Kahan p9 Example

Percentage Accurate: 68.5% → 100.0%

Time: 8.6s

Alternatives: 14

Speedup: 4.9×

Specification

\[\left(0 < x \land x < 1\right) \land y < 1\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (/ (- x y) (* (hypot x y) (/ (hypot x y) (+ x y)))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := N[(N[(x - y), $MachinePrecision] / N[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] * N[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] / N[(x + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.9%

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

   1. associate-/l* 63.4%

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

   2. fma-def 63.4%

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

3. Simplified 63.4%

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

   1. fma-def 63.4%

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

   2. div-inv 63.2%

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

   3. add-sqr-sqrt 63.2%

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

   4. associate-*l* 63.3%

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

   5. hypot-def 63.4%

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

   6. hypot-def 99.7%

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

5. Applied egg-rr 99.7%

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

   1. expm1-log1p-u 98.3%

      \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(x, y\right) \cdot \frac{1}{x + y}\right)\right)}} \]

   2. expm1-udef 98.2%

      \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{hypot}\left(x, y\right) \cdot \frac{1}{x + y}\right)} - 1\right)}} \]

   3. un-div-inv 98.4%

      \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{\mathsf{hypot}\left(x, y\right)}{x + y}}\right)} - 1\right)} \]

7. Applied egg-rr 98.4%

   \[\leadsto \frac{x - y}{\mathsf{hypot}\left(x, y\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\mathsf{hypot}\left(x, y\right)}{x + y}\right)} - 1\right)}} \]
8. Step-by-step derivation

   1. expm1-def 98.4%

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

   2. expm1-log1p 99.9%

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

9. Simplified 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (* (/ (+ x y) (hypot x y)) (/ (- x y) (hypot x y))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := N[(N[(N[(x + y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x - y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.9%

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

   1. add-sqr-sqrt 62.9%

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

   2. times-frac 63.4%

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

   3. hypot-def 63.4%

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

   4. hypot-def 99.9%

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

3. Applied egg-rr 99.9%

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

4. Final simplification 99.9%

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

Alternative 3: 99.9% accurate, 0.1× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (/ (* (- x y) (/ (+ x y) (hypot x y))) (hypot x y)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := N[(N[(N[(x - y), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.9%

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

   1. add-sqr-sqrt 62.9%

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

   2. times-frac 63.4%

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

   3. hypot-def 63.4%

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

   4. hypot-def 99.9%

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

3. Applied egg-rr 99.9%

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

   1. associate-*l/ 100.0%

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

5. Applied egg-rr 100.0%

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

6. Final simplification 100.0%

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

Alternative 4: 93.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{1.5 \cdot \left(x \cdot \frac{x}{y}\right) - y}}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))))
   (if (<= t_0 2.0) t_0 (/ 1.0 (/ (hypot x y) (- (* 1.5 (* x (/ x y))) y))))))

C

y = abs(y);
double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (hypot(x, y) / ((1.5 * (x * (x / y))) - y));
	}
	return tmp;
}

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (Math.hypot(x, y) / ((1.5 * (x * (x / y))) - y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y))
	tmp = 0
	if t_0 <= 2.0:
		tmp = t_0
	else:
		tmp = 1.0 / (math.hypot(x, y) / ((1.5 * (x * (x / y))) - y))
	return tmp

Julia

y = abs(y)
function code(x, y)
	t_0 = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(hypot(x, y) / Float64(Float64(1.5 * Float64(x * Float64(x / y))) - y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = 1.0 / (hypot(x, y) / ((1.5 * (x * (x / y))) - y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], t$95$0, N[(1.0 / N[(N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision] / N[(N[(1.5 * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. times-frac 3.1%

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

      3. hypot-def 3.1%

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

      4. hypot-def 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 13.9%

      \[\leadsto \frac{\color{blue}{1.5 \cdot \frac{{x}^{2}}{y} + -1 \cdot y}}{\mathsf{hypot}\left(x, y\right)} \]
   7. Step-by-step derivation

      1. fma-def 13.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1.5, \frac{{x}^{2}}{y}, -1 \cdot y\right)}}{\mathsf{hypot}\left(x, y\right)} \]

      2. unpow2 13.9%

         \[\leadsto \frac{\mathsf{fma}\left(1.5, \frac{\color{blue}{x \cdot x}}{y}, -1 \cdot y\right)}{\mathsf{hypot}\left(x, y\right)} \]

      3. associate-*r/ 15.1%

         \[\leadsto \frac{\mathsf{fma}\left(1.5, \color{blue}{x \cdot \frac{x}{y}}, -1 \cdot y\right)}{\mathsf{hypot}\left(x, y\right)} \]

      4. mul-1-neg 15.1%

         \[\leadsto \frac{\mathsf{fma}\left(1.5, x \cdot \frac{x}{y}, \color{blue}{-y}\right)}{\mathsf{hypot}\left(x, y\right)} \]

   8. Simplified 15.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1.5, x \cdot \frac{x}{y}, -y\right)}}{\mathsf{hypot}\left(x, y\right)} \]
   9. Step-by-step derivation

      1. clear-num 15.1%

         \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{\mathsf{fma}\left(1.5, x \cdot \frac{x}{y}, -y\right)}}} \]

      2. inv-pow 15.1%

         \[\leadsto \color{blue}{{\left(\frac{\mathsf{hypot}\left(x, y\right)}{\mathsf{fma}\left(1.5, x \cdot \frac{x}{y}, -y\right)}\right)}^{-1}} \]

   10. Applied egg-rr 15.1%

       \[\leadsto \color{blue}{{\left(\frac{\mathsf{hypot}\left(x, y\right)}{\mathsf{fma}\left(1.5, x \cdot \frac{x}{y}, -y\right)}\right)}^{-1}} \]
   11. Step-by-step derivation

       1. unpow-1 15.1%

          \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{\mathsf{fma}\left(1.5, x \cdot \frac{x}{y}, -y\right)}}} \]

       2. fma-neg 15.1%

          \[\leadsto \frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{\color{blue}{1.5 \cdot \left(x \cdot \frac{x}{y}\right) - y}}} \]

   12. Simplified 15.1%

       \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{1.5 \cdot \left(x \cdot \frac{x}{y}\right) - y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(x, y\right)}{1.5 \cdot \left(x \cdot \frac{x}{y}\right) - y}}\\ \end{array} \]

Alternative 5: 93.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1.5 \cdot \left(x \cdot \frac{x}{y}\right) - y}{\mathsf{hypot}\left(x, y\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))))
   (if (<= t_0 2.0) t_0 (/ (- (* 1.5 (* x (/ x y))) y) (hypot x y)))))

C

y = abs(y);
double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = ((1.5 * (x * (x / y))) - y) / hypot(x, y);
	}
	return tmp;
}

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = ((1.5 * (x * (x / y))) - y) / Math.hypot(x, y);
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y))
	tmp = 0
	if t_0 <= 2.0:
		tmp = t_0
	else:
		tmp = ((1.5 * (x * (x / y))) - y) / math.hypot(x, y)
	return tmp

Julia

y = abs(y)
function code(x, y)
	t_0 = Float64(Float64(Float64(x - y) * Float64(x + y)) / Float64(Float64(x * x) + Float64(y * y)))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(1.5 * Float64(x * Float64(x / y))) - y) / hypot(x, y));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = ((1.5 * (x * (x / y))) - y) / hypot(x, y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], t$95$0, N[(N[(N[(1.5 * N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

      1. add-sqr-sqrt 0.0%

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

      2. times-frac 3.1%

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

      3. hypot-def 3.1%

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

      4. hypot-def 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 13.9%

      \[\leadsto \frac{\color{blue}{1.5 \cdot \frac{{x}^{2}}{y} + -1 \cdot y}}{\mathsf{hypot}\left(x, y\right)} \]
   7. Step-by-step derivation

      1. fma-def 13.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1.5, \frac{{x}^{2}}{y}, -1 \cdot y\right)}}{\mathsf{hypot}\left(x, y\right)} \]

      2. unpow2 13.9%

         \[\leadsto \frac{\mathsf{fma}\left(1.5, \frac{\color{blue}{x \cdot x}}{y}, -1 \cdot y\right)}{\mathsf{hypot}\left(x, y\right)} \]

      3. associate-*r/ 15.1%

         \[\leadsto \frac{\mathsf{fma}\left(1.5, \color{blue}{x \cdot \frac{x}{y}}, -1 \cdot y\right)}{\mathsf{hypot}\left(x, y\right)} \]

      4. mul-1-neg 15.1%

         \[\leadsto \frac{\mathsf{fma}\left(1.5, x \cdot \frac{x}{y}, \color{blue}{-y}\right)}{\mathsf{hypot}\left(x, y\right)} \]

   8. Simplified 15.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1.5, x \cdot \frac{x}{y}, -y\right)}}{\mathsf{hypot}\left(x, y\right)} \]

   9. Taylor expanded in x around 0 13.9%

      \[\leadsto \frac{\color{blue}{1.5 \cdot \frac{{x}^{2}}{y} + -1 \cdot y}}{\mathsf{hypot}\left(x, y\right)} \]
   10. Step-by-step derivation

       1. fma-def 13.9%

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1.5, \frac{{x}^{2}}{y}, -1 \cdot y\right)}}{\mathsf{hypot}\left(x, y\right)} \]

       2. unpow2 13.9%

          \[\leadsto \frac{\mathsf{fma}\left(1.5, \frac{\color{blue}{x \cdot x}}{y}, -1 \cdot y\right)}{\mathsf{hypot}\left(x, y\right)} \]

       3. associate-*r/ 15.1%

          \[\leadsto \frac{\mathsf{fma}\left(1.5, \color{blue}{x \cdot \frac{x}{y}}, -1 \cdot y\right)}{\mathsf{hypot}\left(x, y\right)} \]

       4. neg-mul-1 15.1%

          \[\leadsto \frac{\mathsf{fma}\left(1.5, x \cdot \frac{x}{y}, \color{blue}{-y}\right)}{\mathsf{hypot}\left(x, y\right)} \]

       5. fma-neg 15.1%

          \[\leadsto \frac{\color{blue}{1.5 \cdot \left(x \cdot \frac{x}{y}\right) - y}}{\mathsf{hypot}\left(x, y\right)} \]

   11. Simplified 15.1%

       \[\leadsto \frac{\color{blue}{1.5 \cdot \left(x \cdot \frac{x}{y}\right) - y}}{\mathsf{hypot}\left(x, y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{1.5 \cdot \left(x \cdot \frac{x}{y}\right) - y}{\mathsf{hypot}\left(x, y\right)}\\ \end{array} \]

Alternative 6: 92.9% accurate, 0.5× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))))
   (if (<= t_0 2.0) t_0 (+ (/ (* x (/ x y)) y) (+ (* (/ x y) (/ x y)) -1.0)))))

C

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

Fortran

NOTE: y should be positive before calling this function
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 = ((x - y) * (x + y)) / ((x * x) + (y * y))
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = ((x * (x / y)) / y) + (((x / y) * (x / y)) + (-1.0d0))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], t$95$0, N[(N[(N[(x * N[(x / y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + N[(N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in y around inf 46.3%

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

      1. distribute-rgt1-in 46.3%

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

      2. metadata-eval 46.3%

         \[\leadsto \left(\frac{{x}^{2}}{{y}^{2}} + \color{blue}{0} \cdot \frac{x}{y}\right) - \left(-1 \cdot \frac{{x}^{2}}{{y}^{2}} + 1\right) \]

      3. mul0-lft 46.3%

         \[\leadsto \left(\frac{{x}^{2}}{{y}^{2}} + \color{blue}{0}\right) - \left(-1 \cdot \frac{{x}^{2}}{{y}^{2}} + 1\right) \]

      4. +-commutative 46.3%

         \[\leadsto \color{blue}{\left(0 + \frac{{x}^{2}}{{y}^{2}}\right)} - \left(-1 \cdot \frac{{x}^{2}}{{y}^{2}} + 1\right) \]

      5. mul0-lft 46.3%

         \[\leadsto \left(\color{blue}{0 \cdot \frac{x}{y}} + \frac{{x}^{2}}{{y}^{2}}\right) - \left(-1 \cdot \frac{{x}^{2}}{{y}^{2}} + 1\right) \]

      6. associate-*r/ 46.3%

         \[\leadsto \left(\color{blue}{\frac{0 \cdot x}{y}} + \frac{{x}^{2}}{{y}^{2}}\right) - \left(-1 \cdot \frac{{x}^{2}}{{y}^{2}} + 1\right) \]

      7. mul0-lft 46.3%

         \[\leadsto \left(\frac{\color{blue}{0}}{y} + \frac{{x}^{2}}{{y}^{2}}\right) - \left(-1 \cdot \frac{{x}^{2}}{{y}^{2}} + 1\right) \]

      8. metadata-eval 46.3%

         \[\leadsto \left(\frac{\color{blue}{-1 \cdot 0}}{y} + \frac{{x}^{2}}{{y}^{2}}\right) - \left(-1 \cdot \frac{{x}^{2}}{{y}^{2}} + 1\right) \]

      9. mul0-lft 46.3%

         \[\leadsto \left(\frac{-1 \cdot \color{blue}{\left(0 \cdot x\right)}}{y} + \frac{{x}^{2}}{{y}^{2}}\right) - \left(-1 \cdot \frac{{x}^{2}}{{y}^{2}} + 1\right) \]

      10. metadata-eval 46.3%

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

      11. distribute-lft1-in 46.3%

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

      12. associate-*r/ 46.3%

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

      13. +-commutative 46.3%

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

   4. Simplified 77.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{y}, \frac{x}{y}, 0\right) - \left(1 - \frac{x}{y} \cdot \frac{x}{y}\right)} \]

   5. Taylor expanded in x around 0 46.3%

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

      1. unpow2 46.3%

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

      2. unpow2 46.3%

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

      3. times-frac 77.2%

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

      4. unpow2 77.2%

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

   7. Simplified 77.2%

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

      1. pow2 77.2%

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

      2. associate-*l/ 77.2%

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

   9. Applied egg-rr 77.2%

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

4. Final simplification 91.5%

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

Alternative 7: 92.4% accurate, 0.5× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))))
   (if (<= t_0 2.0) t_0 -1.0)))

C

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

Fortran

NOTE: y should be positive before calling this function
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 = ((x - y) * (x + y)) / ((x * x) + (y * y))
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], t$95$0, -1.0]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in x around 0 74.4%

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

4. Final simplification 90.5%

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

Alternative 8: 93.0% accurate, 0.5× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))))
   (if (<= t_0 2.0) t_0 (/ (- x y) (+ y (- (* (/ x y) (+ x x)) x))))))

C

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

Fortran

NOTE: y should be positive before calling this function
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 = ((x - y) * (x + y)) / ((x * x) + (y * y))
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = (x - y) / (y + (((x / y) * (x + x)) - x))
    end if
    code = tmp
end function

Java

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

Python

y = abs(y)
def code(x, y):
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y))
	tmp = 0
	if t_0 <= 2.0:
		tmp = t_0
	else:
		tmp = (x - y) / (y + (((x / y) * (x + x)) - x))
	return tmp

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], t$95$0, N[(N[(x - y), $MachinePrecision] / N[(y + N[(N[(N[(x / y), $MachinePrecision] * N[(x + x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

      1. associate-/l* 3.1%

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

      2. fma-def 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in y around inf 74.4%

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

      1. associate--l+ 74.4%

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

      2. +-commutative 74.4%

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

      3. associate--l+ 74.4%

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

      4. unpow2 74.4%

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

      5. associate-/l* 75.7%

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

      6. fma-neg 75.7%

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

      7. mul-1-neg 75.7%

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

      8. remove-double-neg 75.7%

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

      9. fma-def 75.7%

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

      10. +-commutative 75.7%

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

      11. mul-1-neg 75.7%

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

      12. unsub-neg 75.7%

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

      13. unpow2 75.7%

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

      14. associate-/l* 76.5%

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

   6. Simplified 76.5%

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

   7. Taylor expanded in x around 0 74.4%

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

      1. neg-mul-1 74.4%

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

      2. unsub-neg 74.4%

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

      3. unpow2 74.4%

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

      4. associate-*r/ 76.5%

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

      5. count-2 76.5%

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

      6. distribute-rgt-out 76.5%

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

   9. Simplified 76.5%

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

4. Final simplification 91.3%

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

Alternative 9: 83.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-127}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 5.5e-127) (+ 1.0 (* -2.0 (* (/ y x) (/ y x)))) (/ (- x y) y)))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-127) {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	} else {
		tmp = (x - y) / y;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.5d-127) then
        tmp = 1.0d0 + ((-2.0d0) * ((y / x) * (y / x)))
    else
        tmp = (x - y) / y
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-127) {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	} else {
		tmp = (x - y) / y;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 5.5e-127:
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)))
	else:
		tmp = (x - y) / y
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 5.5e-127)
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y / x) * Float64(y / x))));
	else
		tmp = Float64(Float64(x - y) / y);
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.5e-127)
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	else
		tmp = (x - y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 5.5e-127], N[(1.0 + N[(-2.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.50000000000000036e-127

   1. Initial program 58.1%

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

   2. Taylor expanded in y around 0 27.7%

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

      1. unpow2 27.7%

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

      2. unpow2 27.7%

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

   4. Simplified 27.7%

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

      1. frac-times 38.0%

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

   6. Applied egg-rr 38.0%

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

   if 5.50000000000000036e-127 < y

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

      2. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 77.4%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-127}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \]

Alternative 10: 83.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-127}:\\ \;\;\;\;\left(1 + \frac{y}{x}\right) - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 5.5e-127) (- (+ 1.0 (/ y x)) (/ y x)) (/ (- x y) y)))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-127) {
		tmp = (1.0 + (y / x)) - (y / x);
	} else {
		tmp = (x - y) / y;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5.5d-127) then
        tmp = (1.0d0 + (y / x)) - (y / x)
    else
        tmp = (x - y) / y
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-127) {
		tmp = (1.0 + (y / x)) - (y / x);
	} else {
		tmp = (x - y) / y;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 5.5e-127:
		tmp = (1.0 + (y / x)) - (y / x)
	else:
		tmp = (x - y) / y
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 5.5e-127)
		tmp = Float64(Float64(1.0 + Float64(y / x)) - Float64(y / x));
	else
		tmp = Float64(Float64(x - y) / y);
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.5e-127)
		tmp = (1.0 + (y / x)) - (y / x);
	else
		tmp = (x - y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 5.5e-127], N[(N[(1.0 + N[(y / x), $MachinePrecision]), $MachinePrecision] - N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.50000000000000036e-127

   1. Initial program 58.1%

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

   2. Taylor expanded in x around inf 36.4%

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

      1. associate-+r+ 36.4%

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

      2. associate-*r/ 36.4%

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

      3. mul-1-neg 36.4%

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

   4. Simplified 36.4%

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

   if 5.50000000000000036e-127 < y

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

      2. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 77.4%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-127}:\\ \;\;\;\;\left(1 + \frac{y}{x}\right) - \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \]

Alternative 11: 82.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.22 \cdot 10^{-119}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y) :precision binary64 (if (<= y 1.22e-119) (/ (- x y) x) -1.0))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.22e-119) {
		tmp = (x - y) / x;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.22d-119) then
        tmp = (x - y) / x
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.22e-119) {
		tmp = (x - y) / x;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 1.22e-119:
		tmp = (x - y) / x
	else:
		tmp = -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 1.22e-119)
		tmp = Float64(Float64(x - y) / x);
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.22e-119)
		tmp = (x - y) / x;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 1.22e-119], N[(N[(x - y), $MachinePrecision] / x), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.22e-119

   1. Initial program 58.1%

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

      1. associate-/l* 58.7%

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

      2. fma-def 58.7%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in x around inf 35.3%

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

   if 1.22e-119 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 76.2%

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

4. Final simplification 39.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.22 \cdot 10^{-119}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 12: 82.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.22 \cdot 10^{-119}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 1.22e-119) (/ (- x y) x) (/ (- x y) y)))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.22e-119) {
		tmp = (x - y) / x;
	} else {
		tmp = (x - y) / y;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.22d-119) then
        tmp = (x - y) / x
    else
        tmp = (x - y) / y
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.22e-119) {
		tmp = (x - y) / x;
	} else {
		tmp = (x - y) / y;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 1.22e-119:
		tmp = (x - y) / x
	else:
		tmp = (x - y) / y
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 1.22e-119)
		tmp = Float64(Float64(x - y) / x);
	else
		tmp = Float64(Float64(x - y) / y);
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.22e-119)
		tmp = (x - y) / x;
	else
		tmp = (x - y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 1.22e-119], N[(N[(x - y), $MachinePrecision] / x), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.22e-119

   1. Initial program 58.1%

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

      1. associate-/l* 58.7%

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

      2. fma-def 58.7%

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

   3. Simplified 58.7%

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

   4. Taylor expanded in x around inf 35.3%

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

   if 1.22e-119 < y

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

      2. fma-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 77.4%

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

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.22 \cdot 10^{-119}:\\ \;\;\;\;\frac{x - y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \]

Alternative 13: 82.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-125}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y) :precision binary64 (if (<= y 1.2e-125) 1.0 -1.0))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.2e-125) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.2d-125) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.2e-125) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 1.2e-125:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 1.2e-125)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.2e-125)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 1.2e-125], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.2000000000000001e-125

   1. Initial program 58.1%

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

   2. Taylor expanded in x around inf 36.0%

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

   if 1.2000000000000001e-125 < y

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 76.2%

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

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.2 \cdot 10^{-125}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 14: 66.4% accurate, 15.0× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y) :precision binary64 -1.0)

C

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

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

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

Python

y = abs(y)
def code(x, y):
	return -1.0

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := -1.0

Derivation

1. Initial program 62.9%

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

2. Taylor expanded in x around 0 65.1%

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

3. Final simplification 65.1%

   \[\leadsto -1 \]

Developer target: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023224 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

  :herbie-target
  (if (and (< 0.5 (fabs (/ x y))) (< (fabs (/ x y)) 2.0)) (/ (* (- x y) (+ x y)) (+ (* x x) (* y y))) (- 1.0 (/ 2.0 (+ 1.0 (* (/ x y) (/ x y))))))

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