Kahan p9 Example

Percentage Accurate: 68.7% → 100.0%

Time: 9.9s

Alternatives: 16

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: 68.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 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 67.8%

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

   1. add-sqr-sqrt 67.8%

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

   2. times-frac 68.3%

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

   3. hypot-define 68.4%

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

   4. hypot-define 100.0%

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

4. Applied egg-rr 100.0%

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

   1. *-commutative 100.0%

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

   2. clear-num 100.0%

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

   3. un-div-inv 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 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 67.8%

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

   1. add-sqr-sqrt 67.8%

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

   2. times-frac 68.3%

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

   3. hypot-define 68.4%

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

   4. hypot-define 100.0%

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

4. Applied egg-rr 100.0%

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

   1. *-commutative 100.0%

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

   2. clear-num 100.0%

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

   3. un-div-inv 100.0%

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

6. Applied egg-rr 100.0%

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

   1. div-inv 100.0%

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

   2. div-inv 99.7%

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

   3. clear-num 99.7%

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

   4. associate-*l* 99.7%

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

8. Applied egg-rr 99.7%

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

   1. associate-*l/ 99.7%

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

   2. *-lft-identity 99.7%

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

10. Simplified 99.7%

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

11. Final simplification 99.7%

    \[\leadsto \left(x + y\right) \cdot \frac{\frac{x - y}{\mathsf{hypot}\left(x, y\right)}}{\mathsf{hypot}\left(x, y\right)} \]
12. 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 67.8%

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

   1. add-sqr-sqrt 67.8%

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

   2. times-frac 68.3%

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

   3. hypot-define 68.4%

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

   4. hypot-define 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 4: 92.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{y\_m}{x}\right)}^{2}, -2, 1\right)\\ \mathbf{elif}\;y\_m \leq 10^{-37}:\\ \;\;\;\;\frac{\left(x + y\_m\right) \cdot \left(x - y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.6e-162)
   (fma (pow (/ y_m x) 2.0) -2.0 1.0)
   (if (<= y_m 1e-37)
     (/ (* (+ 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 <= 1.6e-162) {
		tmp = fma(pow((y_m / x), 2.0), -2.0, 1.0);
	} else if (y_m <= 1e-37) {
		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 <= 1.6e-162)
		tmp = fma((Float64(y_m / x) ^ 2.0), -2.0, 1.0);
	elseif (y_m <= 1e-37)
		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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.6e-162], N[(N[Power[N[(y$95$m / x), $MachinePrecision], 2.0], $MachinePrecision] * -2.0 + 1.0), $MachinePrecision], If[LessEqual[y$95$m, 1e-37], 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 < 1.59999999999999988e-162

   1. Initial program 60.7%

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

      1. add-sqr-sqrt 60.7%

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

      2. times-frac 61.5%

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

      3. hypot-define 61.6%

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

      4. hypot-define 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. un-div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0 28.0%

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

      1. +-commutative 28.0%

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

      2. *-commutative 28.0%

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

      3. fma-define 28.0%

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

      4. unpow2 28.0%

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

      5. unpow2 28.0%

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

      6. times-frac 39.7%

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

      7. unpow2 39.7%

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

   9. Simplified 39.7%

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

   if 1.59999999999999988e-162 < y < 1.00000000000000007e-37

   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 1.00000000000000007e-37 < 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* 99.6%

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

      2. +-commutative 99.6%

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

      3. fma-define 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
   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 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\mathsf{fma}\left({\left(\frac{y}{x}\right)}^{2}, -2, 1\right)\\ \mathbf{elif}\;y \leq 10^{-37}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 92.4% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	t_0 = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m));
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = ((x - y_m) / hypot(x, y_m)) * (1.0 + (x / 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[(x - y$95$m), $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[(N[(N[(x - y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.7%

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

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

   1. Initial program 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. times-frac 3.1%

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

      3. hypot-define 3.1%

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

      4. hypot-define 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in x around 0 17.0%

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

4. Final simplification 73.2%

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

Alternative 6: 92.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\left(x + y\_m\right) \cdot \frac{1 - \frac{y\_m}{x}}{\mathsf{hypot}\left(x, y\_m\right)}\\ \mathbf{elif}\;y\_m \leq 1.4 \cdot 10^{-37}:\\ \;\;\;\;\frac{\left(x + y\_m\right) \cdot \left(x - y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.6e-162)
   (* (+ x y_m) (/ (- 1.0 (/ y_m x)) (hypot x y_m)))
   (if (<= y_m 1.4e-37)
     (/ (* (+ 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 <= 1.6e-162) {
		tmp = (x + y_m) * ((1.0 - (y_m / x)) / hypot(x, y_m));
	} else if (y_m <= 1.4e-37) {
		tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m));
	} 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 <= 1.6e-162) {
		tmp = (x + y_m) * ((1.0 - (y_m / x)) / Math.hypot(x, y_m));
	} else if (y_m <= 1.4e-37) {
		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 <= 1.6e-162:
		tmp = (x + y_m) * ((1.0 - (y_m / x)) / math.hypot(x, y_m))
	elif y_m <= 1.4e-37:
		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 <= 1.6e-162)
		tmp = Float64(Float64(x + y_m) * Float64(Float64(1.0 - Float64(y_m / x)) / hypot(x, y_m)));
	elseif (y_m <= 1.4e-37)
		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 <= 1.6e-162)
		tmp = (x + y_m) * ((1.0 - (y_m / x)) / hypot(x, y_m));
	elseif (y_m <= 1.4e-37)
		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, 1.6e-162], N[(N[(x + y$95$m), $MachinePrecision] * N[(N[(1.0 - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1.4e-37], 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 < 1.59999999999999988e-162

   1. Initial program 60.7%

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

      1. add-sqr-sqrt 60.7%

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

      2. times-frac 61.5%

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

      3. hypot-define 61.6%

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

      4. hypot-define 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. un-div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. div-inv 100.0%

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

      2. div-inv 99.7%

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

      3. clear-num 99.7%

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

      4. associate-*l* 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in x around inf 39.8%

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

       1. neg-mul-1 39.8%

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

       2. sub-neg 39.8%

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

   13. Simplified 39.8%

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

   if 1.59999999999999988e-162 < y < 1.4000000000000001e-37

   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 1.4000000000000001e-37 < 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* 99.6%

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

      2. +-commutative 99.6%

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

      3. fma-define 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
   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 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{1 - \frac{y}{x}}{\mathsf{hypot}\left(x, y\right)}\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-37}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{x - y\_m}{\mathsf{hypot}\left(x, y\_m\right)} \cdot \left(\frac{y\_m}{x} + 1\right)\\ \mathbf{elif}\;y\_m \leq 1.2 \cdot 10^{-37}:\\ \;\;\;\;\frac{\left(x + y\_m\right) \cdot \left(x - y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.6e-162)
   (* (/ (- x y_m) (hypot x y_m)) (+ (/ y_m x) 1.0))
   (if (<= y_m 1.2e-37)
     (/ (* (+ 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 <= 1.6e-162) {
		tmp = ((x - y_m) / hypot(x, y_m)) * ((y_m / x) + 1.0);
	} else if (y_m <= 1.2e-37) {
		tmp = ((x + y_m) * (x - y_m)) / ((x * x) + (y_m * y_m));
	} 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 <= 1.6e-162) {
		tmp = ((x - y_m) / Math.hypot(x, y_m)) * ((y_m / x) + 1.0);
	} else if (y_m <= 1.2e-37) {
		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 <= 1.6e-162:
		tmp = ((x - y_m) / math.hypot(x, y_m)) * ((y_m / x) + 1.0)
	elif y_m <= 1.2e-37:
		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 <= 1.6e-162)
		tmp = Float64(Float64(Float64(x - y_m) / hypot(x, y_m)) * Float64(Float64(y_m / x) + 1.0));
	elseif (y_m <= 1.2e-37)
		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 <= 1.6e-162)
		tmp = ((x - y_m) / hypot(x, y_m)) * ((y_m / x) + 1.0);
	elseif (y_m <= 1.2e-37)
		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, 1.6e-162], N[(N[(N[(x - y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1.2e-37], 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 < 1.59999999999999988e-162

   1. Initial program 60.7%

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

      1. add-sqr-sqrt 60.7%

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

      2. times-frac 61.5%

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

      3. hypot-define 61.6%

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

      4. hypot-define 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around inf 39.9%

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

   if 1.59999999999999988e-162 < y < 1.19999999999999995e-37

   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 1.19999999999999995e-37 < 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* 99.6%

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

      2. +-commutative 99.6%

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

      3. fma-define 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
   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 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \left(\frac{y}{x} + 1\right)\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-37}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 92.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{1}{\frac{x}{\left(x - y\_m\right) \cdot \left(\frac{y\_m}{x} + 1\right)}}\\ \mathbf{elif}\;y\_m \leq 1.2 \cdot 10^{-37}:\\ \;\;\;\;\frac{\left(x + y\_m\right) \cdot \left(x - y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.6e-162)
   (/ 1.0 (/ x (* (- x y_m) (+ (/ y_m x) 1.0))))
   (if (<= y_m 1.2e-37)
     (/ (* (+ 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 <= 1.6e-162) {
		tmp = 1.0 / (x / ((x - y_m) * ((y_m / x) + 1.0)));
	} else if (y_m <= 1.2e-37) {
		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 <= 1.6d-162) then
        tmp = 1.0d0 / (x / ((x - y_m) * ((y_m / x) + 1.0d0)))
    else if (y_m <= 1.2d-37) 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 <= 1.6e-162) {
		tmp = 1.0 / (x / ((x - y_m) * ((y_m / x) + 1.0)));
	} else if (y_m <= 1.2e-37) {
		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 <= 1.6e-162:
		tmp = 1.0 / (x / ((x - y_m) * ((y_m / x) + 1.0)))
	elif y_m <= 1.2e-37:
		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 <= 1.6e-162)
		tmp = Float64(1.0 / Float64(x / Float64(Float64(x - y_m) * Float64(Float64(y_m / x) + 1.0))));
	elseif (y_m <= 1.2e-37)
		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 <= 1.6e-162)
		tmp = 1.0 / (x / ((x - y_m) * ((y_m / x) + 1.0)));
	elseif (y_m <= 1.2e-37)
		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, 1.6e-162], N[(1.0 / N[(x / N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1.2e-37], 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 < 1.59999999999999988e-162

   1. Initial program 60.7%

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

      1. associate-/l* 61.3%

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

      2. +-commutative 61.3%

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

      3. fma-define 61.3%

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

   3. Simplified 61.3%

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

   5. Taylor expanded in x around inf 39.2%

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

      1. associate-*r/ 39.4%

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

      2. clear-num 39.4%

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

   7. Applied egg-rr 39.4%

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

   if 1.59999999999999988e-162 < y < 1.19999999999999995e-37

   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 1.19999999999999995e-37 < 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* 99.6%

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

      2. +-commutative 99.6%

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

      3. fma-define 99.6%

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

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
   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 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.6 \cdot 10^{-162}:\\ \;\;\;\;\frac{1}{\frac{x}{\left(x - y\right) \cdot \left(\frac{y}{x} + 1\right)}}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-37}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 3.7 \cdot 10^{-178}:\\ \;\;\;\;\frac{1}{\frac{x}{\left(x - y\_m\right) \cdot \left(\frac{y\_m}{x} + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\_m\right) \cdot \left(1 + \frac{x}{y\_m}\right)}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 3.7e-178)
   (/ 1.0 (/ x (* (- x y_m) (+ (/ y_m x) 1.0))))
   (/ (* (- 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 <= 3.7e-178) {
		tmp = 1.0 / (x / ((x - y_m) * ((y_m / x) + 1.0)));
	} 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 <= 3.7d-178) then
        tmp = 1.0d0 / (x / ((x - y_m) * ((y_m / x) + 1.0d0)))
    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 <= 3.7e-178) {
		tmp = 1.0 / (x / ((x - y_m) * ((y_m / x) + 1.0)));
	} 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 <= 3.7e-178:
		tmp = 1.0 / (x / ((x - y_m) * ((y_m / x) + 1.0)))
	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 <= 3.7e-178)
		tmp = Float64(1.0 / Float64(x / Float64(Float64(x - y_m) * Float64(Float64(y_m / x) + 1.0))));
	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 <= 3.7e-178)
		tmp = 1.0 / (x / ((x - y_m) * ((y_m / x) + 1.0)));
	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, 3.7e-178], N[(1.0 / N[(x / N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $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 < 3.70000000000000004e-178

   1. Initial program 61.1%

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

      1. associate-/l* 61.7%

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

      2. +-commutative 61.7%

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

      3. fma-define 61.7%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in x around inf 39.2%

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

      1. associate-*r/ 39.4%

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

      2. clear-num 39.4%

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

   7. Applied egg-rr 39.4%

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

   if 3.70000000000000004e-178 < y

   1. Initial program 95.9%

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

      1. associate-/l* 95.2%

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

      2. +-commutative 95.2%

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

      3. fma-define 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in y around inf 82.0%

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

      1. associate-*r/ 82.1%

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

   7. Applied egg-rr 82.1%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.7 \cdot 10^{-178}:\\ \;\;\;\;\frac{1}{\frac{x}{\left(x - y\right) \cdot \left(\frac{y}{x} + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(1 + \frac{x}{y}\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 2.4 \cdot 10^{-178}:\\ \;\;\;\;\left(x + y\_m\right) \cdot \frac{1 - \frac{y\_m}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 2.4e-178) (* (+ x y_m) (/ (- 1.0 (/ y_m x)) x)) -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 2.4e-178) {
		tmp = (x + y_m) * ((1.0 - (y_m / x)) / 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 <= 2.4d-178) then
        tmp = (x + y_m) * ((1.0d0 - (y_m / x)) / 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 <= 2.4e-178) {
		tmp = (x + y_m) * ((1.0 - (y_m / x)) / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 2.4e-178:
		tmp = (x + y_m) * ((1.0 - (y_m / x)) / x)
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 2.4e-178)
		tmp = Float64(Float64(x + y_m) * Float64(Float64(1.0 - Float64(y_m / x)) / 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 <= 2.4e-178)
		tmp = (x + y_m) * ((1.0 - (y_m / x)) / 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, 2.4e-178], N[(N[(x + y$95$m), $MachinePrecision] * N[(N[(1.0 - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 2.40000000000000005e-178

   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. Step-by-step derivation

      1. add-sqr-sqrt 61.1%

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

      2. times-frac 61.9%

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

      3. hypot-define 62.0%

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

      4. hypot-define 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. un-div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. div-inv 100.0%

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

      2. div-inv 99.7%

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

      3. clear-num 99.7%

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

      4. associate-*l* 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in x around inf 39.2%

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

       1. mul-1-neg 39.2%

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

       2. sub-neg 39.2%

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

   13. Simplified 39.2%

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

   if 2.40000000000000005e-178 < y

   1. Initial program 95.9%

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

      1. associate-/l* 95.2%

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

      2. +-commutative 95.2%

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

      3. fma-define 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in x around 0 81.2%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.4 \cdot 10^{-178}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{1 - \frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 3 \cdot 10^{-178}:\\ \;\;\;\;\left(x + y\_m\right) \cdot \frac{1 - \frac{y\_m}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\_m\right) \cdot \frac{1 + \frac{x}{y\_m}}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 3e-178)
   (* (+ x y_m) (/ (- 1.0 (/ y_m x)) 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 <= 3e-178) {
		tmp = (x + y_m) * ((1.0 - (y_m / x)) / 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 <= 3d-178) then
        tmp = (x + y_m) * ((1.0d0 - (y_m / x)) / 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 <= 3e-178) {
		tmp = (x + y_m) * ((1.0 - (y_m / x)) / 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 <= 3e-178:
		tmp = (x + y_m) * ((1.0 - (y_m / x)) / 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 <= 3e-178)
		tmp = Float64(Float64(x + y_m) * Float64(Float64(1.0 - Float64(y_m / x)) / 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 <= 3e-178)
		tmp = (x + y_m) * ((1.0 - (y_m / x)) / 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, 3e-178], N[(N[(x + y$95$m), $MachinePrecision] * N[(N[(1.0 - N[(y$95$m / x), $MachinePrecision]), $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 < 2.9999999999999999e-178

   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. Step-by-step derivation

      1. add-sqr-sqrt 61.1%

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

      2. times-frac 61.9%

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

      3. hypot-define 62.0%

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

      4. hypot-define 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. un-div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. div-inv 100.0%

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

      2. div-inv 99.7%

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

      3. clear-num 99.7%

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

      4. associate-*l* 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in x around inf 39.2%

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

       1. mul-1-neg 39.2%

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

       2. sub-neg 39.2%

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

   13. Simplified 39.2%

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

   if 2.9999999999999999e-178 < y

   1. Initial program 95.9%

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

      1. associate-/l* 95.2%

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

      2. +-commutative 95.2%

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

      3. fma-define 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in y around inf 82.0%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3 \cdot 10^{-178}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{1 - \frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 82.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 2.8 \cdot 10^{-178}:\\ \;\;\;\;\left(x + y\_m\right) \cdot \frac{1 - \frac{y\_m}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\_m\right) \cdot \left(1 + \frac{x}{y\_m}\right)}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 2.8e-178)
   (* (+ x y_m) (/ (- 1.0 (/ y_m x)) 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 <= 2.8e-178) {
		tmp = (x + y_m) * ((1.0 - (y_m / x)) / 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 <= 2.8d-178) then
        tmp = (x + y_m) * ((1.0d0 - (y_m / x)) / 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 <= 2.8e-178) {
		tmp = (x + y_m) * ((1.0 - (y_m / x)) / 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 <= 2.8e-178:
		tmp = (x + y_m) * ((1.0 - (y_m / x)) / 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 <= 2.8e-178)
		tmp = Float64(Float64(x + y_m) * Float64(Float64(1.0 - Float64(y_m / x)) / 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 <= 2.8e-178)
		tmp = (x + y_m) * ((1.0 - (y_m / x)) / 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, 2.8e-178], N[(N[(x + y$95$m), $MachinePrecision] * N[(N[(1.0 - N[(y$95$m / x), $MachinePrecision]), $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 < 2.80000000000000019e-178

   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. Step-by-step derivation

      1. add-sqr-sqrt 61.1%

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

      2. times-frac 61.9%

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

      3. hypot-define 62.0%

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

      4. hypot-define 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. *-commutative 100.0%

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

      2. clear-num 100.0%

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

      3. un-div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

      1. div-inv 100.0%

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

      2. div-inv 99.7%

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

      3. clear-num 99.7%

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

      4. associate-*l* 99.7%

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

   8. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

   10. Simplified 99.7%

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

   11. Taylor expanded in x around inf 39.2%

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

       1. mul-1-neg 39.2%

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

       2. sub-neg 39.2%

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

   13. Simplified 39.2%

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

   if 2.80000000000000019e-178 < y

   1. Initial program 95.9%

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

      1. associate-/l* 95.2%

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

      2. +-commutative 95.2%

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

      3. fma-define 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in y around inf 82.0%

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

      1. associate-*r/ 82.1%

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

   7. Applied egg-rr 82.1%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{-178}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{1 - \frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(1 + \frac{x}{y}\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 83.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 2.9 \cdot 10^{-178}:\\ \;\;\;\;\frac{x - y\_m}{\frac{x}{\frac{y\_m}{x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\_m\right) \cdot \left(1 + \frac{x}{y\_m}\right)}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 2.9e-178)
   (/ (- x y_m) (/ x (+ (/ y_m x) 1.0)))
   (/ (* (- 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 <= 2.9e-178) {
		tmp = (x - y_m) / (x / ((y_m / x) + 1.0));
	} 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 <= 2.9d-178) then
        tmp = (x - y_m) / (x / ((y_m / x) + 1.0d0))
    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 <= 2.9e-178) {
		tmp = (x - y_m) / (x / ((y_m / x) + 1.0));
	} 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 <= 2.9e-178:
		tmp = (x - y_m) / (x / ((y_m / x) + 1.0))
	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 <= 2.9e-178)
		tmp = Float64(Float64(x - y_m) / Float64(x / Float64(Float64(y_m / x) + 1.0)));
	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 <= 2.9e-178)
		tmp = (x - y_m) / (x / ((y_m / x) + 1.0));
	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, 2.9e-178], N[(N[(x - y$95$m), $MachinePrecision] / N[(x / N[(N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision]), $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 < 2.8999999999999998e-178

   1. Initial program 61.1%

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

      1. associate-/l* 61.7%

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

      2. +-commutative 61.7%

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

      3. fma-define 61.7%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in x around inf 39.2%

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

      1. clear-num 39.2%

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

      2. un-div-inv 39.4%

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

   7. Applied egg-rr 39.4%

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

   if 2.8999999999999998e-178 < y

   1. Initial program 95.9%

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

      1. associate-/l* 95.2%

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

      2. +-commutative 95.2%

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

      3. fma-define 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in y around inf 82.0%

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

      1. associate-*r/ 82.1%

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

   7. Applied egg-rr 82.1%

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

4. Final simplification 47.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{-178}:\\ \;\;\;\;\frac{x - y}{\frac{x}{\frac{y}{x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(1 + \frac{x}{y}\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 82.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 4 \cdot 10^{-178}:\\ \;\;\;\;\frac{x - y\_m}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (if (<= y_m 4e-178) (/ (- x y_m) x) -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 4e-178) {
		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 <= 4d-178) 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 <= 4e-178) {
		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 <= 4e-178:
		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 <= 4e-178)
		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 <= 4e-178)
		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, 4e-178], N[(N[(x - y$95$m), $MachinePrecision] / x), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 3.9999999999999998e-178

   1. Initial program 61.1%

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

      1. associate-/l* 61.7%

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

      2. +-commutative 61.7%

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

      3. fma-define 61.7%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in x around inf 37.3%

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

      1. un-div-inv 37.4%

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

   7. Applied egg-rr 37.4%

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

   if 3.9999999999999998e-178 < y

   1. Initial program 95.9%

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

      1. associate-/l* 95.2%

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

      2. +-commutative 95.2%

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

      3. fma-define 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in x around 0 81.2%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-178}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 82.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 4 \cdot 10^{-178}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (if (<= y_m 4e-178) 1.0 -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 4e-178) {
		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 <= 4d-178) 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 <= 4e-178) {
		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 <= 4e-178:
		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 <= 4e-178)
		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 <= 4e-178)
		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, 4e-178], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 3.9999999999999998e-178

   1. Initial program 61.1%

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

      1. associate-/l* 61.7%

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

      2. +-commutative 61.7%

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

      3. fma-define 61.7%

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

   3. Simplified 61.7%

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

   5. Taylor expanded in x around inf 38.0%

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

   if 3.9999999999999998e-178 < y

   1. Initial program 95.9%

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

      1. associate-/l* 95.2%

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

      2. +-commutative 95.2%

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

      3. fma-define 95.2%

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

   3. Simplified 95.2%

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

   5. Taylor expanded in x around 0 81.2%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{-178}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 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 67.8%

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

   1. associate-/l* 68.1%

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

   2. +-commutative 68.1%

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

   3. fma-define 68.1%

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

3. Simplified 68.1%

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

5. Taylor expanded in x around 0 65.4%

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

6. Final simplification 65.4%

   \[\leadsto -1 \]
7. 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 2024059 
(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))))