Kahan p9 Example

Percentage Accurate: 68.0% → 100.0%

Time: 13.3s

Alternatives: 12

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: 68.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

   1. fma-def 72.2%

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

   2. add-sqr-sqrt 72.2%

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

   3. times-frac 72.1%

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

   4. fma-def 72.1%

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

   5. hypot-def 72.1%

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

   6. fma-def 72.1%

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

   7. hypot-def 99.9%

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

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. clear-num 99.9%

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

   3. un-div-inv 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

   1. +-commutative 72.2%

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

   2. associate-*r/ 71.8%

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

   3. +-commutative 71.8%

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

   4. fma-def 71.8%

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

3. Simplified 71.8%

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

   1. *-un-lft-identity 71.8%

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

   2. add-sqr-sqrt 71.8%

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

   3. times-frac 71.9%

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

   4. fma-def 71.9%

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

   5. hypot-def 72.0%

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

   6. fma-def 72.0%

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

   7. hypot-def 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-*l/ 99.7%

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

   2. *-un-lft-identity 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 3: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.2%

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

   1. fma-def 72.2%

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

   2. add-sqr-sqrt 72.2%

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

   3. times-frac 72.1%

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

   4. fma-def 72.1%

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

   5. hypot-def 72.1%

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

   6. fma-def 72.1%

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

   7. hypot-def 99.9%

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

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. Final simplification 99.9%

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

Alternative 4: 92.6% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[(x + y$95$m), $MachinePrecision] * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], t$95$0, N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(1.0 + N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $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%

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

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

   1. Initial program 0.0%

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

      1. +-commutative 0.0%

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

      2. associate-*r/ 3.1%

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

      3. +-commutative 3.1%

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

      4. fma-def 3.1%

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

   3. Simplified 3.1%

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

      1. *-un-lft-identity 3.1%

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

      2. add-sqr-sqrt 3.1%

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

      3. times-frac 3.1%

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

      4. fma-def 3.1%

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

      5. hypot-def 3.1%

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

      6. fma-def 3.1%

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

      7. hypot-def 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-un-lft-identity 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0 15.3%

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

4. Final simplification 76.5%

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

Alternative 5: 92.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(N[(x + y$95$m), $MachinePrecision] * N[(x - y$95$m), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y$95$m * y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], t$95$0, N[(N[(N[(x - y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

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

   1. Initial program 0.0%

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

      1. fma-def 0.0%

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

      2. add-sqr-sqrt 0.0%

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

      3. times-frac 3.1%

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

      4. fma-def 3.1%

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

      5. hypot-def 3.1%

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

      6. fma-def 3.1%

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

      7. hypot-def 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in x around 0 15.3%

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

4. Final simplification 76.5%

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

Alternative 6: 92.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := 1 - \frac{y_m}{x}\\ \mathbf{if}\;y_m \leq 1.75 \cdot 10^{-163}:\\ \;\;\;\;t_0 + \frac{y_m}{x} \cdot t_0\\ \mathbf{elif}\;y_m \leq 1.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(x + y_m\right) \cdot \left(x - y_m\right)}{x \cdot x + y_m \cdot y_m}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y_m x))))
   (if (<= y_m 1.75e-163)
     (+ t_0 (* (/ y_m x) t_0))
     (if (<= y_m 1.8e-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 t_0 = 1.0 - (y_m / x);
	double tmp;
	if (y_m <= 1.75e-163) {
		tmp = t_0 + ((y_m / x) * t_0);
	} else if (y_m <= 1.8e-11) {
		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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (y_m / x)
    if (y_m <= 1.75d-163) then
        tmp = t_0 + ((y_m / x) * t_0)
    else if (y_m <= 1.8d-11) 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 t_0 = 1.0 - (y_m / x);
	double tmp;
	if (y_m <= 1.75e-163) {
		tmp = t_0 + ((y_m / x) * t_0);
	} else if (y_m <= 1.8e-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):
	t_0 = 1.0 - (y_m / x)
	tmp = 0
	if y_m <= 1.75e-163:
		tmp = t_0 + ((y_m / x) * t_0)
	elif y_m <= 1.8e-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)
	t_0 = Float64(1.0 - Float64(y_m / x))
	tmp = 0.0
	if (y_m <= 1.75e-163)
		tmp = Float64(t_0 + Float64(Float64(y_m / x) * t_0));
	elseif (y_m <= 1.8e-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)
	t_0 = 1.0 - (y_m / x);
	tmp = 0.0;
	if (y_m <= 1.75e-163)
		tmp = t_0 + ((y_m / x) * t_0);
	elseif (y_m <= 1.8e-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_] := Block[{t$95$0 = N[(1.0 - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$95$m, 1.75e-163], N[(t$95$0 + N[(N[(y$95$m / x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 1.8e-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 3 regimes

2. if y < 1.75000000000000014e-163

   1. Initial program 66.5%

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

      1. +-commutative 66.5%

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

      2. associate-*r/ 66.1%

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

      3. +-commutative 66.1%

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

      4. fma-def 66.1%

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

   3. Simplified 66.1%

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

      1. *-un-lft-identity 66.1%

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

      2. add-sqr-sqrt 66.1%

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

      3. times-frac 66.2%

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

      4. fma-def 66.2%

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

      5. hypot-def 66.2%

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

      6. fma-def 66.2%

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

      7. hypot-def 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf 35.3%

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

   8. Taylor expanded in x around inf 34.6%

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

      1. associate-*r* 34.7%

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

      2. div-inv 34.7%

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

      3. distribute-rgt-in 34.2%

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

      4. *-un-lft-identity 34.2%

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

      5. div-sub 34.2%

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

      6. *-inverses 34.2%

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

      7. div-sub 34.2%

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

      8. *-inverses 34.2%

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

   10. Applied egg-rr 34.2%

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

   if 1.75000000000000014e-163 < y < 1.79999999999999992e-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 1.79999999999999992e-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. +-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

      3. +-commutative 100.0%

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

      4. fma-def 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 100.0%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.75 \cdot 10^{-163}:\\ \;\;\;\;\left(1 - \frac{y}{x}\right) + \frac{y}{x} \cdot \left(1 - \frac{y}{x}\right)\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := 1 - \frac{y_m}{x}\\ \mathbf{if}\;y_m \leq 2.1 \cdot 10^{-153}:\\ \;\;\;\;t_0 + \frac{y_m}{x} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (- 1.0 (/ y_m x))))
   (if (<= y_m 2.1e-153) (+ t_0 (* (/ y_m x) t_0)) -1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = 1.0 - (y_m / x);
	double tmp;
	if (y_m <= 2.1e-153) {
		tmp = t_0 + ((y_m / x) * t_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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 - (y_m / x)
    if (y_m <= 2.1d-153) then
        tmp = t_0 + ((y_m / x) * t_0)
    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 t_0 = 1.0 - (y_m / x);
	double tmp;
	if (y_m <= 2.1e-153) {
		tmp = t_0 + ((y_m / x) * t_0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	t_0 = 1.0 - (y_m / x)
	tmp = 0
	if y_m <= 2.1e-153:
		tmp = t_0 + ((y_m / x) * t_0)
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(1.0 - Float64(y_m / x))
	tmp = 0.0
	if (y_m <= 2.1e-153)
		tmp = Float64(t_0 + Float64(Float64(y_m / x) * t_0));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	t_0 = 1.0 - (y_m / x);
	tmp = 0.0;
	if (y_m <= 2.1e-153)
		tmp = t_0 + ((y_m / x) * t_0);
	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[(1.0 - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$95$m, 2.1e-153], N[(t$95$0 + N[(N[(y$95$m / x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if y < 2.10000000000000004e-153

   1. Initial program 66.7%

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

      1. +-commutative 66.7%

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

      2. associate-*r/ 66.2%

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

      3. +-commutative 66.2%

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

      4. fma-def 66.2%

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

   3. Simplified 66.2%

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

      1. *-un-lft-identity 66.2%

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

      2. add-sqr-sqrt 66.2%

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

      3. times-frac 66.3%

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

      4. fma-def 66.3%

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

      5. hypot-def 66.4%

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

      6. fma-def 66.4%

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

      7. hypot-def 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf 35.6%

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

   8. Taylor expanded in x around inf 34.9%

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

      1. associate-*r* 35.0%

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

      2. div-inv 35.1%

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

      3. distribute-rgt-in 34.5%

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

      4. *-un-lft-identity 34.5%

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

      5. div-sub 34.5%

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

      6. *-inverses 34.5%

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

      7. div-sub 34.5%

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

      8. *-inverses 34.5%

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

   10. Applied egg-rr 34.5%

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

   if 2.10000000000000004e-153 < 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. +-commutative 99.9%

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

      2. associate-*r/ 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 71.5%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{-153}:\\ \;\;\;\;\left(1 - \frac{y}{x}\right) + \frac{y}{x} \cdot \left(1 - \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.1 \cdot 10^{-156}:\\ \;\;\;\;\left(x - y_m\right) \cdot \left(\left(1 + \frac{y_m}{x}\right) \cdot \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.1e-156) (* (- x y_m) (* (+ 1.0 (/ y_m x)) (/ 1.0 x))) -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.1e-156) {
		tmp = (x - y_m) * ((1.0 + (y_m / x)) * (1.0 / x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 1.1d-156) then
        tmp = (x - y_m) * ((1.0d0 + (y_m / x)) * (1.0d0 / x))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.1e-156) {
		tmp = (x - y_m) * ((1.0 + (y_m / x)) * (1.0 / x));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.1e-156:
		tmp = (x - y_m) * ((1.0 + (y_m / x)) * (1.0 / x))
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.1e-156)
		tmp = Float64(Float64(x - y_m) * Float64(Float64(1.0 + Float64(y_m / x)) * Float64(1.0 / x)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 1.1e-156)
		tmp = (x - y_m) * ((1.0 + (y_m / x)) * (1.0 / x));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.1e-156], N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(1.0 + N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.1e-156

   1. Initial program 66.5%

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

      1. +-commutative 66.5%

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

      2. associate-*r/ 66.1%

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

      3. +-commutative 66.1%

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

      4. fma-def 66.1%

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

   3. Simplified 66.1%

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

      1. *-un-lft-identity 66.1%

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

      2. add-sqr-sqrt 66.1%

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

      3. times-frac 66.2%

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

      4. fma-def 66.2%

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

      5. hypot-def 66.2%

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

      6. fma-def 66.2%

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

      7. hypot-def 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf 35.3%

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

   8. Taylor expanded in x around inf 34.6%

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

   if 1.1e-156 < 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. +-commutative 99.9%

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

      2. associate-*r/ 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 69.9%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.1 \cdot 10^{-156}:\\ \;\;\;\;\left(x - y\right) \cdot \left(\left(1 + \frac{y}{x}\right) \cdot \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.85 \cdot 10^{-156}:\\ \;\;\;\;\left(x - y_m\right) \cdot \frac{1}{\frac{x}{1 + \frac{y_m}{x}}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.85e-156) (* (- x y_m) (/ 1.0 (/ x (+ 1.0 (/ y_m x))))) -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.85e-156) {
		tmp = (x - y_m) * (1.0 / (x / (1.0 + (y_m / x))));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 1.85d-156) then
        tmp = (x - y_m) * (1.0d0 / (x / (1.0d0 + (y_m / x))))
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.85e-156) {
		tmp = (x - y_m) * (1.0 / (x / (1.0 + (y_m / x))));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.85e-156:
		tmp = (x - y_m) * (1.0 / (x / (1.0 + (y_m / x))))
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.85e-156)
		tmp = Float64(Float64(x - y_m) * Float64(1.0 / Float64(x / Float64(1.0 + Float64(y_m / x)))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 1.85e-156)
		tmp = (x - y_m) * (1.0 / (x / (1.0 + (y_m / x))));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.85e-156], N[(N[(x - y$95$m), $MachinePrecision] * N[(1.0 / N[(x / N[(1.0 + N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.85e-156

   1. Initial program 66.5%

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

      1. +-commutative 66.5%

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

      2. associate-*r/ 66.1%

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

      3. +-commutative 66.1%

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

      4. fma-def 66.1%

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

   3. Simplified 66.1%

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

      1. *-un-lft-identity 66.1%

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

      2. add-sqr-sqrt 66.1%

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

      3. times-frac 66.2%

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

      4. fma-def 66.2%

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

      5. hypot-def 66.2%

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

      6. fma-def 66.2%

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

      7. hypot-def 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf 35.3%

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

   8. Taylor expanded in x around inf 34.6%

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

      1. associate-*l/ 34.6%

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

      2. *-un-lft-identity 34.6%

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

      3. clear-num 34.6%

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

   10. Applied egg-rr 34.6%

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

   if 1.85e-156 < 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. +-commutative 99.9%

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

      2. associate-*r/ 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 69.9%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.85 \cdot 10^{-156}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1}{\frac{x}{1 + \frac{y}{x}}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 2.25 \cdot 10^{-156}:\\ \;\;\;\;\left(x - y_m\right) \cdot \frac{1 + \frac{y_m}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 2.25e-156) (* (- x y_m) (/ (+ 1.0 (/ y_m x)) x)) -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 2.25e-156) {
		tmp = (x - y_m) * ((1.0 + (y_m / x)) / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 2.25d-156) then
        tmp = (x - y_m) * ((1.0d0 + (y_m / x)) / x)
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 2.25e-156) {
		tmp = (x - y_m) * ((1.0 + (y_m / x)) / x);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 2.25e-156:
		tmp = (x - y_m) * ((1.0 + (y_m / x)) / x)
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 2.25e-156)
		tmp = Float64(Float64(x - y_m) * Float64(Float64(1.0 + Float64(y_m / x)) / x));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 2.25e-156)
		tmp = (x - y_m) * ((1.0 + (y_m / x)) / x);
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 2.25e-156], N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(1.0 + N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 2.24999999999999993e-156

   1. Initial program 66.5%

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

      1. +-commutative 66.5%

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

      2. associate-*r/ 66.1%

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

      3. +-commutative 66.1%

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

      4. fma-def 66.1%

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

   3. Simplified 66.1%

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

      1. *-un-lft-identity 66.1%

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

      2. add-sqr-sqrt 66.1%

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

      3. times-frac 66.2%

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

      4. fma-def 66.2%

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

      5. hypot-def 66.2%

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

      6. fma-def 66.2%

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

      7. hypot-def 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around inf 35.3%

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

   8. Taylor expanded in x around inf 34.6%

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

      1. *-commutative 34.6%

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

      2. sub-neg 34.6%

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

      3. distribute-lft-in 34.5%

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

      4. associate-*l/ 34.5%

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

      5. *-un-lft-identity 34.5%

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

      6. associate-*l/ 34.5%

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

      7. *-un-lft-identity 34.5%

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

   10. Applied egg-rr 34.5%

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

       1. distribute-lft-out 34.6%

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

       2. sub-neg 34.6%

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

   12. Simplified 34.6%

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

   if 2.24999999999999993e-156 < 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. +-commutative 99.9%

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

      2. associate-*r/ 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 69.9%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.25 \cdot 10^{-156}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.05 \cdot 10^{-153}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (if (<= y_m 1.05e-153) 1.0 -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.05e-153) {
		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 <= 1.05d-153) 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 <= 1.05e-153) {
		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 <= 1.05e-153:
		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 <= 1.05e-153)
		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 <= 1.05e-153)
		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, 1.05e-153], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.05000000000000002e-153

   1. Initial program 66.7%

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

      1. +-commutative 66.7%

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

      2. associate-*r/ 66.2%

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

      3. +-commutative 66.2%

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

      4. fma-def 66.2%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in x around inf 33.3%

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

   if 1.05000000000000002e-153 < 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. +-commutative 99.9%

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

      2. associate-*r/ 99.5%

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

      3. +-commutative 99.5%

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

      4. fma-def 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 71.5%

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

4. Final simplification 39.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.05 \cdot 10^{-153}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.9% accurate, 15.0× speedup

Math

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

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 -1.0)

C

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

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    code = -1.0d0
end function

Java

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

Python

y_m = math.fabs(y)
def code(x, y_m):
	return -1.0

Julia

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

MATLAB

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

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := -1.0

Derivation

1. Initial program 72.2%

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

   1. +-commutative 72.2%

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

   2. associate-*r/ 71.8%

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

   3. +-commutative 71.8%

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

   4. fma-def 71.8%

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

3. Simplified 71.8%

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

5. Taylor expanded in x around 0 67.5%

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

6. Final simplification 67.5%

   \[\leadsto -1 \]
7. Add Preprocessing

Developer target: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024017 
(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))))