Kahan p9 Example

Percentage Accurate: 68.8% → 100.0%

Time: 9.7s

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: 68.8% 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 72.6%

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

   1. add-sqr-sqrt 72.6%

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

   2. times-frac 72.8%

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

   3. hypot-define 72.8%

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

   4. hypot-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. clear-num 100.0%

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

   2. un-div-inv 100.0%

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

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. Add Preprocessing

Alternative 2: 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 72.6%

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

   1. add-sqr-sqrt 72.6%

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

   2. times-frac 72.8%

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

   3. hypot-define 72.8%

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

   4. hypot-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. Add Preprocessing

Alternative 3: 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 72.6%

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

   1. associate-/l* 72.5%

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

   2. +-commutative 72.5%

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

   3. +-commutative 72.5%

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

   4. +-commutative 72.5%

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

   5. fma-define 72.5%

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

3. Simplified 72.5%

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

   1. fma-undefine 72.5%

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

   2. +-commutative 72.5%

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

   3. *-un-lft-identity 72.5%

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

   4. add-sqr-sqrt 72.5%

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

   5. times-frac 72.6%

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

   6. hypot-define 72.6%

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

   7. hypot-define 99.7%

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

6. Applied egg-rr 99.7%

   \[\leadsto \left(x - y\right) \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}\right)} \]
7. 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. *-un-lft-identity 99.7%

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

8. Applied egg-rr 99.7%

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

Alternative 4: 93.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\mathsf{fma}\left(-2, {\left(\frac{y\_m}{x}\right)}^{2}, 1\right)\\ \mathbf{elif}\;y\_m \leq 0.002:\\ \;\;\;\;\frac{\left(x - y\_m\right) \cdot \left(x + y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.55e-162)
   (fma -2.0 (pow (/ y_m x) 2.0) 1.0)
   (if (<= y_m 0.002)
     (/ (* (- 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 <= 1.55e-162) {
		tmp = fma(-2.0, pow((y_m / x), 2.0), 1.0);
	} else if (y_m <= 0.002) {
		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 <= 1.55e-162)
		tmp = fma(-2.0, (Float64(y_m / x) ^ 2.0), 1.0);
	elseif (y_m <= 0.002)
		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, 1.55e-162], N[(-2.0 * N[Power[N[(y$95$m / x), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[y$95$m, 0.002], 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 < 1.5499999999999999e-162

   1. Initial program 65.2%

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

      1. associate-/l* 66.0%

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

      2. +-commutative 66.0%

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

      3. +-commutative 66.0%

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

      4. +-commutative 66.0%

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

      5. fma-define 66.0%

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

   3. Simplified 66.0%

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

      1. fma-undefine 66.0%

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

      2. +-commutative 66.0%

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

      3. *-un-lft-identity 66.0%

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

      4. add-sqr-sqrt 66.0%

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

      5. times-frac 66.1%

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

      6. hypot-define 66.1%

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

      7. hypot-define 99.7%

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

   6. Applied egg-rr 99.7%

      \[\leadsto \left(x - y\right) \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}\right)} \]
   7. 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. *-un-lft-identity 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in y around 0 29.9%

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

       1. +-commutative 29.9%

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

       2. fma-define 29.9%

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

       3. unpow2 29.9%

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

       4. unpow2 29.9%

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

       5. times-frac 38.8%

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

       6. unpow2 38.8%

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

   11. Simplified 38.8%

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

   if 1.5499999999999999e-162 < y < 2e-3

   1. Initial program 99.9%

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

   if 2e-3 < 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* 100.0%

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

      2. +-commutative 100.0%

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

      3. +-commutative 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

Alternative 5: 93.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{x - y\_m}{x \cdot \frac{\mathsf{hypot}\left(x, y\_m\right)}{x + y\_m}}\\ \mathbf{elif}\;y\_m \leq 0.002:\\ \;\;\;\;\frac{\left(x - y\_m\right) \cdot \left(x + y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.55e-162)
   (/ (- x y_m) (* x (/ (hypot x y_m) (+ x y_m))))
   (if (<= y_m 0.002)
     (/ (* (- 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 <= 1.55e-162) {
		tmp = (x - y_m) / (x * (hypot(x, y_m) / (x + y_m)));
	} else if (y_m <= 0.002) {
		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 <= 1.55e-162) {
		tmp = (x - y_m) / (x * (Math.hypot(x, y_m) / (x + y_m)));
	} else if (y_m <= 0.002) {
		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 <= 1.55e-162:
		tmp = (x - y_m) / (x * (math.hypot(x, y_m) / (x + y_m)))
	elif y_m <= 0.002:
		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 <= 1.55e-162)
		tmp = Float64(Float64(x - y_m) / Float64(x * Float64(hypot(x, y_m) / Float64(x + y_m))));
	elseif (y_m <= 0.002)
		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 <= 1.55e-162)
		tmp = (x - y_m) / (x * (hypot(x, y_m) / (x + y_m)));
	elseif (y_m <= 0.002)
		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, 1.55e-162], N[(N[(x - y$95$m), $MachinePrecision] / N[(x * N[(N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision] / N[(x + y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 0.002], 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 < 1.5499999999999999e-162

   1. Initial program 65.2%

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

      1. add-sqr-sqrt 65.2%

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

      2. times-frac 66.3%

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

      3. hypot-define 66.3%

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

      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 39.1%

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

      1. *-commutative 39.1%

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

      2. clear-num 39.1%

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

      3. frac-times 39.1%

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

      4. *-un-lft-identity 39.1%

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

   7. Applied egg-rr 39.1%

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

   if 1.5499999999999999e-162 < y < 2e-3

   1. Initial program 99.9%

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

   if 2e-3 < 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* 100.0%

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

      2. +-commutative 100.0%

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

      3. +-commutative 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
   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 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{x - y}{x \cdot \frac{\mathsf{hypot}\left(x, y\right)}{x + y}}\\ \mathbf{elif}\;y \leq 0.002:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{x + y\_m}{\mathsf{hypot}\left(x, y\_m\right)} \cdot \frac{x - y\_m}{x}\\ \mathbf{elif}\;y\_m \leq 0.002:\\ \;\;\;\;\frac{\left(x - y\_m\right) \cdot \left(x + y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.55e-162)
   (* (/ (+ x y_m) (hypot x y_m)) (/ (- x y_m) x))
   (if (<= y_m 0.002)
     (/ (* (- 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 <= 1.55e-162) {
		tmp = ((x + y_m) / hypot(x, y_m)) * ((x - y_m) / x);
	} else if (y_m <= 0.002) {
		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 <= 1.55e-162) {
		tmp = ((x + y_m) / Math.hypot(x, y_m)) * ((x - y_m) / x);
	} else if (y_m <= 0.002) {
		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 <= 1.55e-162:
		tmp = ((x + y_m) / math.hypot(x, y_m)) * ((x - y_m) / x)
	elif y_m <= 0.002:
		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 <= 1.55e-162)
		tmp = Float64(Float64(Float64(x + y_m) / hypot(x, y_m)) * Float64(Float64(x - y_m) / x));
	elseif (y_m <= 0.002)
		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 <= 1.55e-162)
		tmp = ((x + y_m) / hypot(x, y_m)) * ((x - y_m) / x);
	elseif (y_m <= 0.002)
		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, 1.55e-162], 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] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 0.002], 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 < 1.5499999999999999e-162

   1. Initial program 65.2%

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

      1. add-sqr-sqrt 65.2%

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

      2. times-frac 66.3%

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

      3. hypot-define 66.3%

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

      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 39.1%

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

   if 1.5499999999999999e-162 < y < 2e-3

   1. Initial program 99.9%

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

   if 2e-3 < 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* 100.0%

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

      2. +-commutative 100.0%

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

      3. +-commutative 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
   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 52.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{x + y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x - y}{x}\\ \mathbf{elif}\;y \leq 0.002:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\left(\frac{y\_m}{x} + 1\right) \cdot \left(1 - \frac{y\_m}{x}\right)\\ \mathbf{elif}\;y\_m \leq 0.005:\\ \;\;\;\;\frac{\left(x - y\_m\right) \cdot \left(x + y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.55e-162)
   (* (+ (/ y_m x) 1.0) (- 1.0 (/ y_m x)))
   (if (<= y_m 0.005)
     (/ (* (- 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 <= 1.55e-162) {
		tmp = ((y_m / x) + 1.0) * (1.0 - (y_m / x));
	} else if (y_m <= 0.005) {
		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 <= 1.55d-162) then
        tmp = ((y_m / x) + 1.0d0) * (1.0d0 - (y_m / x))
    else if (y_m <= 0.005d0) 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 <= 1.55e-162) {
		tmp = ((y_m / x) + 1.0) * (1.0 - (y_m / x));
	} else if (y_m <= 0.005) {
		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 <= 1.55e-162:
		tmp = ((y_m / x) + 1.0) * (1.0 - (y_m / x))
	elif y_m <= 0.005:
		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 <= 1.55e-162)
		tmp = Float64(Float64(Float64(y_m / x) + 1.0) * Float64(1.0 - Float64(y_m / x)));
	elseif (y_m <= 0.005)
		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 <= 1.55e-162)
		tmp = ((y_m / x) + 1.0) * (1.0 - (y_m / x));
	elseif (y_m <= 0.005)
		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, 1.55e-162], N[(N[(N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 0.005], 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 < 1.5499999999999999e-162

   1. Initial program 65.2%

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

      1. associate-/l* 66.0%

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

      2. +-commutative 66.0%

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

      3. +-commutative 66.0%

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

      4. +-commutative 66.0%

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

      5. fma-define 66.0%

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

   3. Simplified 66.0%

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

   5. Taylor expanded in x around inf 38.5%

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

      1. clear-num 38.5%

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

      2. un-div-inv 38.6%

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

   7. Applied egg-rr 38.6%

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

      1. associate-/r/ 38.6%

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

      2. *-commutative 38.6%

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

      3. div-sub 38.6%

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

      4. *-inverses 38.6%

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

   9. Simplified 38.6%

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

   if 1.5499999999999999e-162 < y < 0.0050000000000000001

   1. Initial program 99.9%

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

   if 0.0050000000000000001 < 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* 100.0%

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

      2. +-commutative 100.0%

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

      3. +-commutative 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-define 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(x - y\right) \cdot \frac{x + y}{\mathsf{fma}\left(y, y, x \cdot x\right)}} \]
   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 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\left(\frac{y}{x} + 1\right) \cdot \left(1 - \frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 0.005:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.75 \cdot 10^{-168}:\\ \;\;\;\;\left(\frac{y\_m}{x} + 1\right) \cdot \left(1 - \frac{y\_m}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y\_m}{y\_m} \cdot \left(1 + \frac{x}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.75e-168)
   (* (+ (/ y_m x) 1.0) (- 1.0 (/ y_m x)))
   (* (/ (- x y_m) y_m) (+ 1.0 (/ x y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.75e-168) {
		tmp = ((y_m / x) + 1.0) * (1.0 - (y_m / x));
	} else {
		tmp = ((x - y_m) / y_m) * (1.0 + (x / 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 <= 1.75d-168) then
        tmp = ((y_m / x) + 1.0d0) * (1.0d0 - (y_m / x))
    else
        tmp = ((x - y_m) / y_m) * (1.0d0 + (x / 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 <= 1.75e-168) {
		tmp = ((y_m / x) + 1.0) * (1.0 - (y_m / x));
	} else {
		tmp = ((x - y_m) / y_m) * (1.0 + (x / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.75e-168:
		tmp = ((y_m / x) + 1.0) * (1.0 - (y_m / x))
	else:
		tmp = ((x - y_m) / y_m) * (1.0 + (x / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.75e-168)
		tmp = Float64(Float64(Float64(y_m / x) + 1.0) * Float64(1.0 - Float64(y_m / x)));
	else
		tmp = Float64(Float64(Float64(x - y_m) / 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)
	tmp = 0.0;
	if (y_m <= 1.75e-168)
		tmp = ((y_m / x) + 1.0) * (1.0 - (y_m / x));
	else
		tmp = ((x - y_m) / y_m) * (1.0 + (x / y_m));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.75e-168], N[(N[(N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x - y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] * N[(1.0 + N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.74999999999999991e-168

   1. Initial program 65.5%

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

      1. associate-/l* 66.3%

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

      2. +-commutative 66.3%

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

      3. +-commutative 66.3%

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

      4. +-commutative 66.3%

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

      5. fma-define 66.3%

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

   3. Simplified 66.3%

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

   5. Taylor expanded in x around inf 38.6%

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

      1. clear-num 38.6%

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

      2. un-div-inv 38.8%

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

   7. Applied egg-rr 38.8%

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

      1. associate-/r/ 38.8%

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

      2. *-commutative 38.8%

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

      3. div-sub 38.8%

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

      4. *-inverses 38.8%

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

   9. Simplified 38.8%

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

   if 1.74999999999999991e-168 < y

   1. Initial program 98.2%

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

      1. associate-/l* 94.5%

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

      2. +-commutative 94.5%

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

      3. +-commutative 94.5%

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

      4. +-commutative 94.5%

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

      5. fma-define 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in y around inf 73.4%

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

      1. clear-num 73.4%

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

      2. un-div-inv 73.5%

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

   7. Applied egg-rr 73.5%

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

      1. associate-/r/ 73.6%

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

   9. Simplified 73.6%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{-168}:\\ \;\;\;\;\left(\frac{y}{x} + 1\right) \cdot \left(1 - \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y} \cdot \left(1 + \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 8.4 \cdot 10^{-171}:\\ \;\;\;\;\left(\frac{y\_m}{x} + 1\right) \cdot \left(1 - \frac{y\_m}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y\_m}\right) \cdot \left(-1 + \frac{x}{y\_m}\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 8.4e-171)
   (* (+ (/ y_m x) 1.0) (- 1.0 (/ y_m x)))
   (* (+ 1.0 (/ x y_m)) (+ -1.0 (/ x y_m)))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 8.4e-171) {
		tmp = ((y_m / x) + 1.0) * (1.0 - (y_m / x));
	} else {
		tmp = (1.0 + (x / y_m)) * (-1.0 + (x / 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 <= 8.4d-171) then
        tmp = ((y_m / x) + 1.0d0) * (1.0d0 - (y_m / x))
    else
        tmp = (1.0d0 + (x / y_m)) * ((-1.0d0) + (x / 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 <= 8.4e-171) {
		tmp = ((y_m / x) + 1.0) * (1.0 - (y_m / x));
	} else {
		tmp = (1.0 + (x / y_m)) * (-1.0 + (x / y_m));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 8.4e-171:
		tmp = ((y_m / x) + 1.0) * (1.0 - (y_m / x))
	else:
		tmp = (1.0 + (x / y_m)) * (-1.0 + (x / y_m))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 8.4e-171)
		tmp = Float64(Float64(Float64(y_m / x) + 1.0) * Float64(1.0 - Float64(y_m / x)));
	else
		tmp = Float64(Float64(1.0 + Float64(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)
	tmp = 0.0;
	if (y_m <= 8.4e-171)
		tmp = ((y_m / x) + 1.0) * (1.0 - (y_m / x));
	else
		tmp = (1.0 + (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_] := If[LessEqual[y$95$m, 8.4e-171], N[(N[(N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] * N[(-1.0 + N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.4e-171

   1. Initial program 65.5%

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

      1. associate-/l* 66.3%

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

      2. +-commutative 66.3%

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

      3. +-commutative 66.3%

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

      4. +-commutative 66.3%

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

      5. fma-define 66.3%

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

   3. Simplified 66.3%

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

   5. Taylor expanded in x around inf 38.6%

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

      1. clear-num 38.6%

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

      2. un-div-inv 38.8%

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

   7. Applied egg-rr 38.8%

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

      1. associate-/r/ 38.8%

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

      2. *-commutative 38.8%

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

      3. div-sub 38.8%

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

      4. *-inverses 38.8%

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

   9. Simplified 38.8%

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

   if 8.4e-171 < y

   1. Initial program 98.2%

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

      1. associate-/l* 94.5%

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

      2. +-commutative 94.5%

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

      3. +-commutative 94.5%

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

      4. +-commutative 94.5%

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

      5. fma-define 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in y around inf 73.4%

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

      1. clear-num 73.4%

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

      2. un-div-inv 73.5%

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

   7. Applied egg-rr 73.5%

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

      1. associate-/r/ 73.6%

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

   9. Simplified 73.6%

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

   10. Taylor expanded in x around 0 73.6%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.4 \cdot 10^{-171}:\\ \;\;\;\;\left(\frac{y}{x} + 1\right) \cdot \left(1 - \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{x}{y}\right) \cdot \left(-1 + \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 3.05 \cdot 10^{-170}:\\ \;\;\;\;\left(\frac{y\_m}{x} + 1\right) \cdot \left(1 - \frac{y\_m}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\_m\right) \cdot \frac{1 + \frac{x}{y\_m}}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 3.05e-170)
   (* (+ (/ y_m x) 1.0) (- 1.0 (/ 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 <= 3.05e-170) {
		tmp = ((y_m / x) + 1.0) * (1.0 - (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 <= 3.05d-170) then
        tmp = ((y_m / x) + 1.0d0) * (1.0d0 - (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 <= 3.05e-170) {
		tmp = ((y_m / x) + 1.0) * (1.0 - (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 <= 3.05e-170:
		tmp = ((y_m / x) + 1.0) * (1.0 - (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 <= 3.05e-170)
		tmp = Float64(Float64(Float64(y_m / x) + 1.0) * Float64(1.0 - Float64(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 <= 3.05e-170)
		tmp = ((y_m / x) + 1.0) * (1.0 - (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, 3.05e-170], N[(N[(N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 - N[(y$95$m / x), $MachinePrecision]), $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 < 3.05e-170

   1. Initial program 65.5%

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

      1. associate-/l* 66.3%

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

      2. +-commutative 66.3%

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

      3. +-commutative 66.3%

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

      4. +-commutative 66.3%

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

      5. fma-define 66.3%

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

   3. Simplified 66.3%

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

   5. Taylor expanded in x around inf 38.6%

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

      1. clear-num 38.6%

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

      2. un-div-inv 38.8%

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

   7. Applied egg-rr 38.8%

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

      1. associate-/r/ 38.8%

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

      2. *-commutative 38.8%

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

      3. div-sub 38.8%

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

      4. *-inverses 38.8%

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

   9. Simplified 38.8%

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

   if 3.05e-170 < y

   1. Initial program 98.2%

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

      1. associate-/l* 94.5%

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

      2. +-commutative 94.5%

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

      3. +-commutative 94.5%

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

      4. +-commutative 94.5%

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

      5. fma-define 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in y around inf 73.4%

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

4. Final simplification 46.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.05 \cdot 10^{-170}:\\ \;\;\;\;\left(\frac{y}{x} + 1\right) \cdot \left(1 - \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 2.25 \cdot 10^{-169}:\\ \;\;\;\;\left(\frac{y\_m}{x} + 1\right) \cdot \left(1 - \frac{y\_m}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 2.25e-169) (* (+ (/ 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 <= 2.25e-169) {
		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 <= 2.25d-169) 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 <= 2.25e-169) {
		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 <= 2.25e-169:
		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 <= 2.25e-169)
		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 <= 2.25e-169)
		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, 2.25e-169], 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 < 2.2499999999999999e-169

   1. Initial program 65.5%

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

      1. associate-/l* 66.3%

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

      2. +-commutative 66.3%

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

      3. +-commutative 66.3%

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

      4. +-commutative 66.3%

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

      5. fma-define 66.3%

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

   3. Simplified 66.3%

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

   5. Taylor expanded in x around inf 38.6%

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

      1. clear-num 38.6%

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

      2. un-div-inv 38.8%

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

   7. Applied egg-rr 38.8%

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

      1. associate-/r/ 38.8%

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

      2. *-commutative 38.8%

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

      3. div-sub 38.8%

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

      4. *-inverses 38.8%

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

   9. Simplified 38.8%

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

   if 2.2499999999999999e-169 < y

   1. Initial program 98.2%

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

      1. associate-/l* 94.5%

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

      2. +-commutative 94.5%

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

      3. +-commutative 94.5%

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

      4. +-commutative 94.5%

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

      5. fma-define 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in x around 0 71.9%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{-169}:\\ \;\;\;\;\left(\frac{y}{x} + 1\right) \cdot \left(1 - \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 82.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 5 \cdot 10^{-170}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (if (<= y_m 5e-170) 1.0 -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 5e-170) {
		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 <= 5d-170) 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 <= 5e-170) {
		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 <= 5e-170:
		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 <= 5e-170)
		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 <= 5e-170)
		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, 5e-170], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 5.0000000000000001e-170

   1. Initial program 65.5%

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

      1. associate-/l* 66.3%

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

      2. +-commutative 66.3%

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

      3. +-commutative 66.3%

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

      4. +-commutative 66.3%

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

      5. fma-define 66.3%

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

   3. Simplified 66.3%

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

   5. Taylor expanded in x around inf 37.3%

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

   if 5.0000000000000001e-170 < y

   1. Initial program 98.2%

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

      1. associate-/l* 94.5%

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

      2. +-commutative 94.5%

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

      3. +-commutative 94.5%

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

      4. +-commutative 94.5%

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

      5. fma-define 94.5%

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

   3. Simplified 94.5%

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

   5. Taylor expanded in x around 0 71.9%

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

Alternative 13: 65.9% 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 72.6%

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

   1. associate-/l* 72.5%

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

   2. +-commutative 72.5%

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

   3. +-commutative 72.5%

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

   4. +-commutative 72.5%

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

   5. fma-define 72.5%

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

3. Simplified 72.5%

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

5. Taylor expanded in x around 0 65.1%

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

Developer Target 1: 99.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|\frac{x}{y}\right|\\ \mathbf{if}\;0.5 < t\_0 \land t\_0 < 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{2}{1 + \frac{x}{y} \cdot \frac{x}{y}}\\ \end{array} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = abs((x / y))
    if ((0.5d0 < t_0) .and. (t_0 < 2.0d0)) then
        tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
    else
        tmp = 1.0d0 - (2.0d0 / (1.0d0 + ((x / y) * (x / y))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.abs((x / y));
	double tmp;
	if ((0.5 < t_0) && (t_0 < 2.0)) {
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	} else {
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.fabs((x / y))
	tmp = 0
	if (0.5 < t_0) and (t_0 < 2.0):
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y))
	else:
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))))
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x, y)
	t_0 = abs((x / y));
	tmp = 0.0;
	if ((0.5 < t_0) && (t_0 < 2.0))
		tmp = ((x - y) * (x + y)) / ((x * x) + (y * y));
	else
		tmp = 1.0 - (2.0 / (1.0 + ((x / y) * (x / y))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[Abs[N[(x / y), $MachinePrecision]], $MachinePrecision]}, If[And[Less[0.5, t$95$0], Less[t$95$0, 2.0]], N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(2.0 / N[(1.0 + N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024185 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

  :alt
  (! :herbie-platform default (if (< 1/2 (fabs (/ x y)) 2) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1 (/ 2 (+ 1 (* (/ x y) (/ x y)))))))

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