Percentage Accurate: 68.5% → 97.5%

Time: 7.7s

Alternatives: 6

Speedup: 2.5×

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: 68.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{\frac{y\_m}{x}}{\frac{x}{y\_m}}\\ t_1 := 1 + \frac{\frac{y\_m}{x}}{\frac{x}{y\_m \cdot 2}}\\ \mathbf{if}\;y\_m \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{t\_1 - t\_0 \cdot \left(4 \cdot t\_0\right)}{t\_1 \cdot t\_1}\\ \mathbf{elif}\;y\_m \leq 6.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{\left(x - y\_m\right) \cdot \left(y\_m + x\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (/ (/ y_m x) (/ x y_m)))
        (t_1 (+ 1.0 (/ (/ y_m x) (/ x (* y_m 2.0))))))
   (if (<= y_m 1.6e-162)
     (/ (- t_1 (* t_0 (* 4.0 t_0))) (* t_1 t_1))
     (if (<= y_m 6.5e-24)
       (/ (* (- x y_m) (+ y_m x)) (+ (* x x) (* y_m y_m)))
       -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = (y_m / x) / (x / y_m);
	double t_1 = 1.0 + ((y_m / x) / (x / (y_m * 2.0)));
	double tmp;
	if (y_m <= 1.6e-162) {
		tmp = (t_1 - (t_0 * (4.0 * t_0))) / (t_1 * t_1);
	} else if (y_m <= 6.5e-24) {
		tmp = ((x - y_m) * (y_m + x)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y_m / x) / (x / y_m)
    t_1 = 1.0d0 + ((y_m / x) / (x / (y_m * 2.0d0)))
    if (y_m <= 1.6d-162) then
        tmp = (t_1 - (t_0 * (4.0d0 * t_0))) / (t_1 * t_1)
    else if (y_m <= 6.5d-24) then
        tmp = ((x - y_m) * (y_m + x)) / ((x * x) + (y_m * y_m))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double t_0 = (y_m / x) / (x / y_m);
	double t_1 = 1.0 + ((y_m / x) / (x / (y_m * 2.0)));
	double tmp;
	if (y_m <= 1.6e-162) {
		tmp = (t_1 - (t_0 * (4.0 * t_0))) / (t_1 * t_1);
	} else if (y_m <= 6.5e-24) {
		tmp = ((x - y_m) * (y_m + x)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	t_0 = (y_m / x) / (x / y_m)
	t_1 = 1.0 + ((y_m / x) / (x / (y_m * 2.0)))
	tmp = 0
	if y_m <= 1.6e-162:
		tmp = (t_1 - (t_0 * (4.0 * t_0))) / (t_1 * t_1)
	elif y_m <= 6.5e-24:
		tmp = ((x - y_m) * (y_m + x)) / ((x * x) + (y_m * y_m))
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(Float64(y_m / x) / Float64(x / y_m))
	t_1 = Float64(1.0 + Float64(Float64(y_m / x) / Float64(x / Float64(y_m * 2.0))))
	tmp = 0.0
	if (y_m <= 1.6e-162)
		tmp = Float64(Float64(t_1 - Float64(t_0 * Float64(4.0 * t_0))) / Float64(t_1 * t_1));
	elseif (y_m <= 6.5e-24)
		tmp = Float64(Float64(Float64(x - y_m) * Float64(y_m + x)) / Float64(Float64(x * x) + Float64(y_m * y_m)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	t_0 = (y_m / x) / (x / y_m);
	t_1 = 1.0 + ((y_m / x) / (x / (y_m * 2.0)));
	tmp = 0.0;
	if (y_m <= 1.6e-162)
		tmp = (t_1 - (t_0 * (4.0 * t_0))) / (t_1 * t_1);
	elseif (y_m <= 6.5e-24)
		tmp = ((x - y_m) * (y_m + x)) / ((x * x) + (y_m * y_m));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(y$95$m / x), $MachinePrecision] / N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[(y$95$m / x), $MachinePrecision] / N[(x / N[(y$95$m * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$95$m, 1.6e-162], N[(N[(t$95$1 - N[(t$95$0 * N[(4.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 6.5e-24], N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(y$95$m + x), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.59999999999999988e-162

   1. Initial program 61.0%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 25.4%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{y \cdot \left(y \cdot -2\right)}{\color{blue}{x} \cdot x}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\left(y \cdot -2\right) \cdot y}{\color{blue}{x} \cdot x}\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{y \cdot -2}{x} \cdot \color{blue}{\frac{y}{x}}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{y \cdot -2}{x}\right), \color{blue}{\left(\frac{y}{x}\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(y \cdot -2\right), x\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, -2\right), x\right), \left(\frac{y}{x}\right)\right)\right) \]

      7. /-lowering-/.f64 34.3%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, -2\right), x\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right)\right) \]

   7. Applied egg-rr 34.3%

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

      1. flip-+ N/A

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

      2. metadata-eval N/A

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

      3. div-sub N/A

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

      4. frac-sub N/A

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

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(1 \cdot \left(1 - \frac{y \cdot -2}{x} \cdot \frac{y}{x}\right) - \left(1 - \frac{y \cdot -2}{x} \cdot \frac{y}{x}\right) \cdot \left(\left(\frac{y \cdot -2}{x} \cdot \frac{y}{x}\right) \cdot \left(\frac{y \cdot -2}{x} \cdot \frac{y}{x}\right)\right)\right), \color{blue}{\left(\left(1 - \frac{y \cdot -2}{x} \cdot \frac{y}{x}\right) \cdot \left(1 - \frac{y \cdot -2}{x} \cdot \frac{y}{x}\right)\right)}\right) \]

   9. Applied egg-rr 32.4%

      \[\leadsto \color{blue}{\frac{1 \cdot \left(1 + \frac{\frac{y}{x}}{\frac{x}{2 \cdot y}}\right) - \left(1 + \frac{\frac{y}{x}}{\frac{x}{2 \cdot y}}\right) \cdot \left(\left(4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\right) \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\right)}{\left(1 + \frac{\frac{y}{x}}{\frac{x}{2 \cdot y}}\right) \cdot \left(1 + \frac{\frac{y}{x}}{\frac{x}{2 \cdot y}}\right)}} \]

   10. Taylor expanded in y around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(1, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(2, y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \mathsf{/.f64}\left(x, y\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \mathsf{/.f64}\left(x, y\right)\right)\right)\right)\right), \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(2, y\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), \mathsf{/.f64}\left(x, \mathsf{*.f64}\left(2, y\right)\right)\right)\right)\right)\right) \]
   11. Step-by-step derivation

   12. Simplified 44.8%

       \[\leadsto \frac{1 \cdot \left(1 + \frac{\frac{y}{x}}{\frac{x}{2 \cdot y}}\right) - \color{blue}{1} \cdot \left(\left(4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\right) \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\right)}{\left(1 + \frac{\frac{y}{x}}{\frac{x}{2 \cdot y}}\right) \cdot \left(1 + \frac{\frac{y}{x}}{\frac{x}{2 \cdot y}}\right)} \]

   if 1.59999999999999988e-162 < y < 6.5e-24

   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 6.5e-24 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 100.0%

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

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{\left(1 + \frac{\frac{y}{x}}{\frac{x}{y \cdot 2}}\right) - \frac{\frac{y}{x}}{\frac{x}{y}} \cdot \left(4 \cdot \frac{\frac{y}{x}}{\frac{x}{y}}\right)}{\left(1 + \frac{\frac{y}{x}}{\frac{x}{y \cdot 2}}\right) \cdot \left(1 + \frac{\frac{y}{x}}{\frac{x}{y \cdot 2}}\right)}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(y + x\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.1% accurate, 0.4× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (/ (* (- x y_m) (+ y_m x)) (+ (* x x) (* y_m y_m)))))
   (if (<= t_0 2.0) t_0 (+ -1.0 (/ (* (/ x y_m) (* x 2.0)) y_m)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   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 x around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left({x}^{2} \cdot \left(2 \cdot \frac{1}{{y}^{2}}\right)\right)}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({x}^{2} \cdot \frac{2 \cdot 1}{\color{blue}{{y}^{2}}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({x}^{2} \cdot \frac{2}{{\color{blue}{y}}^{2}}\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} \cdot 2}{\color{blue}{{y}^{2}}}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{2 \cdot {x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{\left(1 + 1\right) \cdot {x}^{2}}{{y}^{2}}\right)\right) \]

      15. distribute-rgt1-in N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} + 1 \cdot {x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} + \left(\mathsf{neg}\left(-1\right)\right) \cdot {x}^{2}}{{y}^{2}}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} - -1 \cdot {x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} - -1 \cdot {x}^{2}}{y \cdot \color{blue}{y}}\right)\right) \]

      19. associate-/r* N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{\frac{{x}^{2} - -1 \cdot {x}^{2}}{y}}{\color{blue}{y}}\right)\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{{x}^{2} - -1 \cdot {x}^{2}}{y}\right), \color{blue}{y}\right)\right) \]

   5. Simplified 76.2%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{\left(2 \cdot x\right) \cdot x}{y}\right), y\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(2 \cdot x\right) \cdot \frac{x}{y}\right), y\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot x\right), \left(\frac{x}{y}\right)\right), y\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, x\right), \left(\frac{x}{y}\right)\right), y\right)\right) \]

      5. /-lowering-/.f64 77.8%

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, x\right), \mathsf{/.f64}\left(x, y\right)\right), y\right)\right) \]

   7. Applied egg-rr 77.8%

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

4. Final simplification 92.8%

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

Alternative 3: 84.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 2.9 \cdot 10^{-173}:\\ \;\;\;\;1 + \frac{y\_m}{x} \cdot \frac{y\_m \cdot -2}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{\frac{x}{y\_m} \cdot \left(x \cdot 2\right)}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 2.9e-173)
   (+ 1.0 (* (/ y_m x) (/ (* y_m -2.0) x)))
   (+ -1.0 (/ (* (/ x y_m) (* x 2.0)) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 2.9e-173) {
		tmp = 1.0 + ((y_m / x) * ((y_m * -2.0) / x));
	} else {
		tmp = -1.0 + (((x / y_m) * (x * 2.0)) / y_m);
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 2.9d-173) then
        tmp = 1.0d0 + ((y_m / x) * ((y_m * (-2.0d0)) / x))
    else
        tmp = (-1.0d0) + (((x / y_m) * (x * 2.0d0)) / y_m)
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 2.9e-173) {
		tmp = 1.0 + ((y_m / x) * ((y_m * -2.0) / x));
	} else {
		tmp = -1.0 + (((x / y_m) * (x * 2.0)) / y_m);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 2.9e-173:
		tmp = 1.0 + ((y_m / x) * ((y_m * -2.0) / x))
	else:
		tmp = -1.0 + (((x / y_m) * (x * 2.0)) / y_m)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 2.9e-173)
		tmp = Float64(1.0 + Float64(Float64(y_m / x) * Float64(Float64(y_m * -2.0) / x)));
	else
		tmp = Float64(-1.0 + Float64(Float64(Float64(x / y_m) * Float64(x * 2.0)) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 2.9e-173)
		tmp = 1.0 + ((y_m / x) * ((y_m * -2.0) / x));
	else
		tmp = -1.0 + (((x / y_m) * (x * 2.0)) / y_m);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 2.9e-173], N[(1.0 + N[(N[(y$95$m / x), $MachinePrecision] * N[(N[(y$95$m * -2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(N[(x / y$95$m), $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.8999999999999998e-173

   1. Initial program 61.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 25.2%

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

      1. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{y \cdot \left(y \cdot -2\right)}{\color{blue}{x} \cdot x}\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\left(y \cdot -2\right) \cdot y}{\color{blue}{x} \cdot x}\right)\right) \]

      3. times-frac N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{y \cdot -2}{x} \cdot \color{blue}{\frac{y}{x}}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{y \cdot -2}{x}\right), \color{blue}{\left(\frac{y}{x}\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(y \cdot -2\right), x\right), \left(\frac{\color{blue}{y}}{x}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, -2\right), x\right), \left(\frac{y}{x}\right)\right)\right) \]

      7. /-lowering-/.f64 34.1%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y, -2\right), x\right), \mathsf{/.f64}\left(y, \color{blue}{x}\right)\right)\right) \]

   7. Applied egg-rr 34.1%

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

   if 2.8999999999999998e-173 < y

   1. Initial program 97.8%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left({x}^{2} \cdot \left(2 \cdot \frac{1}{{y}^{2}}\right)\right)}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({x}^{2} \cdot \frac{2 \cdot 1}{\color{blue}{{y}^{2}}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({x}^{2} \cdot \frac{2}{{\color{blue}{y}}^{2}}\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} \cdot 2}{\color{blue}{{y}^{2}}}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{2 \cdot {x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{\left(1 + 1\right) \cdot {x}^{2}}{{y}^{2}}\right)\right) \]

      15. distribute-rgt1-in N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} + 1 \cdot {x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} + \left(\mathsf{neg}\left(-1\right)\right) \cdot {x}^{2}}{{y}^{2}}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} - -1 \cdot {x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} - -1 \cdot {x}^{2}}{y \cdot \color{blue}{y}}\right)\right) \]

      19. associate-/r* N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{\frac{{x}^{2} - -1 \cdot {x}^{2}}{y}}{\color{blue}{y}}\right)\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{{x}^{2} - -1 \cdot {x}^{2}}{y}\right), \color{blue}{y}\right)\right) \]

   5. Simplified 75.8%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{\left(2 \cdot x\right) \cdot x}{y}\right), y\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(2 \cdot x\right) \cdot \frac{x}{y}\right), y\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot x\right), \left(\frac{x}{y}\right)\right), y\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, x\right), \left(\frac{x}{y}\right)\right), y\right)\right) \]

      5. /-lowering-/.f64 75.8%

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, x\right), \mathsf{/.f64}\left(x, y\right)\right), y\right)\right) \]

   7. Applied egg-rr 75.8%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{-173}:\\ \;\;\;\;1 + \frac{y}{x} \cdot \frac{y \cdot -2}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{\frac{x}{y} \cdot \left(x \cdot 2\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 3.2 \cdot 10^{-173}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{\frac{x}{y\_m} \cdot \left(x \cdot 2\right)}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 3.2e-173) 1.0 (+ -1.0 (/ (* (/ x y_m) (* x 2.0)) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 3.2e-173) {
		tmp = 1.0;
	} else {
		tmp = -1.0 + (((x / y_m) * (x * 2.0)) / y_m);
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 3.2d-173) then
        tmp = 1.0d0
    else
        tmp = (-1.0d0) + (((x / y_m) * (x * 2.0d0)) / y_m)
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 3.2e-173) {
		tmp = 1.0;
	} else {
		tmp = -1.0 + (((x / y_m) * (x * 2.0)) / y_m);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 3.2e-173:
		tmp = 1.0
	else:
		tmp = -1.0 + (((x / y_m) * (x * 2.0)) / y_m)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 3.2e-173)
		tmp = 1.0;
	else
		tmp = Float64(-1.0 + Float64(Float64(Float64(x / y_m) * Float64(x * 2.0)) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 3.2e-173)
		tmp = 1.0;
	else
		tmp = -1.0 + (((x / y_m) * (x * 2.0)) / y_m);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 3.2e-173], 1.0, N[(-1.0 + N[(N[(N[(x / y$95$m), $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.2e-173

   1. Initial program 61.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 32.1%

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

   if 3.2e-173 < y

   1. Initial program 97.8%

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

   3. Taylor expanded in x around 0

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

      1. sub-neg N/A

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

      2. *-commutative N/A

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

      3. associate-*l/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. +-commutative N/A

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

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \color{blue}{\left({x}^{2} \cdot \left(2 \cdot \frac{1}{{y}^{2}}\right)\right)}\right) \]

      10. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({x}^{2} \cdot \frac{2 \cdot 1}{\color{blue}{{y}^{2}}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left({x}^{2} \cdot \frac{2}{{\color{blue}{y}}^{2}}\right)\right) \]

      12. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} \cdot 2}{\color{blue}{{y}^{2}}}\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{2 \cdot {x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{\left(1 + 1\right) \cdot {x}^{2}}{{y}^{2}}\right)\right) \]

      15. distribute-rgt1-in N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} + 1 \cdot {x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} + \left(\mathsf{neg}\left(-1\right)\right) \cdot {x}^{2}}{{y}^{2}}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} - -1 \cdot {x}^{2}}{{\color{blue}{y}}^{2}}\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{{x}^{2} - -1 \cdot {x}^{2}}{y \cdot \color{blue}{y}}\right)\right) \]

      19. associate-/r* N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \left(\frac{\frac{{x}^{2} - -1 \cdot {x}^{2}}{y}}{\color{blue}{y}}\right)\right) \]

      20. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{{x}^{2} - -1 \cdot {x}^{2}}{y}\right), \color{blue}{y}\right)\right) \]

   5. Simplified 75.8%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{\left(2 \cdot x\right) \cdot x}{y}\right), y\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\left(\left(2 \cdot x\right) \cdot \frac{x}{y}\right), y\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(2 \cdot x\right), \left(\frac{x}{y}\right)\right), y\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, x\right), \left(\frac{x}{y}\right)\right), y\right)\right) \]

      5. /-lowering-/.f64 75.8%

         \[\leadsto \mathsf{+.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(2, x\right), \mathsf{/.f64}\left(x, y\right)\right), y\right)\right) \]

   7. Applied egg-rr 75.8%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{-173}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{\frac{x}{y} \cdot \left(x \cdot 2\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 2.5 \cdot 10^{-173}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (if (<= y_m 2.5e-173) 1.0 -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 2.5e-173) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 2.5d-173) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 2.5e-173) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 2.5e-173:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 2.5e-173)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 2.5e-173)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 2.5e-173], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 2.5000000000000001e-173

   1. Initial program 61.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 32.1%

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

   if 2.5000000000000001e-173 < y

   1. Initial program 97.8%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 74.4%

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

Alternative 6: 66.2% accurate, 15.0× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ -1 \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 -1.0)

C

y_m = fabs(y);
double code(double x, double y_m) {
	return -1.0;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = -1.0d0
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	return -1.0;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	return -1.0

Julia

y_m = abs(y)
function code(x, y_m)
	return -1.0
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m)
	tmp = -1.0;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := -1.0

Derivation

1. Initial program 67.6%

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

3. Taylor expanded in x around 0

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

5. Simplified 68.9%

   \[\leadsto \color{blue}{-1} \]
6. Add Preprocessing

Developer Target 1: 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 2024158 
(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))))