Kahan p9 Example

Percentage Accurate: 68.2% → 100.0%

Time: 9.3s

Alternatives: 13

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: 68.2% 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: 100.0% accurate, 0.1× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (/ (/ (- x y_m) (hypot x y_m)) (/ (hypot x y_m) (+ x y_m))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	return ((x - y_m) / hypot(x, y_m)) / (hypot(x, y_m) / (x + y_m));
}

Java

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

Python

y_m = math.fabs(y)
def code(x, y_m):
	return ((x - y_m) / math.hypot(x, y_m)) / (math.hypot(x, y_m) / (x + y_m))

Julia

y_m = abs(y)
function code(x, y_m)
	return Float64(Float64(Float64(x - y_m) / hypot(x, y_m)) / Float64(hypot(x, y_m) / Float64(x + y_m)))
end

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[(N[(x - y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision] / N[(x + y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.9%

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

   1. associate-/l* 70.1%

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

   2. +-commutative 70.1%

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

   3. fma-define 70.1%

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

3. Simplified 70.1%

   \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r/ 69.9%

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

   2. fma-undefine 69.9%

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

   3. +-commutative 69.9%

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

   4. add-sqr-sqrt 69.9%

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

   5. times-frac 70.4%

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

   6. hypot-define 70.4%

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

   7. hypot-define 100.0%

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

6. Applied egg-rr 100.0%

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

   1. clear-num 99.9%

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

   2. un-div-inv 100.0%

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

8. Applied egg-rr 100.0%

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

Alternative 2: 92.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 8.5 \cdot 10^{-163}:\\ \;\;\;\;\frac{x - y\_m}{x + y\_m \cdot \left(2 \cdot \frac{y\_m}{x} + -1\right)}\\ \mathbf{elif}\;y\_m \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\left(x - y\_m\right) \cdot \left(x + y\_m\right)}{{\left(\mathsf{hypot}\left(x, y\_m\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 8.5e-163)
   (/ (- x y_m) (+ x (* y_m (+ (* 2.0 (/ y_m x)) -1.0))))
   (if (<= y_m 5e-23)
     (/ (* (- x y_m) (+ x y_m)) (pow (hypot x y_m) 2.0))
     -1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 8.5e-163) {
		tmp = (x - y_m) / (x + (y_m * ((2.0 * (y_m / x)) + -1.0)));
	} else if (y_m <= 5e-23) {
		tmp = ((x - y_m) * (x + y_m)) / pow(hypot(x, y_m), 2.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 8.5e-163) {
		tmp = (x - y_m) / (x + (y_m * ((2.0 * (y_m / x)) + -1.0)));
	} else if (y_m <= 5e-23) {
		tmp = ((x - y_m) * (x + y_m)) / Math.pow(Math.hypot(x, y_m), 2.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 8.5e-163:
		tmp = (x - y_m) / (x + (y_m * ((2.0 * (y_m / x)) + -1.0)))
	elif y_m <= 5e-23:
		tmp = ((x - y_m) * (x + y_m)) / math.pow(math.hypot(x, y_m), 2.0)
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 8.5e-163)
		tmp = Float64(Float64(x - y_m) / Float64(x + Float64(y_m * Float64(Float64(2.0 * Float64(y_m / x)) + -1.0))));
	elseif (y_m <= 5e-23)
		tmp = Float64(Float64(Float64(x - y_m) * Float64(x + y_m)) / (hypot(x, y_m) ^ 2.0));
	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 <= 8.5e-163)
		tmp = (x - y_m) / (x + (y_m * ((2.0 * (y_m / x)) + -1.0)));
	elseif (y_m <= 5e-23)
		tmp = ((x - y_m) * (x + y_m)) / (hypot(x, y_m) ^ 2.0);
	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, 8.5e-163], N[(N[(x - y$95$m), $MachinePrecision] / N[(x + N[(y$95$m * N[(N[(2.0 * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 5e-23], N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(x + y$95$m), $MachinePrecision]), $MachinePrecision] / N[Power[N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 8.5e-163

   1. Initial program 63.8%

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

      1. associate-/l* 64.2%

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

      2. +-commutative 64.2%

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

      3. fma-define 64.2%

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

   3. Simplified 64.2%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 64.3%

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

      2. un-div-inv 64.4%

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

      3. fma-undefine 64.4%

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

      4. +-commutative 64.4%

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

      5. add-sqr-sqrt 64.4%

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

      6. pow2 64.4%

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

      7. hypot-define 64.4%

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

   6. Applied egg-rr 64.4%

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

   7. Taylor expanded in y around 0 35.5%

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

   if 8.5e-163 < y < 5.0000000000000002e-23

   1. Initial program 99.8%

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

      1. associate-/l* 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-define 99.4%

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

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*r/ 99.8%

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

      2. fma-undefine 99.8%

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

      3. +-commutative 99.8%

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

      4. add-sqr-sqrt 99.9%

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

      5. times-frac 99.7%

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

      6. hypot-define 99.8%

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

      7. hypot-define 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. log1p-expm1-u 99.8%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x - y}{\mathsf{hypot}\left(x, y\right)}\right)\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)} \]

   8. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x - y}{\mathsf{hypot}\left(x, y\right)}\right)\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)} \]
   9. Step-by-step derivation

      1. log1p-expm1-u 99.9%

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

      2. frac-times 100.0%

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

      3. pow2 100.0%

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

   10. Applied egg-rr 100.0%

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

   if 5.0000000000000002e-23 < y

   1. Initial program 100.0%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-163}:\\ \;\;\;\;\frac{x - y}{x + y \cdot \left(2 \cdot \frac{y}{x} + -1\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{{\left(\mathsf{hypot}\left(x, y\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.1× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (* (/ (- x y_m) (hypot x y_m)) (/ (+ x y_m) (hypot x y_m))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[(N[(x - y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x + y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.9%

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

   1. associate-/l* 70.1%

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

   2. +-commutative 70.1%

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

   3. fma-define 70.1%

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

3. Simplified 70.1%

   \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. associate-*r/ 69.9%

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

   2. fma-undefine 69.9%

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

   3. +-commutative 69.9%

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

   4. add-sqr-sqrt 69.9%

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

   5. times-frac 70.4%

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

   6. hypot-define 70.4%

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

   7. hypot-define 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 4: 99.7% accurate, 0.1× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (* (- x y_m) (/ (/ (+ x y_m) (hypot x y_m)) (hypot x y_m))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(N[(x + y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.9%

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

   1. associate-/l* 70.1%

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

   2. +-commutative 70.1%

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

   3. fma-define 70.1%

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

3. Simplified 70.1%

   \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 70.1%

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

   2. +-commutative 70.1%

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

   3. add-sqr-sqrt 70.1%

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

   4. associate-/r* 70.2%

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

   5. hypot-define 70.3%

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

   6. hypot-define 99.7%

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

6. Applied egg-rr 99.7%

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

Alternative 5: 92.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 8.5 \cdot 10^{-163}:\\ \;\;\;\;\frac{x - y\_m}{x + y\_m \cdot \left(2 \cdot \frac{y\_m}{x} + -1\right)}\\ \mathbf{elif}\;y\_m \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\left(x - y\_m\right) \cdot \left(x + y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 8.5e-163)
   (/ (- x y_m) (+ x (* y_m (+ (* 2.0 (/ y_m x)) -1.0))))
   (if (<= y_m 5e-23)
     (/ (* (- x y_m) (+ x y_m)) (+ (* x x) (* y_m y_m)))
     -1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 8.5e-163) {
		tmp = (x - y_m) / (x + (y_m * ((2.0 * (y_m / x)) + -1.0)));
	} else if (y_m <= 5e-23) {
		tmp = ((x - y_m) * (x + y_m)) / ((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) :: tmp
    if (y_m <= 8.5d-163) then
        tmp = (x - y_m) / (x + (y_m * ((2.0d0 * (y_m / x)) + (-1.0d0))))
    else if (y_m <= 5d-23) then
        tmp = ((x - y_m) * (x + y_m)) / ((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 tmp;
	if (y_m <= 8.5e-163) {
		tmp = (x - y_m) / (x + (y_m * ((2.0 * (y_m / x)) + -1.0)));
	} else if (y_m <= 5e-23) {
		tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 8.5e-163:
		tmp = (x - y_m) / (x + (y_m * ((2.0 * (y_m / x)) + -1.0)))
	elif y_m <= 5e-23:
		tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m))
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 8.5e-163)
		tmp = Float64(Float64(x - y_m) / Float64(x + Float64(y_m * Float64(Float64(2.0 * Float64(y_m / x)) + -1.0))));
	elseif (y_m <= 5e-23)
		tmp = Float64(Float64(Float64(x - y_m) * Float64(x + y_m)) / 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)
	tmp = 0.0;
	if (y_m <= 8.5e-163)
		tmp = (x - y_m) / (x + (y_m * ((2.0 * (y_m / x)) + -1.0)));
	elseif (y_m <= 5e-23)
		tmp = ((x - y_m) * (x + y_m)) / ((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_] := If[LessEqual[y$95$m, 8.5e-163], N[(N[(x - y$95$m), $MachinePrecision] / N[(x + N[(y$95$m * N[(N[(2.0 * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 5e-23], N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(x + y$95$m), $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 < 8.5e-163

   1. Initial program 63.8%

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

      1. associate-/l* 64.2%

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

      2. +-commutative 64.2%

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

      3. fma-define 64.2%

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

   3. Simplified 64.2%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 64.3%

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

      2. un-div-inv 64.4%

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

      3. fma-undefine 64.4%

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

      4. +-commutative 64.4%

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

      5. add-sqr-sqrt 64.4%

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

      6. pow2 64.4%

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

      7. hypot-define 64.4%

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

   6. Applied egg-rr 64.4%

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

   7. Taylor expanded in y around 0 35.5%

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

   if 8.5e-163 < y < 5.0000000000000002e-23

   1. Initial program 99.8%

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

   if 5.0000000000000002e-23 < y

   1. Initial program 100.0%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-163}:\\ \;\;\;\;\frac{x - y}{x + y \cdot \left(2 \cdot \frac{y}{x} + -1\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 8.5 \cdot 10^{-163}:\\ \;\;\;\;\left(\frac{y\_m}{x} + 1\right) \cdot \frac{x - y\_m}{x}\\ \mathbf{elif}\;y\_m \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\left(x - y\_m\right) \cdot \left(x + y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 8.5e-163)
   (* (+ (/ y_m x) 1.0) (/ (- x y_m) x))
   (if (<= y_m 5e-23)
     (/ (* (- x y_m) (+ x y_m)) (+ (* x x) (* y_m y_m)))
     -1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 8.5e-163) {
		tmp = ((y_m / x) + 1.0) * ((x - y_m) / x);
	} else if (y_m <= 5e-23) {
		tmp = ((x - y_m) * (x + y_m)) / ((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) :: tmp
    if (y_m <= 8.5d-163) then
        tmp = ((y_m / x) + 1.0d0) * ((x - y_m) / x)
    else if (y_m <= 5d-23) then
        tmp = ((x - y_m) * (x + y_m)) / ((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 tmp;
	if (y_m <= 8.5e-163) {
		tmp = ((y_m / x) + 1.0) * ((x - y_m) / x);
	} else if (y_m <= 5e-23) {
		tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 8.5e-163:
		tmp = ((y_m / x) + 1.0) * ((x - y_m) / x)
	elif y_m <= 5e-23:
		tmp = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m))
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 8.5e-163)
		tmp = Float64(Float64(Float64(y_m / x) + 1.0) * Float64(Float64(x - y_m) / x));
	elseif (y_m <= 5e-23)
		tmp = Float64(Float64(Float64(x - y_m) * Float64(x + y_m)) / 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)
	tmp = 0.0;
	if (y_m <= 8.5e-163)
		tmp = ((y_m / x) + 1.0) * ((x - y_m) / x);
	elseif (y_m <= 5e-23)
		tmp = ((x - y_m) * (x + y_m)) / ((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_] := If[LessEqual[y$95$m, 8.5e-163], N[(N[(N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x - y$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 5e-23], N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(x + y$95$m), $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 < 8.5e-163

   1. Initial program 63.8%

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

      1. associate-/l* 64.2%

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

      2. +-commutative 64.2%

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

      3. fma-define 64.2%

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

   3. Simplified 64.2%

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

   5. Taylor expanded in x around inf 35.7%

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

      1. *-commutative 35.7%

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

      2. div-inv 35.6%

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

      3. associate-*l* 35.8%

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

      4. *-commutative 35.8%

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

      5. un-div-inv 35.8%

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

   7. Applied egg-rr 35.8%

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

   if 8.5e-163 < y < 5.0000000000000002e-23

   1. Initial program 99.8%

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

   if 5.0000000000000002e-23 < y

   1. Initial program 100.0%

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

      1. associate-/l* 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-define 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-163}:\\ \;\;\;\;\left(\frac{y}{x} + 1\right) \cdot \frac{x - y}{x}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-23}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 4.05 \cdot 10^{-159}:\\ \;\;\;\;\left(\frac{y\_m}{x} + 1\right) \cdot \frac{x - y\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\_m\right) \cdot \left(1 + \frac{x}{y\_m}\right)}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 4.05e-159)
   (* (+ (/ y_m x) 1.0) (/ (- x y_m) x))
   (/ (* (- x y_m) (+ 1.0 (/ x y_m))) y_m)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 4.05e-159) {
		tmp = ((y_m / x) + 1.0) * ((x - y_m) / x);
	} else {
		tmp = ((x - y_m) * (1.0 + (x / y_m))) / 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 <= 4.05d-159) then
        tmp = ((y_m / x) + 1.0d0) * ((x - y_m) / x)
    else
        tmp = ((x - y_m) * (1.0d0 + (x / y_m))) / 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 <= 4.05e-159) {
		tmp = ((y_m / x) + 1.0) * ((x - y_m) / x);
	} else {
		tmp = ((x - y_m) * (1.0 + (x / y_m))) / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 4.05e-159:
		tmp = ((y_m / x) + 1.0) * ((x - y_m) / x)
	else:
		tmp = ((x - y_m) * (1.0 + (x / y_m))) / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 4.05e-159)
		tmp = Float64(Float64(Float64(y_m / x) + 1.0) * Float64(Float64(x - y_m) / x));
	else
		tmp = Float64(Float64(Float64(x - y_m) * Float64(1.0 + Float64(x / y_m))) / y_m);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 4.05e-159)
		tmp = ((y_m / x) + 1.0) * ((x - y_m) / x);
	else
		tmp = ((x - y_m) * (1.0 + (x / y_m))) / y_m;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 4.05e-159], N[(N[(N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x - y$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(1.0 + N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.05000000000000005e-159

   1. Initial program 64.0%

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

      1. associate-/l* 64.4%

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

      2. +-commutative 64.4%

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

      3. fma-define 64.4%

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

   3. Simplified 64.4%

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

   5. Taylor expanded in x around inf 36.0%

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

      1. *-commutative 36.0%

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

      2. div-inv 35.9%

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

      3. associate-*l* 36.1%

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

      4. *-commutative 36.1%

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

      5. un-div-inv 36.1%

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

   7. Applied egg-rr 36.1%

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

   if 4.05000000000000005e-159 < y

   1. Initial program 99.9%

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

      1. associate-/l* 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 66.9%

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

      1. associate-*r/ 67.2%

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

   7. Applied egg-rr 67.2%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.05 \cdot 10^{-159}:\\ \;\;\;\;\left(\frac{y}{x} + 1\right) \cdot \frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(1 + \frac{x}{y}\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.25 \cdot 10^{-159}:\\ \;\;\;\;\left(\frac{y\_m}{x} + 1\right) \cdot \frac{x - y\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\_m\right) \cdot \frac{1 + \frac{x}{y\_m}}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.25e-159)
   (* (+ (/ y_m x) 1.0) (/ (- x y_m) x))
   (* (- x y_m) (/ (+ 1.0 (/ x y_m)) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.25e-159) {
		tmp = ((y_m / x) + 1.0) * ((x - y_m) / x);
	} else {
		tmp = (x - y_m) * ((1.0 + (x / y_m)) / 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 <= 1.25d-159) then
        tmp = ((y_m / x) + 1.0d0) * ((x - y_m) / x)
    else
        tmp = (x - y_m) * ((1.0d0 + (x / y_m)) / 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 <= 1.25e-159) {
		tmp = ((y_m / x) + 1.0) * ((x - y_m) / x);
	} else {
		tmp = (x - y_m) * ((1.0 + (x / y_m)) / y_m);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.25e-159:
		tmp = ((y_m / x) + 1.0) * ((x - y_m) / x)
	else:
		tmp = (x - y_m) * ((1.0 + (x / y_m)) / y_m)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.25e-159)
		tmp = Float64(Float64(Float64(y_m / x) + 1.0) * Float64(Float64(x - y_m) / x));
	else
		tmp = Float64(Float64(x - y_m) * Float64(Float64(1.0 + Float64(x / y_m)) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 1.25e-159)
		tmp = ((y_m / x) + 1.0) * ((x - y_m) / x);
	else
		tmp = (x - y_m) * ((1.0 + (x / y_m)) / y_m);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.25e-159], N[(N[(N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x - y$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(1.0 + N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.25000000000000008e-159

   1. Initial program 64.0%

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

      1. associate-/l* 64.4%

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

      2. +-commutative 64.4%

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

      3. fma-define 64.4%

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

   3. Simplified 64.4%

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

   5. Taylor expanded in x around inf 36.0%

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

      1. *-commutative 36.0%

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

      2. div-inv 35.9%

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

      3. associate-*l* 36.1%

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

      4. *-commutative 36.1%

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

      5. un-div-inv 36.1%

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

   7. Applied egg-rr 36.1%

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

   if 1.25000000000000008e-159 < y

   1. Initial program 99.9%

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

      1. associate-/l* 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 66.9%

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

4. Final simplification 41.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{-159}:\\ \;\;\;\;\left(\frac{y}{x} + 1\right) \cdot \frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.5 \cdot 10^{-158}:\\ \;\;\;\;\left(\frac{y\_m}{x} + 1\right) \cdot \frac{x - y\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.5e-158) (* (+ (/ y_m x) 1.0) (/ (- x y_m) x)) -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.5e-158) {
		tmp = ((y_m / x) + 1.0) * ((x - y_m) / x);
	} 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 <= 1.5d-158) then
        tmp = ((y_m / x) + 1.0d0) * ((x - y_m) / x)
    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 <= 1.5e-158) {
		tmp = ((y_m / x) + 1.0) * ((x - y_m) / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.5e-158:
		tmp = ((y_m / x) + 1.0) * ((x - y_m) / x)
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.5e-158)
		tmp = Float64(Float64(Float64(y_m / x) + 1.0) * Float64(Float64(x - y_m) / x));
	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 <= 1.5e-158)
		tmp = ((y_m / x) + 1.0) * ((x - y_m) / x);
	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, 1.5e-158], N[(N[(N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x - y$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.5e-158

   1. Initial program 64.0%

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

      1. associate-/l* 64.4%

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

      2. +-commutative 64.4%

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

      3. fma-define 64.4%

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

   3. Simplified 64.4%

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

   5. Taylor expanded in x around inf 36.0%

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

      1. *-commutative 36.0%

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

      2. div-inv 35.9%

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

      3. associate-*l* 36.1%

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

      4. *-commutative 36.1%

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

      5. un-div-inv 36.1%

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

   7. Applied egg-rr 36.1%

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

   if 1.5e-158 < y

   1. Initial program 99.9%

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

      1. associate-/l* 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 64.8%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-158}:\\ \;\;\;\;\left(\frac{y}{x} + 1\right) \cdot \frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.35 \cdot 10^{-158}:\\ \;\;\;\;\left(\frac{y\_m}{x} + 1\right) \cdot \left(1 - \frac{y\_m}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.35e-158) (* (+ (/ y_m x) 1.0) (- 1.0 (/ y_m x))) -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.35e-158) {
		tmp = ((y_m / x) + 1.0) * (1.0 - (y_m / x));
	} 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 <= 1.35d-158) then
        tmp = ((y_m / x) + 1.0d0) * (1.0d0 - (y_m / x))
    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 <= 1.35e-158) {
		tmp = ((y_m / x) + 1.0) * (1.0 - (y_m / x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.35e-158:
		tmp = ((y_m / x) + 1.0) * (1.0 - (y_m / x))
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.35e-158)
		tmp = Float64(Float64(Float64(y_m / x) + 1.0) * Float64(1.0 - Float64(y_m / x)));
	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 <= 1.35e-158)
		tmp = ((y_m / x) + 1.0) * (1.0 - (y_m / x));
	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, 1.35e-158], N[(N[(N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.3499999999999999e-158

   1. Initial program 64.0%

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

      1. associate-/l* 64.4%

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

      2. +-commutative 64.4%

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

      3. fma-define 64.4%

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

   3. Simplified 64.4%

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

   5. Taylor expanded in x around inf 36.0%

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

      1. *-commutative 36.0%

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

      2. sub-neg 36.0%

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

      3. distribute-lft-in 35.8%

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

   7. Applied egg-rr 35.8%

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

      1. distribute-lft-out 36.0%

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

      2. sub-neg 36.0%

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

      3. associate-*l/ 36.2%

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

      4. associate-*r/ 36.1%

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

      5. div-sub 36.2%

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

      6. *-inverses 36.2%

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

   9. Simplified 36.2%

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

   if 1.3499999999999999e-158 < y

   1. Initial program 99.9%

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

      1. associate-/l* 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 64.8%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-158}:\\ \;\;\;\;\left(\frac{y}{x} + 1\right) \cdot \left(1 - \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.4 \cdot 10^{-158}:\\ \;\;\;\;\frac{x - y\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.4e-158) (/ (- x y_m) x) -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.4e-158) {
		tmp = (x - y_m) / x;
	} 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 <= 1.4d-158) then
        tmp = (x - y_m) / x
    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 <= 1.4e-158) {
		tmp = (x - y_m) / x;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.4e-158:
		tmp = (x - y_m) / x
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.4e-158)
		tmp = Float64(Float64(x - y_m) / x);
	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 <= 1.4e-158)
		tmp = (x - y_m) / x;
	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, 1.4e-158], N[(N[(x - y$95$m), $MachinePrecision] / x), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.40000000000000001e-158

   1. Initial program 64.0%

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

      1. associate-/l* 64.4%

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

      2. +-commutative 64.4%

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

      3. fma-define 64.4%

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

   3. Simplified 64.4%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 64.4%

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

      2. un-div-inv 64.6%

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

      3. fma-undefine 64.6%

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

      4. +-commutative 64.6%

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

      5. add-sqr-sqrt 64.6%

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

      6. pow2 64.6%

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

      7. hypot-define 64.6%

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

   6. Applied egg-rr 64.6%

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

   7. Taylor expanded in x around inf 33.5%

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

   if 1.40000000000000001e-158 < y

   1. Initial program 99.9%

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

      1. associate-/l* 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 64.8%

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

Alternative 12: 82.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.72 \cdot 10^{-159}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (if (<= y_m 1.72e-159) 1.0 -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.72e-159) {
		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 <= 1.72d-159) 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 <= 1.72e-159) {
		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 <= 1.72e-159:
		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 <= 1.72e-159)
		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 <= 1.72e-159)
		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, 1.72e-159], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.72000000000000006e-159

   1. Initial program 64.0%

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

      1. associate-/l* 64.4%

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

      2. +-commutative 64.4%

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

      3. fma-define 64.4%

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

   3. Simplified 64.4%

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

   5. Taylor expanded in x around inf 34.4%

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

   if 1.72000000000000006e-159 < y

   1. Initial program 99.9%

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

      1. associate-/l* 99.5%

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

      2. +-commutative 99.5%

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

      3. fma-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 64.8%

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

Alternative 13: 66.0% 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 69.9%

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

   1. associate-/l* 70.1%

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

   2. +-commutative 70.1%

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

   3. fma-define 70.1%

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

3. Simplified 70.1%

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

5. Taylor expanded in x around 0 65.3%

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

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 2024096 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

  :alt
  (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))))