Kahan p9 Example

Percentage Accurate: 68.7% → 100.0%

Time: 9.0s

Alternatives: 12

Speedup: 0.9×

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: 68.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

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(x - y_m) / Float64(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[(x - y$95$m), $MachinePrecision] / N[(N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision] * N[(N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision] / N[(x + y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 70.3%

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

   1. add-sqr-sqrt 70.3%

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

   2. times-frac 71.1%

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

   3. hypot-define 71.1%

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

   4. hypot-define 100.0%

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

4. Applied egg-rr 100.0%

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

   1. *-commutative 100.0%

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

   2. clear-num 100.0%

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

   3. frac-times 99.9%

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

   4. *-un-lft-identity 99.9%

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

6. Applied egg-rr 99.9%

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

7. Final simplification 99.9%

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

Alternative 2: 92.5% accurate, 0.1× speedup

Math

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

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 (fma 2.0 (pow (/ x y_m) 2.0) -1.0))))

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 = fma(2.0, pow((x / y_m), 2.0), -1.0);
	}
	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 = fma(2.0, (Float64(x / y_m) ^ 2.0), -1.0);
	end
	return 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[(2.0 * N[Power[N[(x / y$95$m), $MachinePrecision], 2.0], $MachinePrecision] + -1.0), $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. add-sqr-sqrt 0.0%

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

      2. times-frac 3.1%

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

      3. hypot-define 3.1%

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

      4. hypot-define 99.9%

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

   4. Applied egg-rr 99.9%

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

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

      3. frac-times 99.8%

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

      4. *-un-lft-identity 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 47.4%

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

      1. fma-neg 47.4%

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

      2. unpow2 47.4%

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

      3. unpow2 47.4%

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

      4. times-frac 77.6%

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

      5. unpow2 77.6%

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

      6. metadata-eval 77.6%

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

   9. Simplified 77.6%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, {\left(\frac{x}{y}\right)}^{2}, -1\right)\\ \end{array} \]
5. 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 70.3%

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

   1. add-sqr-sqrt 70.3%

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

   2. times-frac 71.1%

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

   3. hypot-define 71.1%

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

   4. hypot-define 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 4: 92.3% accurate, 0.4× speedup

Math

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

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) (+ (/ x y_m) 1.0)) 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) * ((x / y_m) + 1.0)) / y_m;
	}
	return tmp;
}

Fortran

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

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) * ((x / y_m) + 1.0)) / 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) * ((x / y_m) + 1.0)) / 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) * Float64(Float64(x / y_m) + 1.0)) / 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) * ((x / y_m) + 1.0)) / 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[(N[(x / y$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / y$95$m), $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. associate-/l* 3.1%

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

      2. +-commutative 3.1%

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

      3. fma-define 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in y around inf 76.6%

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

      1. associate-*r/ 76.8%

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

   7. Applied egg-rr 76.8%

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

4. Final simplification 93.1%

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

Alternative 5: 92.6% accurate, 0.4× speedup

Math

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

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) (+ y_m (* x (+ -1.0 (* 2.0 (/ 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) / (y_m + (x * (-1.0 + (2.0 * (x / y_m)))));
	}
	return tmp;
}

Fortran

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

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double 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) / (y_m + (x * (-1.0 + (2.0 * (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) / (y_m + (x * (-1.0 + (2.0 * (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(x - y_m) / Float64(y_m + Float64(x * Float64(-1.0 + Float64(2.0 * 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) / (y_m + (x * (-1.0 + (2.0 * (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[(x - y$95$m), $MachinePrecision] / N[(y$95$m + N[(x * N[(-1.0 + N[(2.0 * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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. add-sqr-sqrt 0.0%

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

      2. times-frac 3.1%

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

      3. hypot-define 3.1%

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

      4. hypot-define 99.9%

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

   4. Applied egg-rr 99.9%

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

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

      3. frac-times 99.8%

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

      4. *-un-lft-identity 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in x around 0 76.6%

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

4. Final simplification 93.0%

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

Alternative 6: 82.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.3 \cdot 10^{-197}:\\ \;\;\;\;1\\ \mathbf{elif}\;y\_m \leq 6.9 \cdot 10^{-183} \lor \neg \left(y\_m \leq 2.9 \cdot 10^{-156}\right):\\ \;\;\;\;\left(x - y\_m\right) \cdot \frac{\frac{x}{y\_m} + 1}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.3e-197)
   1.0
   (if (or (<= y_m 6.9e-183) (not (<= y_m 2.9e-156)))
     (* (- x y_m) (/ (+ (/ x y_m) 1.0) y_m))
     1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.3e-197) {
		tmp = 1.0;
	} else if ((y_m <= 6.9e-183) || !(y_m <= 2.9e-156)) {
		tmp = (x - y_m) * (((x / y_m) + 1.0) / y_m);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 1.3d-197) then
        tmp = 1.0d0
    else if ((y_m <= 6.9d-183) .or. (.not. (y_m <= 2.9d-156))) then
        tmp = (x - y_m) * (((x / y_m) + 1.0d0) / y_m)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.3e-197) {
		tmp = 1.0;
	} else if ((y_m <= 6.9e-183) || !(y_m <= 2.9e-156)) {
		tmp = (x - y_m) * (((x / y_m) + 1.0) / y_m);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.3e-197:
		tmp = 1.0
	elif (y_m <= 6.9e-183) or not (y_m <= 2.9e-156):
		tmp = (x - y_m) * (((x / y_m) + 1.0) / y_m)
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.3e-197)
		tmp = 1.0;
	elseif ((y_m <= 6.9e-183) || !(y_m <= 2.9e-156))
		tmp = Float64(Float64(x - y_m) * Float64(Float64(Float64(x / y_m) + 1.0) / y_m));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 1.3e-197)
		tmp = 1.0;
	elseif ((y_m <= 6.9e-183) || ~((y_m <= 2.9e-156)))
		tmp = (x - y_m) * (((x / y_m) + 1.0) / y_m);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.3e-197], 1.0, If[Or[LessEqual[y$95$m, 6.9e-183], N[Not[LessEqual[y$95$m, 2.9e-156]], $MachinePrecision]], N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(N[(x / y$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3000000000000001e-197 or 6.9000000000000001e-183 < y < 2.90000000000000021e-156

   1. Initial program 65.9%

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

      1. associate-/l* 66.5%

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

      2. +-commutative 66.5%

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

      3. fma-define 66.5%

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

   3. Simplified 66.5%

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

   5. Taylor expanded in x around inf 37.1%

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

   if 1.3000000000000001e-197 < y < 6.9000000000000001e-183 or 2.90000000000000021e-156 < y

   1. Initial program 89.6%

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

      1. associate-/l* 89.2%

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

      2. +-commutative 89.2%

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

      3. fma-define 89.2%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in y around inf 80.2%

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

4. Final simplification 45.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.3 \cdot 10^{-197}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{-183} \lor \neg \left(y \leq 2.9 \cdot 10^{-156}\right):\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.15 \cdot 10^{-197} \lor \neg \left(y\_m \leq 3.5 \cdot 10^{-181}\right) \land y\_m \leq 3.2 \cdot 10^{-156}:\\ \;\;\;\;\left(x - y\_m\right) \cdot \frac{1 + \frac{y\_m}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\_m\right) \cdot \frac{\frac{x}{y\_m} + 1}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (or (<= y_m 1.15e-197) (and (not (<= y_m 3.5e-181)) (<= y_m 3.2e-156)))
   (* (- x y_m) (/ (+ 1.0 (/ y_m x)) x))
   (* (- x y_m) (/ (+ (/ x y_m) 1.0) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((y_m <= 1.15e-197) || (!(y_m <= 3.5e-181) && (y_m <= 3.2e-156))) {
		tmp = (x - y_m) * ((1.0 + (y_m / x)) / x);
	} else {
		tmp = (x - y_m) * (((x / y_m) + 1.0) / y_m);
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((y_m <= 1.15d-197) .or. (.not. (y_m <= 3.5d-181)) .and. (y_m <= 3.2d-156)) then
        tmp = (x - y_m) * ((1.0d0 + (y_m / x)) / x)
    else
        tmp = (x - y_m) * (((x / y_m) + 1.0d0) / y_m)
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if ((y_m <= 1.15e-197) || (!(y_m <= 3.5e-181) && (y_m <= 3.2e-156))) {
		tmp = (x - y_m) * ((1.0 + (y_m / x)) / x);
	} else {
		tmp = (x - y_m) * (((x / y_m) + 1.0) / y_m);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if (y_m <= 1.15e-197) or (not (y_m <= 3.5e-181) and (y_m <= 3.2e-156)):
		tmp = (x - y_m) * ((1.0 + (y_m / x)) / x)
	else:
		tmp = (x - y_m) * (((x / y_m) + 1.0) / y_m)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if ((y_m <= 1.15e-197) || (!(y_m <= 3.5e-181) && (y_m <= 3.2e-156)))
		tmp = Float64(Float64(x - y_m) * Float64(Float64(1.0 + Float64(y_m / x)) / x));
	else
		tmp = Float64(Float64(x - y_m) * Float64(Float64(Float64(x / y_m) + 1.0) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if ((y_m <= 1.15e-197) || (~((y_m <= 3.5e-181)) && (y_m <= 3.2e-156)))
		tmp = (x - y_m) * ((1.0 + (y_m / x)) / x);
	else
		tmp = (x - y_m) * (((x / y_m) + 1.0) / y_m);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[Or[LessEqual[y$95$m, 1.15e-197], And[N[Not[LessEqual[y$95$m, 3.5e-181]], $MachinePrecision], LessEqual[y$95$m, 3.2e-156]]], N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(1.0 + N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(N[(x / y$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.15e-197 or 3.49999999999999996e-181 < y < 3.19999999999999982e-156

   1. Initial program 65.9%

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

      1. associate-/l* 66.5%

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

      2. +-commutative 66.5%

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

      3. fma-define 66.5%

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

   3. Simplified 66.5%

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

   5. Taylor expanded in x around inf 38.5%

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

   if 1.15e-197 < y < 3.49999999999999996e-181 or 3.19999999999999982e-156 < y

   1. Initial program 89.6%

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

      1. associate-/l* 89.2%

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

      2. +-commutative 89.2%

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

      3. fma-define 89.2%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in y around inf 80.2%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.15 \cdot 10^{-197} \lor \neg \left(y \leq 3.5 \cdot 10^{-181}\right) \land y \leq 3.2 \cdot 10^{-156}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 9.2 \cdot 10^{-198} \lor \neg \left(y\_m \leq 10^{-182}\right) \land y\_m \leq 3.1 \cdot 10^{-156}:\\ \;\;\;\;1 - \frac{2 \cdot \left(y\_m \cdot \frac{y\_m}{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\_m\right) \cdot \frac{\frac{x}{y\_m} + 1}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (or (<= y_m 9.2e-198) (and (not (<= y_m 1e-182)) (<= y_m 3.1e-156)))
   (- 1.0 (/ (* 2.0 (* y_m (/ y_m x))) x))
   (* (- x y_m) (/ (+ (/ x y_m) 1.0) y_m))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((y_m <= 9.2e-198) || (!(y_m <= 1e-182) && (y_m <= 3.1e-156))) {
		tmp = 1.0 - ((2.0 * (y_m * (y_m / x))) / x);
	} else {
		tmp = (x - y_m) * (((x / y_m) + 1.0) / y_m);
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if ((y_m <= 9.2d-198) .or. (.not. (y_m <= 1d-182)) .and. (y_m <= 3.1d-156)) then
        tmp = 1.0d0 - ((2.0d0 * (y_m * (y_m / x))) / x)
    else
        tmp = (x - y_m) * (((x / y_m) + 1.0d0) / y_m)
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if ((y_m <= 9.2e-198) || (!(y_m <= 1e-182) && (y_m <= 3.1e-156))) {
		tmp = 1.0 - ((2.0 * (y_m * (y_m / x))) / x);
	} else {
		tmp = (x - y_m) * (((x / y_m) + 1.0) / y_m);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if (y_m <= 9.2e-198) or (not (y_m <= 1e-182) and (y_m <= 3.1e-156)):
		tmp = 1.0 - ((2.0 * (y_m * (y_m / x))) / x)
	else:
		tmp = (x - y_m) * (((x / y_m) + 1.0) / y_m)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if ((y_m <= 9.2e-198) || (!(y_m <= 1e-182) && (y_m <= 3.1e-156)))
		tmp = Float64(1.0 - Float64(Float64(2.0 * Float64(y_m * Float64(y_m / x))) / x));
	else
		tmp = Float64(Float64(x - y_m) * Float64(Float64(Float64(x / y_m) + 1.0) / y_m));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if ((y_m <= 9.2e-198) || (~((y_m <= 1e-182)) && (y_m <= 3.1e-156)))
		tmp = 1.0 - ((2.0 * (y_m * (y_m / x))) / x);
	else
		tmp = (x - y_m) * (((x / y_m) + 1.0) / y_m);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[Or[LessEqual[y$95$m, 9.2e-198], And[N[Not[LessEqual[y$95$m, 1e-182]], $MachinePrecision], LessEqual[y$95$m, 3.1e-156]]], N[(1.0 - N[(N[(2.0 * N[(y$95$m * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(N[(x / y$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.20000000000000053e-198 or 1e-182 < y < 3.0999999999999998e-156

   1. Initial program 65.9%

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

      1. associate-/l* 66.5%

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

      2. +-commutative 66.5%

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

      3. fma-define 66.5%

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

   3. Simplified 66.5%

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

   5. Taylor expanded in x around -inf 37.7%

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

   7. Simplified 38.2%

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

   8. Taylor expanded in y around 0 38.2%

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

      1. div-inv 38.2%

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

      2. unpow2 38.2%

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

      3. associate-*r* 39.1%

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

      4. div-inv 39.1%

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

   10. Applied egg-rr 39.1%

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

   if 9.20000000000000053e-198 < y < 1e-182 or 3.0999999999999998e-156 < y

   1. Initial program 89.6%

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

      1. associate-/l* 89.2%

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

      2. +-commutative 89.2%

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

      3. fma-define 89.2%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in y around inf 80.2%

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

4. Final simplification 46.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9.2 \cdot 10^{-198} \lor \neg \left(y \leq 10^{-182}\right) \land y \leq 3.1 \cdot 10^{-156}:\\ \;\;\;\;1 - \frac{2 \cdot \left(y \cdot \frac{y}{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} t_0 := \frac{x}{y\_m} + 1\\ t_1 := 1 - \frac{2 \cdot \left(y\_m \cdot \frac{y\_m}{x}\right)}{x}\\ \mathbf{if}\;y\_m \leq 1.26 \cdot 10^{-197}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y\_m \leq 1.55 \cdot 10^{-181}:\\ \;\;\;\;\left(x - y\_m\right) \cdot \frac{t\_0}{y\_m}\\ \mathbf{elif}\;y\_m \leq 3.1 \cdot 10^{-156}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\_m\right) \cdot t\_0}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (let* ((t_0 (+ (/ x y_m) 1.0)) (t_1 (- 1.0 (/ (* 2.0 (* y_m (/ y_m x))) x))))
   (if (<= y_m 1.26e-197)
     t_1
     (if (<= y_m 1.55e-181)
       (* (- x y_m) (/ t_0 y_m))
       (if (<= y_m 3.1e-156) t_1 (/ (* (- x y_m) t_0) y_m))))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double t_0 = (x / y_m) + 1.0;
	double t_1 = 1.0 - ((2.0 * (y_m * (y_m / x))) / x);
	double tmp;
	if (y_m <= 1.26e-197) {
		tmp = t_1;
	} else if (y_m <= 1.55e-181) {
		tmp = (x - y_m) * (t_0 / y_m);
	} else if (y_m <= 3.1e-156) {
		tmp = t_1;
	} else {
		tmp = ((x - y_m) * t_0) / y_m;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x / y_m) + 1.0d0
    t_1 = 1.0d0 - ((2.0d0 * (y_m * (y_m / x))) / x)
    if (y_m <= 1.26d-197) then
        tmp = t_1
    else if (y_m <= 1.55d-181) then
        tmp = (x - y_m) * (t_0 / y_m)
    else if (y_m <= 3.1d-156) then
        tmp = t_1
    else
        tmp = ((x - y_m) * t_0) / y_m
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double t_0 = (x / y_m) + 1.0;
	double t_1 = 1.0 - ((2.0 * (y_m * (y_m / x))) / x);
	double tmp;
	if (y_m <= 1.26e-197) {
		tmp = t_1;
	} else if (y_m <= 1.55e-181) {
		tmp = (x - y_m) * (t_0 / y_m);
	} else if (y_m <= 3.1e-156) {
		tmp = t_1;
	} else {
		tmp = ((x - y_m) * t_0) / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	t_0 = (x / y_m) + 1.0
	t_1 = 1.0 - ((2.0 * (y_m * (y_m / x))) / x)
	tmp = 0
	if y_m <= 1.26e-197:
		tmp = t_1
	elif y_m <= 1.55e-181:
		tmp = (x - y_m) * (t_0 / y_m)
	elif y_m <= 3.1e-156:
		tmp = t_1
	else:
		tmp = ((x - y_m) * t_0) / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	t_0 = Float64(Float64(x / y_m) + 1.0)
	t_1 = Float64(1.0 - Float64(Float64(2.0 * Float64(y_m * Float64(y_m / x))) / x))
	tmp = 0.0
	if (y_m <= 1.26e-197)
		tmp = t_1;
	elseif (y_m <= 1.55e-181)
		tmp = Float64(Float64(x - y_m) * Float64(t_0 / y_m));
	elseif (y_m <= 3.1e-156)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(x - y_m) * t_0) / y_m);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	t_0 = (x / y_m) + 1.0;
	t_1 = 1.0 - ((2.0 * (y_m * (y_m / x))) / x);
	tmp = 0.0;
	if (y_m <= 1.26e-197)
		tmp = t_1;
	elseif (y_m <= 1.55e-181)
		tmp = (x - y_m) * (t_0 / y_m);
	elseif (y_m <= 3.1e-156)
		tmp = t_1;
	else
		tmp = ((x - y_m) * t_0) / 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[(x / y$95$m), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 - N[(N[(2.0 * N[(y$95$m * N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$95$m, 1.26e-197], t$95$1, If[LessEqual[y$95$m, 1.55e-181], N[(N[(x - y$95$m), $MachinePrecision] * N[(t$95$0 / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 3.1e-156], t$95$1, N[(N[(N[(x - y$95$m), $MachinePrecision] * t$95$0), $MachinePrecision] / y$95$m), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < 1.26000000000000003e-197 or 1.55000000000000011e-181 < y < 3.0999999999999998e-156

   1. Initial program 65.9%

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

      1. associate-/l* 66.5%

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

      2. +-commutative 66.5%

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

      3. fma-define 66.5%

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

   3. Simplified 66.5%

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

   5. Taylor expanded in x around -inf 37.7%

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

   7. Simplified 38.2%

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

   8. Taylor expanded in y around 0 38.2%

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

      1. div-inv 38.2%

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

      2. unpow2 38.2%

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

      3. associate-*r* 39.1%

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

      4. div-inv 39.1%

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

   10. Applied egg-rr 39.1%

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

   if 1.26000000000000003e-197 < y < 1.55000000000000011e-181

   1. Initial program 28.6%

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

      1. associate-/l* 30.8%

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

      2. +-commutative 30.8%

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

      3. fma-define 30.8%

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

   3. Simplified 30.8%

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

   5. Taylor expanded in y around inf 72.8%

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

   if 3.0999999999999998e-156 < y

   1. Initial program 100.0%

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

      1. associate-/l* 99.2%

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

      2. +-commutative 99.2%

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

      3. fma-define 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in y around inf 81.5%

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

      1. associate-*r/ 81.7%

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

   7. Applied egg-rr 81.7%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.26 \cdot 10^{-197}:\\ \;\;\;\;1 - \frac{2 \cdot \left(y \cdot \frac{y}{x}\right)}{x}\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-181}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-156}:\\ \;\;\;\;1 - \frac{2 \cdot \left(y \cdot \frac{y}{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(\frac{x}{y} + 1\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 3.5 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;y\_m \leq 5 \cdot 10^{-183}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y\_m \leq 3 \cdot 10^{-156}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 3.5e-189)
   1.0
   (if (<= y_m 5e-183) -1.0 (if (<= y_m 3e-156) 1.0 -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 3.5e-189) {
		tmp = 1.0;
	} else if (y_m <= 5e-183) {
		tmp = -1.0;
	} else if (y_m <= 3e-156) {
		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 <= 3.5d-189) then
        tmp = 1.0d0
    else if (y_m <= 5d-183) then
        tmp = -1.0d0
    else if (y_m <= 3d-156) 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 <= 3.5e-189) {
		tmp = 1.0;
	} else if (y_m <= 5e-183) {
		tmp = -1.0;
	} else if (y_m <= 3e-156) {
		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 <= 3.5e-189:
		tmp = 1.0
	elif y_m <= 5e-183:
		tmp = -1.0
	elif y_m <= 3e-156:
		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 <= 3.5e-189)
		tmp = 1.0;
	elseif (y_m <= 5e-183)
		tmp = -1.0;
	elseif (y_m <= 3e-156)
		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 <= 3.5e-189)
		tmp = 1.0;
	elseif (y_m <= 5e-183)
		tmp = -1.0;
	elseif (y_m <= 3e-156)
		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, 3.5e-189], 1.0, If[LessEqual[y$95$m, 5e-183], -1.0, If[LessEqual[y$95$m, 3e-156], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if y < 3.5000000000000001e-189 or 5.0000000000000002e-183 < y < 3e-156

   1. Initial program 65.6%

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

      1. associate-/l* 66.2%

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

      2. +-commutative 66.2%

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

      3. fma-define 66.2%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in x around inf 37.3%

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

   if 3.5000000000000001e-189 < y < 5.0000000000000002e-183 or 3e-156 < y

   1. Initial program 93.1%

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

      1. associate-/l* 92.7%

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

      2. +-commutative 92.7%

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

      3. fma-define 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in x around 0 82.1%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-183}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-156}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 82.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 3.1 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;y\_m \leq 1.4 \cdot 10^{-182}:\\ \;\;\;\;y\_m \cdot \frac{-1}{y\_m}\\ \mathbf{elif}\;y\_m \leq 3.1 \cdot 10^{-156}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 3.1e-189)
   1.0
   (if (<= y_m 1.4e-182)
     (* y_m (/ -1.0 y_m))
     (if (<= y_m 3.1e-156) 1.0 -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 3.1e-189) {
		tmp = 1.0;
	} else if (y_m <= 1.4e-182) {
		tmp = y_m * (-1.0 / y_m);
	} else if (y_m <= 3.1e-156) {
		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 <= 3.1d-189) then
        tmp = 1.0d0
    else if (y_m <= 1.4d-182) then
        tmp = y_m * ((-1.0d0) / y_m)
    else if (y_m <= 3.1d-156) 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 <= 3.1e-189) {
		tmp = 1.0;
	} else if (y_m <= 1.4e-182) {
		tmp = y_m * (-1.0 / y_m);
	} else if (y_m <= 3.1e-156) {
		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 <= 3.1e-189:
		tmp = 1.0
	elif y_m <= 1.4e-182:
		tmp = y_m * (-1.0 / y_m)
	elif y_m <= 3.1e-156:
		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 <= 3.1e-189)
		tmp = 1.0;
	elseif (y_m <= 1.4e-182)
		tmp = Float64(y_m * Float64(-1.0 / y_m));
	elseif (y_m <= 3.1e-156)
		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 <= 3.1e-189)
		tmp = 1.0;
	elseif (y_m <= 1.4e-182)
		tmp = y_m * (-1.0 / y_m);
	elseif (y_m <= 3.1e-156)
		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, 3.1e-189], 1.0, If[LessEqual[y$95$m, 1.4e-182], N[(y$95$m * N[(-1.0 / y$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$95$m, 3.1e-156], 1.0, -1.0]]]

Derivation

1. Split input into 3 regimes

2. if y < 3.1e-189 or 1.39999999999999997e-182 < y < 3.0999999999999998e-156

   1. Initial program 65.6%

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

      1. associate-/l* 66.2%

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

      2. +-commutative 66.2%

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

      3. fma-define 66.2%

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

   3. Simplified 66.2%

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

   5. Taylor expanded in x around inf 37.3%

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

   if 3.1e-189 < y < 1.39999999999999997e-182

   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%

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

      2. +-commutative 3.1%

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

      3. fma-define 3.1%

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

   3. Simplified 3.1%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. associate-*r/ 100.0%

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

   7. Applied egg-rr 100.0%

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

      1. clear-num 100.0%

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

      2. associate-/r/ 100.0%

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

   9. Applied egg-rr 100.0%

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

   10. Taylor expanded in x around 0 100.0%

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

       1. neg-mul-1 100.0%

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

   12. Simplified 100.0%

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

   if 3.0999999999999998e-156 < y

   1. Initial program 100.0%

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

      1. associate-/l* 99.2%

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

      2. +-commutative 99.2%

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

      3. fma-define 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in x around 0 80.8%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-189}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-182}:\\ \;\;\;\;y \cdot \frac{-1}{y}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-156}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.3% 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 70.3%

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

   1. associate-/l* 70.8%

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

   2. +-commutative 70.8%

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

   3. fma-define 70.8%

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

3. Simplified 70.8%

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

5. Taylor expanded in x around 0 65.8%

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

6. Final simplification 65.8%

   \[\leadsto -1 \]
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 2024067 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

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

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