Kahan p9 Example

Percentage Accurate: 68.4% → 99.9%

Time: 12.3s

Alternatives: 18

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: 68.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.1× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (/ (* (/ (+ x y_m) (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(Float64(x + y_m) / hypot(x, y_m)) * 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[(N[(x + y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.3%

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

   1. add-sqr-sqrt 61.3%

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

   2. times-frac 62.0%

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

   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. associate-*r/ 99.9%

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

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{\frac{x + y}{\mathsf{hypot}\left(x, y\right)} \cdot \left(x - y\right)}{\mathsf{hypot}\left(x, y\right)}} \]
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 61.3%

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

   1. add-sqr-sqrt 61.3%

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

   2. times-frac 62.0%

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

   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. 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 3: 90.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.25 \cdot 10^{-203}:\\ \;\;\;\;\mathsf{fma}\left(-2, {\left(\frac{y\_m}{x}\right)}^{2}, 1\right)\\ \mathbf{elif}\;y\_m \leq 1.2 \cdot 10^{-163}:\\ \;\;\;\;\left(\frac{x}{y\_m} + -1\right) \cdot \frac{1}{\frac{\mathsf{hypot}\left(x, y\_m\right)}{x + y\_m}}\\ \mathbf{elif}\;y\_m \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(x + y\_m\right) \cdot \left(x - y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.25e-203)
   (fma -2.0 (pow (/ y_m x) 2.0) 1.0)
   (if (<= y_m 1.2e-163)
     (* (+ (/ x y_m) -1.0) (/ 1.0 (/ (hypot x y_m) (+ x y_m))))
     (if (<= y_m 5e-15)
       (/ (* (+ 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 <= 1.25e-203) {
		tmp = fma(-2.0, pow((y_m / x), 2.0), 1.0);
	} else if (y_m <= 1.2e-163) {
		tmp = ((x / y_m) + -1.0) * (1.0 / (hypot(x, y_m) / (x + y_m)));
	} else if (y_m <= 5e-15) {
		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 <= 1.25e-203)
		tmp = fma(-2.0, (Float64(y_m / x) ^ 2.0), 1.0);
	elseif (y_m <= 1.2e-163)
		tmp = Float64(Float64(Float64(x / y_m) + -1.0) * Float64(1.0 / Float64(hypot(x, y_m) / Float64(x + y_m))));
	elseif (y_m <= 5e-15)
		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, 1.25e-203], N[(-2.0 * N[Power[N[(y$95$m / x), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision], If[LessEqual[y$95$m, 1.2e-163], N[(N[(N[(x / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * N[(1.0 / N[(N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision] / N[(x + y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 5e-15], 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 4 regimes

2. if y < 1.25e-203

   1. Initial program 54.4%

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

      1. add-sqr-sqrt 54.3%

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

      2. times-frac 55.1%

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

      3. hypot-define 55.2%

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

      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. Step-by-step derivation

      1. *-commutative 99.9%

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

      2. associate-*l/ 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around 0 19.6%

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

      1. +-commutative 19.6%

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

      2. fma-define 19.6%

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

      3. unpow2 19.6%

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

      4. unpow2 19.6%

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

      5. times-frac 31.5%

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

      6. unpow2 31.5%

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

   9. Simplified 31.5%

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

   if 1.25e-203 < y < 1.2e-163

   1. Initial program 37.5%

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

      1. add-sqr-sqrt 37.5%

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

      2. times-frac 39.5%

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

      3. hypot-define 39.5%

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

      4. hypot-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 0 63.9%

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

      1. clear-num 64.1%

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

      2. inv-pow 64.1%

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

   7. Applied egg-rr 64.1%

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

      1. unpow-1 64.1%

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

   9. Simplified 64.1%

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

   if 1.2e-163 < y < 4.99999999999999999e-15

   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 4.99999999999999999e-15 < 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%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. 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 4 regimes into one program.

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{-203}:\\ \;\;\;\;\mathsf{fma}\left(-2, {\left(\frac{y}{x}\right)}^{2}, 1\right)\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{-163}:\\ \;\;\;\;\left(\frac{x}{y} + -1\right) \cdot \frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.3 \cdot 10^{-203}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(x, y\_m\right)}{\left(x - y\_m\right) \cdot \frac{x + y\_m}{x}}}\\ \mathbf{elif}\;y\_m \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\left(\frac{x}{y\_m} + -1\right) \cdot \frac{1}{\frac{\mathsf{hypot}\left(x, y\_m\right)}{x + y\_m}}\\ \mathbf{elif}\;y\_m \leq 5.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(x + y\_m\right) \cdot \left(x - y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.3e-203)
   (/ 1.0 (/ (hypot x y_m) (* (- x y_m) (/ (+ x y_m) x))))
   (if (<= y_m 1.55e-162)
     (* (+ (/ x y_m) -1.0) (/ 1.0 (/ (hypot x y_m) (+ x y_m))))
     (if (<= y_m 5.8e-11)
       (/ (* (+ 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 <= 1.3e-203) {
		tmp = 1.0 / (hypot(x, y_m) / ((x - y_m) * ((x + y_m) / x)));
	} else if (y_m <= 1.55e-162) {
		tmp = ((x / y_m) + -1.0) * (1.0 / (hypot(x, y_m) / (x + y_m)));
	} else if (y_m <= 5.8e-11) {
		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 <= 1.3e-203) {
		tmp = 1.0 / (Math.hypot(x, y_m) / ((x - y_m) * ((x + y_m) / x)));
	} else if (y_m <= 1.55e-162) {
		tmp = ((x / y_m) + -1.0) * (1.0 / (Math.hypot(x, y_m) / (x + y_m)));
	} else if (y_m <= 5.8e-11) {
		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 <= 1.3e-203:
		tmp = 1.0 / (math.hypot(x, y_m) / ((x - y_m) * ((x + y_m) / x)))
	elif y_m <= 1.55e-162:
		tmp = ((x / y_m) + -1.0) * (1.0 / (math.hypot(x, y_m) / (x + y_m)))
	elif y_m <= 5.8e-11:
		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 <= 1.3e-203)
		tmp = Float64(1.0 / Float64(hypot(x, y_m) / Float64(Float64(x - y_m) * Float64(Float64(x + y_m) / x))));
	elseif (y_m <= 1.55e-162)
		tmp = Float64(Float64(Float64(x / y_m) + -1.0) * Float64(1.0 / Float64(hypot(x, y_m) / Float64(x + y_m))));
	elseif (y_m <= 5.8e-11)
		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 <= 1.3e-203)
		tmp = 1.0 / (hypot(x, y_m) / ((x - y_m) * ((x + y_m) / x)));
	elseif (y_m <= 1.55e-162)
		tmp = ((x / y_m) + -1.0) * (1.0 / (hypot(x, y_m) / (x + y_m)));
	elseif (y_m <= 5.8e-11)
		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, 1.3e-203], N[(1.0 / N[(N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision] / N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(x + y$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1.55e-162], N[(N[(N[(x / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * N[(1.0 / N[(N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision] / N[(x + y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 5.8e-11], 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 4 regimes

2. if y < 1.29999999999999988e-203

   1. Initial program 54.4%

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

      1. add-sqr-sqrt 54.3%

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

      2. times-frac 55.1%

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

      3. hypot-define 55.2%

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

      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. Step-by-step derivation

      1. associate-*l/ 99.9%

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

      2. clear-num 99.9%

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in x around inf 31.1%

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

   if 1.29999999999999988e-203 < y < 1.5499999999999999e-162

   1. Initial program 37.5%

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

      1. add-sqr-sqrt 37.5%

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

      2. times-frac 39.5%

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

      3. hypot-define 39.5%

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

      4. hypot-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 0 63.9%

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

      1. clear-num 64.1%

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

      2. inv-pow 64.1%

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

   7. Applied egg-rr 64.1%

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

      1. unpow-1 64.1%

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

   9. Simplified 64.1%

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

   if 1.5499999999999999e-162 < y < 5.8e-11

   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 5.8e-11 < 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%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. 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 4 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-203}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{\left(x - y\right) \cdot \frac{x + y}{x}}}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\left(\frac{x}{y} + -1\right) \cdot \frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.16 \cdot 10^{-203}:\\ \;\;\;\;\frac{x + y\_m}{\mathsf{hypot}\left(x, y\_m\right)} \cdot \frac{x - y\_m}{x}\\ \mathbf{elif}\;y\_m \leq 1.56 \cdot 10^{-162}:\\ \;\;\;\;\left(\frac{x}{y\_m} + -1\right) \cdot \frac{1}{\frac{\mathsf{hypot}\left(x, y\_m\right)}{x + y\_m}}\\ \mathbf{elif}\;y\_m \leq 10^{-12}:\\ \;\;\;\;\frac{\left(x + y\_m\right) \cdot \left(x - y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.16e-203)
   (* (/ (+ x y_m) (hypot x y_m)) (/ (- x y_m) x))
   (if (<= y_m 1.56e-162)
     (* (+ (/ x y_m) -1.0) (/ 1.0 (/ (hypot x y_m) (+ x y_m))))
     (if (<= y_m 1e-12)
       (/ (* (+ 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 <= 1.16e-203) {
		tmp = ((x + y_m) / hypot(x, y_m)) * ((x - y_m) / x);
	} else if (y_m <= 1.56e-162) {
		tmp = ((x / y_m) + -1.0) * (1.0 / (hypot(x, y_m) / (x + y_m)));
	} else if (y_m <= 1e-12) {
		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 <= 1.16e-203) {
		tmp = ((x + y_m) / Math.hypot(x, y_m)) * ((x - y_m) / x);
	} else if (y_m <= 1.56e-162) {
		tmp = ((x / y_m) + -1.0) * (1.0 / (Math.hypot(x, y_m) / (x + y_m)));
	} else if (y_m <= 1e-12) {
		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 <= 1.16e-203:
		tmp = ((x + y_m) / math.hypot(x, y_m)) * ((x - y_m) / x)
	elif y_m <= 1.56e-162:
		tmp = ((x / y_m) + -1.0) * (1.0 / (math.hypot(x, y_m) / (x + y_m)))
	elif y_m <= 1e-12:
		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 <= 1.16e-203)
		tmp = Float64(Float64(Float64(x + y_m) / hypot(x, y_m)) * Float64(Float64(x - y_m) / x));
	elseif (y_m <= 1.56e-162)
		tmp = Float64(Float64(Float64(x / y_m) + -1.0) * Float64(1.0 / Float64(hypot(x, y_m) / Float64(x + y_m))));
	elseif (y_m <= 1e-12)
		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 <= 1.16e-203)
		tmp = ((x + y_m) / hypot(x, y_m)) * ((x - y_m) / x);
	elseif (y_m <= 1.56e-162)
		tmp = ((x / y_m) + -1.0) * (1.0 / (hypot(x, y_m) / (x + y_m)));
	elseif (y_m <= 1e-12)
		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, 1.16e-203], 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, 1.56e-162], N[(N[(N[(x / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * N[(1.0 / N[(N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision] / N[(x + y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1e-12], 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 4 regimes

2. if y < 1.16000000000000004e-203

   1. Initial program 54.4%

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

      1. add-sqr-sqrt 54.3%

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

      2. times-frac 55.1%

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

      3. hypot-define 55.2%

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

      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 inf 31.2%

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

   if 1.16000000000000004e-203 < y < 1.5600000000000001e-162

   1. Initial program 37.5%

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

      1. add-sqr-sqrt 37.5%

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

      2. times-frac 39.5%

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

      3. hypot-define 39.5%

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

      4. hypot-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 0 63.9%

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

      1. clear-num 64.1%

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

      2. inv-pow 64.1%

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

   7. Applied egg-rr 64.1%

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

      1. unpow-1 64.1%

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

   9. Simplified 64.1%

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

   if 1.5600000000000001e-162 < y < 9.9999999999999998e-13

   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 9.9999999999999998e-13 < 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%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. 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 4 regimes into one program.

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.16 \cdot 10^{-203}:\\ \;\;\;\;\frac{x + y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x - y}{x}\\ \mathbf{elif}\;y \leq 1.56 \cdot 10^{-162}:\\ \;\;\;\;\left(\frac{x}{y} + -1\right) \cdot \frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}\\ \mathbf{elif}\;y \leq 10^{-12}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.3 \cdot 10^{-203}:\\ \;\;\;\;\frac{x + y\_m}{\mathsf{hypot}\left(x, y\_m\right)} \cdot \frac{x - y\_m}{x}\\ \mathbf{elif}\;y\_m \leq 1.56 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{x}{y\_m} + -1}{\frac{\mathsf{hypot}\left(x, y\_m\right)}{x + y\_m}}\\ \mathbf{elif}\;y\_m \leq 10^{-12}:\\ \;\;\;\;\frac{\left(x + y\_m\right) \cdot \left(x - y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.3e-203)
   (* (/ (+ x y_m) (hypot x y_m)) (/ (- x y_m) x))
   (if (<= y_m 1.56e-162)
     (/ (+ (/ x y_m) -1.0) (/ (hypot x y_m) (+ x y_m)))
     (if (<= y_m 1e-12)
       (/ (* (+ 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 <= 1.3e-203) {
		tmp = ((x + y_m) / hypot(x, y_m)) * ((x - y_m) / x);
	} else if (y_m <= 1.56e-162) {
		tmp = ((x / y_m) + -1.0) / (hypot(x, y_m) / (x + y_m));
	} else if (y_m <= 1e-12) {
		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 <= 1.3e-203) {
		tmp = ((x + y_m) / Math.hypot(x, y_m)) * ((x - y_m) / x);
	} else if (y_m <= 1.56e-162) {
		tmp = ((x / y_m) + -1.0) / (Math.hypot(x, y_m) / (x + y_m));
	} else if (y_m <= 1e-12) {
		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 <= 1.3e-203:
		tmp = ((x + y_m) / math.hypot(x, y_m)) * ((x - y_m) / x)
	elif y_m <= 1.56e-162:
		tmp = ((x / y_m) + -1.0) / (math.hypot(x, y_m) / (x + y_m))
	elif y_m <= 1e-12:
		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 <= 1.3e-203)
		tmp = Float64(Float64(Float64(x + y_m) / hypot(x, y_m)) * Float64(Float64(x - y_m) / x));
	elseif (y_m <= 1.56e-162)
		tmp = Float64(Float64(Float64(x / y_m) + -1.0) / Float64(hypot(x, y_m) / Float64(x + y_m)));
	elseif (y_m <= 1e-12)
		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 <= 1.3e-203)
		tmp = ((x + y_m) / hypot(x, y_m)) * ((x - y_m) / x);
	elseif (y_m <= 1.56e-162)
		tmp = ((x / y_m) + -1.0) / (hypot(x, y_m) / (x + y_m));
	elseif (y_m <= 1e-12)
		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, 1.3e-203], 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, 1.56e-162], N[(N[(N[(x / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision] / N[(N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision] / N[(x + y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1e-12], 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 4 regimes

2. if y < 1.29999999999999988e-203

   1. Initial program 54.4%

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

      1. add-sqr-sqrt 54.3%

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

      2. times-frac 55.1%

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

      3. hypot-define 55.2%

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

      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 inf 31.2%

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

   if 1.29999999999999988e-203 < y < 1.5600000000000001e-162

   1. Initial program 37.5%

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

      1. add-sqr-sqrt 37.5%

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

      2. times-frac 39.5%

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

      3. hypot-define 39.5%

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

      4. hypot-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 0 63.9%

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

      1. clear-num 64.1%

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

      2. un-div-inv 64.0%

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

      3. sub-neg 64.0%

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

      4. metadata-eval 64.0%

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

   7. Applied egg-rr 64.0%

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

   if 1.5600000000000001e-162 < y < 9.9999999999999998e-13

   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 9.9999999999999998e-13 < 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%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. 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 4 regimes into one program.

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-203}:\\ \;\;\;\;\frac{x + y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x - y}{x}\\ \mathbf{elif}\;y \leq 1.56 \cdot 10^{-162}:\\ \;\;\;\;\frac{\frac{x}{y} + -1}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}\\ \mathbf{elif}\;y \leq 10^{-12}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x + y\_m}{\mathsf{hypot}\left(x, y\_m\right)}\\ \mathbf{if}\;y\_m \leq 1.3 \cdot 10^{-203}:\\ \;\;\;\;t\_0 \cdot \frac{x - y\_m}{x}\\ \mathbf{elif}\;y\_m \leq 1.45 \cdot 10^{-162}:\\ \;\;\;\;t\_0 \cdot \left(\frac{x}{y\_m} + -1\right)\\ \mathbf{elif}\;y\_m \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(x + y\_m\right) \cdot \left(x - y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (/ (+ x y_m) (hypot x y_m))))
   (if (<= y_m 1.3e-203)
     (* t_0 (/ (- x y_m) x))
     (if (<= y_m 1.45e-162)
       (* t_0 (+ (/ x y_m) -1.0))
       (if (<= y_m 5e-11)
         (/ (* (+ 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 t_0 = (x + y_m) / hypot(x, y_m);
	double tmp;
	if (y_m <= 1.3e-203) {
		tmp = t_0 * ((x - y_m) / x);
	} else if (y_m <= 1.45e-162) {
		tmp = t_0 * ((x / y_m) + -1.0);
	} else if (y_m <= 5e-11) {
		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 t_0 = (x + y_m) / Math.hypot(x, y_m);
	double tmp;
	if (y_m <= 1.3e-203) {
		tmp = t_0 * ((x - y_m) / x);
	} else if (y_m <= 1.45e-162) {
		tmp = t_0 * ((x / y_m) + -1.0);
	} else if (y_m <= 5e-11) {
		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):
	t_0 = (x + y_m) / math.hypot(x, y_m)
	tmp = 0
	if y_m <= 1.3e-203:
		tmp = t_0 * ((x - y_m) / x)
	elif y_m <= 1.45e-162:
		tmp = t_0 * ((x / y_m) + -1.0)
	elif y_m <= 5e-11:
		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)
	t_0 = Float64(Float64(x + y_m) / hypot(x, y_m))
	tmp = 0.0
	if (y_m <= 1.3e-203)
		tmp = Float64(t_0 * Float64(Float64(x - y_m) / x));
	elseif (y_m <= 1.45e-162)
		tmp = Float64(t_0 * Float64(Float64(x / y_m) + -1.0));
	elseif (y_m <= 5e-11)
		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)
	t_0 = (x + y_m) / hypot(x, y_m);
	tmp = 0.0;
	if (y_m <= 1.3e-203)
		tmp = t_0 * ((x - y_m) / x);
	elseif (y_m <= 1.45e-162)
		tmp = t_0 * ((x / y_m) + -1.0);
	elseif (y_m <= 5e-11)
		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_] := Block[{t$95$0 = N[(N[(x + y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$95$m, 1.3e-203], N[(t$95$0 * N[(N[(x - y$95$m), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1.45e-162], N[(t$95$0 * N[(N[(x / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 5e-11], 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 4 regimes

2. if y < 1.29999999999999988e-203

   1. Initial program 54.4%

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

      1. add-sqr-sqrt 54.3%

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

      2. times-frac 55.1%

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

      3. hypot-define 55.2%

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

      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 inf 31.2%

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

   if 1.29999999999999988e-203 < y < 1.4500000000000001e-162

   1. Initial program 37.5%

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

      1. add-sqr-sqrt 37.5%

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

      2. times-frac 39.5%

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

      3. hypot-define 39.5%

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

      4. hypot-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 0 63.9%

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

   if 1.4500000000000001e-162 < y < 5.00000000000000018e-11

   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 5.00000000000000018e-11 < 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%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. 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 4 regimes into one program.

4. Final simplification 43.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-203}:\\ \;\;\;\;\frac{x + y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x - y}{x}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-162}:\\ \;\;\;\;\frac{x + y}{\mathsf{hypot}\left(x, y\right)} \cdot \left(\frac{x}{y} + -1\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.3 \cdot 10^{-203}:\\ \;\;\;\;\frac{1}{\frac{x}{\left(x - y\_m\right) \cdot \left(\frac{y\_m}{x} + 1\right)}}\\ \mathbf{elif}\;y\_m \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{x + y\_m}{\mathsf{hypot}\left(x, y\_m\right)} \cdot \left(\frac{x}{y\_m} + -1\right)\\ \mathbf{elif}\;y\_m \leq 5.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(x + y\_m\right) \cdot \left(x - y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.3e-203)
   (/ 1.0 (/ x (* (- x y_m) (+ (/ y_m x) 1.0))))
   (if (<= y_m 1.55e-162)
     (* (/ (+ x y_m) (hypot x y_m)) (+ (/ x y_m) -1.0))
     (if (<= y_m 5.5e-11)
       (/ (* (+ 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 <= 1.3e-203) {
		tmp = 1.0 / (x / ((x - y_m) * ((y_m / x) + 1.0)));
	} else if (y_m <= 1.55e-162) {
		tmp = ((x + y_m) / hypot(x, y_m)) * ((x / y_m) + -1.0);
	} else if (y_m <= 5.5e-11) {
		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 <= 1.3e-203) {
		tmp = 1.0 / (x / ((x - y_m) * ((y_m / x) + 1.0)));
	} else if (y_m <= 1.55e-162) {
		tmp = ((x + y_m) / Math.hypot(x, y_m)) * ((x / y_m) + -1.0);
	} else if (y_m <= 5.5e-11) {
		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 <= 1.3e-203:
		tmp = 1.0 / (x / ((x - y_m) * ((y_m / x) + 1.0)))
	elif y_m <= 1.55e-162:
		tmp = ((x + y_m) / math.hypot(x, y_m)) * ((x / y_m) + -1.0)
	elif y_m <= 5.5e-11:
		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 <= 1.3e-203)
		tmp = Float64(1.0 / Float64(x / Float64(Float64(x - y_m) * Float64(Float64(y_m / x) + 1.0))));
	elseif (y_m <= 1.55e-162)
		tmp = Float64(Float64(Float64(x + y_m) / hypot(x, y_m)) * Float64(Float64(x / y_m) + -1.0));
	elseif (y_m <= 5.5e-11)
		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 <= 1.3e-203)
		tmp = 1.0 / (x / ((x - y_m) * ((y_m / x) + 1.0)));
	elseif (y_m <= 1.55e-162)
		tmp = ((x + y_m) / hypot(x, y_m)) * ((x / y_m) + -1.0);
	elseif (y_m <= 5.5e-11)
		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, 1.3e-203], 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, 1.55e-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] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 5.5e-11], 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 4 regimes

2. if y < 1.29999999999999988e-203

   1. Initial program 54.4%

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

      1. associate-/l* 54.9%

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

      2. +-commutative 54.9%

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

      3. +-commutative 54.9%

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

      4. +-commutative 54.9%

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

      5. fma-define 54.9%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in x around inf 30.5%

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

      1. associate-*r/ 30.6%

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

      2. clear-num 30.6%

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

   7. Applied egg-rr 30.6%

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

   if 1.29999999999999988e-203 < y < 1.5499999999999999e-162

   1. Initial program 37.5%

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

      1. add-sqr-sqrt 37.5%

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

      2. times-frac 39.5%

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

      3. hypot-define 39.5%

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

      4. hypot-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 0 63.9%

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

   if 1.5499999999999999e-162 < y < 5.49999999999999975e-11

   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 5.49999999999999975e-11 < 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%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. 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 4 regimes into one program.

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-203}:\\ \;\;\;\;\frac{1}{\frac{x}{\left(x - y\right) \cdot \left(\frac{y}{x} + 1\right)}}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-162}:\\ \;\;\;\;\frac{x + y}{\mathsf{hypot}\left(x, y\right)} \cdot \left(\frac{x}{y} + -1\right)\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.3 \cdot 10^{-203}:\\ \;\;\;\;\frac{1}{\frac{x}{\left(x - y\_m\right) \cdot \left(\frac{y\_m}{x} + 1\right)}}\\ \mathbf{elif}\;y\_m \leq 1.15 \cdot 10^{-163}:\\ \;\;\;\;\left(x - y\_m\right) \cdot \frac{1}{y\_m + x \cdot \left(\frac{x}{y\_m} + -1\right)}\\ \mathbf{elif}\;y\_m \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(x + y\_m\right) \cdot \left(x - y\_m\right)}{x \cdot x + y\_m \cdot y\_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.3e-203)
   (/ 1.0 (/ x (* (- x y_m) (+ (/ y_m x) 1.0))))
   (if (<= y_m 1.15e-163)
     (* (- x y_m) (/ 1.0 (+ y_m (* x (+ (/ x y_m) -1.0)))))
     (if (<= y_m 5e-15)
       (/ (* (+ 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 <= 1.3e-203) {
		tmp = 1.0 / (x / ((x - y_m) * ((y_m / x) + 1.0)));
	} else if (y_m <= 1.15e-163) {
		tmp = (x - y_m) * (1.0 / (y_m + (x * ((x / y_m) + -1.0))));
	} else if (y_m <= 5e-15) {
		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 <= 1.3d-203) then
        tmp = 1.0d0 / (x / ((x - y_m) * ((y_m / x) + 1.0d0)))
    else if (y_m <= 1.15d-163) then
        tmp = (x - y_m) * (1.0d0 / (y_m + (x * ((x / y_m) + (-1.0d0)))))
    else if (y_m <= 5d-15) 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 <= 1.3e-203) {
		tmp = 1.0 / (x / ((x - y_m) * ((y_m / x) + 1.0)));
	} else if (y_m <= 1.15e-163) {
		tmp = (x - y_m) * (1.0 / (y_m + (x * ((x / y_m) + -1.0))));
	} else if (y_m <= 5e-15) {
		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 <= 1.3e-203:
		tmp = 1.0 / (x / ((x - y_m) * ((y_m / x) + 1.0)))
	elif y_m <= 1.15e-163:
		tmp = (x - y_m) * (1.0 / (y_m + (x * ((x / y_m) + -1.0))))
	elif y_m <= 5e-15:
		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 <= 1.3e-203)
		tmp = Float64(1.0 / Float64(x / Float64(Float64(x - y_m) * Float64(Float64(y_m / x) + 1.0))));
	elseif (y_m <= 1.15e-163)
		tmp = Float64(Float64(x - y_m) * Float64(1.0 / Float64(y_m + Float64(x * Float64(Float64(x / y_m) + -1.0)))));
	elseif (y_m <= 5e-15)
		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 <= 1.3e-203)
		tmp = 1.0 / (x / ((x - y_m) * ((y_m / x) + 1.0)));
	elseif (y_m <= 1.15e-163)
		tmp = (x - y_m) * (1.0 / (y_m + (x * ((x / y_m) + -1.0))));
	elseif (y_m <= 5e-15)
		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, 1.3e-203], 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, 1.15e-163], N[(N[(x - y$95$m), $MachinePrecision] * N[(1.0 / N[(y$95$m + N[(x * N[(N[(x / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 5e-15], 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 4 regimes

2. if y < 1.29999999999999988e-203

   1. Initial program 54.4%

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

      1. associate-/l* 54.9%

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

      2. +-commutative 54.9%

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

      3. +-commutative 54.9%

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

      4. +-commutative 54.9%

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

      5. fma-define 54.9%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in x around inf 30.5%

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

      1. associate-*r/ 30.6%

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

      2. clear-num 30.6%

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

   7. Applied egg-rr 30.6%

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

   if 1.29999999999999988e-203 < y < 1.15e-163

   1. Initial program 37.5%

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

      1. associate-/l* 39.3%

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

      2. +-commutative 39.3%

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

      3. +-commutative 39.3%

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

      4. +-commutative 39.3%

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

      5. fma-define 39.3%

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

   3. Simplified 39.3%

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

   5. Taylor expanded in y around inf 63.2%

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

   6. Taylor expanded in y around 0 63.2%

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

      1. clear-num 63.2%

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

      2. inv-pow 63.2%

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

   8. Applied egg-rr 63.2%

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

      1. unpow-1 63.2%

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

      2. associate-/r/ 63.2%

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

   10. Simplified 63.2%

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

   11. Taylor expanded in x around 0 63.7%

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

   if 1.15e-163 < y < 4.99999999999999999e-15

   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 4.99999999999999999e-15 < 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%

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

      3. +-commutative 99.6%

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

      4. +-commutative 99.6%

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

      5. 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 4 regimes into one program.

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-203}:\\ \;\;\;\;\frac{1}{\frac{x}{\left(x - y\right) \cdot \left(\frac{y}{x} + 1\right)}}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-163}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1}{y + x \cdot \left(\frac{x}{y} + -1\right)}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.05 \cdot 10^{-203}:\\ \;\;\;\;\frac{1}{\frac{x}{\left(x - y\_m\right) \cdot \left(\frac{y\_m}{x} + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y\_m} + -1\right) \cdot \frac{x + y\_m}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.05e-203)
   (/ 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 <= 1.05e-203) {
		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 <= 1.05d-203) 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 <= 1.05e-203) {
		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 <= 1.05e-203:
		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 <= 1.05e-203)
		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) + -1.0) * Float64(Float64(x + y_m) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 1.05e-203)
		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, 1.05e-203], 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] + -1.0), $MachinePrecision] * N[(N[(x + y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.05000000000000001e-203

   1. Initial program 54.4%

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

      1. associate-/l* 54.9%

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

      2. +-commutative 54.9%

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

      3. +-commutative 54.9%

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

      4. +-commutative 54.9%

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

      5. fma-define 54.9%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in x around inf 30.5%

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

      1. associate-*r/ 30.6%

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

      2. clear-num 30.6%

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

   7. Applied egg-rr 30.6%

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

   if 1.05000000000000001e-203 < y

   1. Initial program 89.9%

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

      1. add-sqr-sqrt 89.9%

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

      2. times-frac 90.2%

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

      3. hypot-define 90.2%

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

      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 0 79.7%

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

   6. Taylor expanded in x around 0 79.3%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-203}:\\ \;\;\;\;\frac{1}{\frac{x}{\left(x - y\right) \cdot \left(\frac{y}{x} + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} + -1\right) \cdot \frac{x + y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 9 \cdot 10^{-204}:\\ \;\;\;\;\frac{x - y\_m}{\frac{x}{\frac{y\_m}{x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y\_m} + -1\right) \cdot \frac{x + y\_m}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 9e-204)
   (/ (- 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 <= 9e-204) {
		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 <= 9d-204) 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 <= 9e-204) {
		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 <= 9e-204:
		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 <= 9e-204)
		tmp = Float64(Float64(x - y_m) / Float64(x / Float64(Float64(y_m / x) + 1.0)));
	else
		tmp = Float64(Float64(Float64(x / y_m) + -1.0) * Float64(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 <= 9e-204)
		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, 9e-204], 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] + -1.0), $MachinePrecision] * N[(N[(x + y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.99999999999999948e-204

   1. Initial program 54.4%

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

      1. associate-/l* 54.9%

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

      2. +-commutative 54.9%

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

      3. +-commutative 54.9%

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

      4. +-commutative 54.9%

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

      5. fma-define 54.9%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in x around inf 30.5%

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

      1. clear-num 30.5%

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

      2. un-div-inv 30.5%

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

   7. Applied egg-rr 30.5%

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

   if 8.99999999999999948e-204 < y

   1. Initial program 89.9%

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

      1. add-sqr-sqrt 89.9%

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

      2. times-frac 90.2%

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

      3. hypot-define 90.2%

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

      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 0 79.7%

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

   6. Taylor expanded in x around 0 79.3%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-204}:\\ \;\;\;\;\frac{x - y}{\frac{x}{\frac{y}{x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} + -1\right) \cdot \frac{x + y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 82.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 9 \cdot 10^{-204}:\\ \;\;\;\;\left(x - y\_m\right) \cdot \frac{\frac{x + y\_m}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y\_m} + -1\right) \cdot \frac{x + y\_m}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 9e-204)
   (* (- x y_m) (/ (/ (+ x 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 <= 9e-204) {
		tmp = (x - y_m) * (((x + 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 <= 9d-204) then
        tmp = (x - y_m) * (((x + 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 <= 9e-204) {
		tmp = (x - y_m) * (((x + 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 <= 9e-204:
		tmp = (x - y_m) * (((x + 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 <= 9e-204)
		tmp = Float64(Float64(x - y_m) * Float64(Float64(Float64(x + y_m) / x) / x));
	else
		tmp = Float64(Float64(Float64(x / y_m) + -1.0) * Float64(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 <= 9e-204)
		tmp = (x - y_m) * (((x + 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, 9e-204], N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(N[(x + y$95$m), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[(x + y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.99999999999999948e-204

   1. Initial program 54.4%

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

      1. associate-/l* 54.9%

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

      2. +-commutative 54.9%

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

      3. +-commutative 54.9%

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

      4. +-commutative 54.9%

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

      5. fma-define 54.9%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in x around inf 30.5%

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

   6. Taylor expanded in x around 0 30.5%

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

   if 8.99999999999999948e-204 < y

   1. Initial program 89.9%

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

      1. add-sqr-sqrt 89.9%

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

      2. times-frac 90.2%

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

      3. hypot-define 90.2%

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

      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 0 79.7%

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

   6. Taylor expanded in x around 0 79.3%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-204}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x + y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} + -1\right) \cdot \frac{x + y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 82.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.3 \cdot 10^{-203}:\\ \;\;\;\;\left(x - y\_m\right) \cdot \frac{\frac{x + y\_m}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\_m\right) \cdot \frac{\frac{x + y\_m}{y\_m}}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.3e-203)
   (* (- x y_m) (/ (/ (+ x y_m) x) x))
   (* (- x y_m) (/ (/ (+ x y_m) y_m) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.3e-203) {
		tmp = (x - y_m) * (((x + y_m) / x) / x);
	} else {
		tmp = (x - y_m) * (((x + y_m) / y_m) / y_m);
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 1.3d-203) then
        tmp = (x - y_m) * (((x + y_m) / x) / x)
    else
        tmp = (x - y_m) * (((x + y_m) / y_m) / y_m)
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.3e-203) {
		tmp = (x - y_m) * (((x + y_m) / x) / x);
	} else {
		tmp = (x - y_m) * (((x + y_m) / y_m) / y_m);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.3e-203:
		tmp = (x - y_m) * (((x + y_m) / x) / x)
	else:
		tmp = (x - y_m) * (((x + y_m) / y_m) / y_m)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.3e-203)
		tmp = Float64(Float64(x - y_m) * Float64(Float64(Float64(x + y_m) / x) / x));
	else
		tmp = Float64(Float64(x - y_m) * Float64(Float64(Float64(x + y_m) / y_m) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 1.3e-203)
		tmp = (x - y_m) * (((x + y_m) / x) / x);
	else
		tmp = (x - y_m) * (((x + y_m) / y_m) / y_m);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.3e-203], N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(N[(x + y$95$m), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(N[(x + y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.29999999999999988e-203

   1. Initial program 54.4%

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

      1. associate-/l* 54.9%

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

      2. +-commutative 54.9%

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

      3. +-commutative 54.9%

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

      4. +-commutative 54.9%

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

      5. fma-define 54.9%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in x around inf 30.5%

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

   6. Taylor expanded in x around 0 30.5%

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

   if 1.29999999999999988e-203 < y

   1. Initial program 89.9%

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

      1. associate-/l* 89.9%

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

      2. +-commutative 89.9%

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

      3. +-commutative 89.9%

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

      4. +-commutative 89.9%

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

      5. fma-define 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in y around inf 79.1%

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

   6. Taylor expanded in y around 0 79.1%

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

Alternative 14: 82.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.3 \cdot 10^{-203}:\\ \;\;\;\;\left(x - y\_m\right) \cdot \frac{\frac{x + y\_m}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\_m\right) \cdot \frac{1 + \frac{x}{y\_m}}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.3e-203)
   (* (- x y_m) (/ (/ (+ x 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 <= 1.3e-203) {
		tmp = (x - y_m) * (((x + 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 <= 1.3d-203) then
        tmp = (x - y_m) * (((x + 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 <= 1.3e-203) {
		tmp = (x - y_m) * (((x + 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 <= 1.3e-203:
		tmp = (x - y_m) * (((x + 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 <= 1.3e-203)
		tmp = Float64(Float64(x - y_m) * Float64(Float64(Float64(x + 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 <= 1.3e-203)
		tmp = (x - y_m) * (((x + 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, 1.3e-203], N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(N[(x + y$95$m), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(1.0 + N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.29999999999999988e-203

   1. Initial program 54.4%

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

      1. associate-/l* 54.9%

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

      2. +-commutative 54.9%

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

      3. +-commutative 54.9%

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

      4. +-commutative 54.9%

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

      5. fma-define 54.9%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in x around inf 30.5%

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

   6. Taylor expanded in x around 0 30.5%

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

   if 1.29999999999999988e-203 < y

   1. Initial program 89.9%

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

      1. associate-/l* 89.9%

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

      2. +-commutative 89.9%

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

      3. +-commutative 89.9%

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

      4. +-commutative 89.9%

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

      5. fma-define 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in y around inf 79.1%

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

Alternative 15: 82.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.3 \cdot 10^{-203}:\\ \;\;\;\;\left(x - y\_m\right) \cdot \frac{\frac{y\_m}{x} + 1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\_m\right) \cdot \frac{1 + \frac{x}{y\_m}}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.3e-203)
   (* (- x y_m) (/ (+ (/ y_m x) 1.0) x))
   (* (- x y_m) (/ (+ 1.0 (/ x y_m)) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.3e-203) {
		tmp = (x - y_m) * (((y_m / x) + 1.0) / x);
	} else {
		tmp = (x - y_m) * ((1.0 + (x / y_m)) / y_m);
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 1.3d-203) then
        tmp = (x - y_m) * (((y_m / x) + 1.0d0) / x)
    else
        tmp = (x - y_m) * ((1.0d0 + (x / y_m)) / y_m)
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.3e-203) {
		tmp = (x - y_m) * (((y_m / x) + 1.0) / x);
	} else {
		tmp = (x - y_m) * ((1.0 + (x / y_m)) / y_m);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.3e-203:
		tmp = (x - y_m) * (((y_m / x) + 1.0) / x)
	else:
		tmp = (x - y_m) * ((1.0 + (x / y_m)) / y_m)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.3e-203)
		tmp = Float64(Float64(x - y_m) * Float64(Float64(Float64(y_m / x) + 1.0) / x));
	else
		tmp = Float64(Float64(x - y_m) * Float64(Float64(1.0 + Float64(x / y_m)) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 1.3e-203)
		tmp = (x - y_m) * (((y_m / x) + 1.0) / x);
	else
		tmp = (x - y_m) * ((1.0 + (x / y_m)) / y_m);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.3e-203], N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(1.0 + N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.29999999999999988e-203

   1. Initial program 54.4%

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

      1. associate-/l* 54.9%

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

      2. +-commutative 54.9%

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

      3. +-commutative 54.9%

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

      4. +-commutative 54.9%

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

      5. fma-define 54.9%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in x around inf 30.5%

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

   if 1.29999999999999988e-203 < y

   1. Initial program 89.9%

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

      1. associate-/l* 89.9%

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

      2. +-commutative 89.9%

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

      3. +-commutative 89.9%

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

      4. +-commutative 89.9%

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

      5. fma-define 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in y around inf 79.1%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-203}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{y}{x} + 1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{x}{y}}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.3 \cdot 10^{-203}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\_m\right) \cdot \frac{1 + \frac{x}{y\_m}}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.3e-203) 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 <= 1.3e-203) {
		tmp = 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 <= 1.3d-203) then
        tmp = 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 <= 1.3e-203) {
		tmp = 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 <= 1.3e-203:
		tmp = 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 <= 1.3e-203)
		tmp = 1.0;
	else
		tmp = Float64(Float64(x - y_m) * Float64(Float64(1.0 + Float64(x / y_m)) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 1.3e-203)
		tmp = 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, 1.3e-203], 1.0, N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(1.0 + N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.29999999999999988e-203

   1. Initial program 54.4%

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

      1. associate-/l* 54.9%

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

      2. +-commutative 54.9%

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

      3. +-commutative 54.9%

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

      4. +-commutative 54.9%

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

      5. fma-define 54.9%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in x around inf 28.9%

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

   if 1.29999999999999988e-203 < y

   1. Initial program 89.9%

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

      1. associate-/l* 89.9%

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

      2. +-commutative 89.9%

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

      3. +-commutative 89.9%

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

      4. +-commutative 89.9%

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

      5. fma-define 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in y around inf 79.1%

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

Alternative 17: 81.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.16 \cdot 10^{-203}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (if (<= y_m 1.16e-203) 1.0 -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.16e-203) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 1.16d-203) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.16e-203) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.16e-203:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.16e-203)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 1.16e-203)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.16e-203], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.16000000000000004e-203

   1. Initial program 54.4%

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

      1. associate-/l* 54.9%

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

      2. +-commutative 54.9%

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

      3. +-commutative 54.9%

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

      4. +-commutative 54.9%

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

      5. fma-define 54.9%

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

   3. Simplified 54.9%

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

   5. Taylor expanded in x around inf 28.9%

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

   if 1.16000000000000004e-203 < y

   1. Initial program 89.9%

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

      1. associate-/l* 89.9%

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

      2. +-commutative 89.9%

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

      3. +-commutative 89.9%

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

      4. +-commutative 89.9%

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

      5. fma-define 89.9%

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

   3. Simplified 89.9%

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

   5. Taylor expanded in x around 0 78.6%

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

Alternative 18: 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 61.3%

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

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

   3. +-commutative 61.7%

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

   4. +-commutative 61.7%

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

   5. 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 0 72.1%

   \[\leadsto \color{blue}{-1} \]
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 2024152 
(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))))