Kahan p9 Example

Percentage Accurate: 69.4% → 100.0%

Time: 9.7s

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: 69.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 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 70.6%

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

   3. hypot-define 70.6%

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

   4. hypot-define 100.0%

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

4. Applied egg-rr 100.0%

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

   1. clear-num 100.0%

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

   2. frac-times 100.0%

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

   3. *-commutative 100.0%

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

   4. *-un-lft-identity 100.0%

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

6. Applied egg-rr 100.0%

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

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 70.6%

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

   3. hypot-define 70.6%

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

   4. hypot-define 100.0%

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

4. Applied egg-rr 100.0%

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

Alternative 3: 99.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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* 70.3%

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

   2. +-commutative 70.3%

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

   3. fma-define 70.3%

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

3. Simplified 70.3%

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

   1. fma-undefine 70.3%

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

   2. +-commutative 70.3%

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

   3. *-un-lft-identity 70.3%

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

   4. add-sqr-sqrt 70.3%

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

   5. times-frac 70.4%

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

   6. hypot-define 70.5%

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

   7. hypot-define 99.7%

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

6. Applied egg-rr 99.7%

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

   1. associate-*l/ 99.7%

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

   2. *-un-lft-identity 99.7%

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

8. Applied egg-rr 99.7%

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

Alternative 4: 92.9% accurate, 0.4× speedup

Math

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

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
     (/ (- x y_m) (+ y_m (* x (+ (* 2.0 (/ x y_m)) -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 = (x - y_m) / (y_m + (x * ((2.0 * (x / y_m)) + -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 = ((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 * ((2.0d0 * (x / y_m)) + (-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 = ((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 * ((2.0 * (x / y_m)) + -1.0)));
	}
	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 * ((2.0 * (x / y_m)) + -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 = Float64(Float64(x - y_m) / Float64(y_m + Float64(x * Float64(Float64(2.0 * Float64(x / y_m)) + -1.0))));
	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 * ((2.0 * (x / y_m)) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(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[(N[(2.0 * N[(x / y$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $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. clear-num 99.9%

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

      2. frac-times 100.0%

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

      3. *-commutative 100.0%

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

      4. *-un-lft-identity 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 80.8%

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

4. Final simplification 94.3%

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

Alternative 5: 92.9% accurate, 0.4× speedup

Math

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

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 (/ (- x y_m) (+ y_m (* x (+ (/ x y_m) -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 = (x - y_m) / (y_m + (x * ((x / y_m) + -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 = ((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 * ((x / y_m) + (-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 = ((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 * ((x / y_m) + -1.0)));
	}
	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 * ((x / y_m) + -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 = Float64(Float64(x - y_m) / Float64(y_m + Float64(x * Float64(Float64(x / y_m) + -1.0))));
	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 * ((x / y_m) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := Block[{t$95$0 = N[(N[(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[(N[(x / y$95$m), $MachinePrecision] + -1.0), $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. 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 80.8%

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

      1. clear-num 80.8%

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

      2. un-div-inv 81.0%

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

   7. Applied egg-rr 81.0%

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

   8. Taylor expanded in x around 0 80.5%

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

4. Final simplification 94.2%

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

Alternative 6: 84.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 3.8 \cdot 10^{-150}:\\ \;\;\;\;\frac{\left(x - y\_m\right) \cdot \left(1 + \frac{y\_m}{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y\_m}{y\_m + x \cdot \left(\frac{x}{y\_m} + -1\right)}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 3.8e-150)
   (/ (* (- x y_m) (+ 1.0 (/ y_m x))) x)
   (/ (- x y_m) (+ y_m (* x (+ (/ x y_m) -1.0))))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 3.8e-150) {
		tmp = ((x - y_m) * (1.0 + (y_m / x))) / x;
	} else {
		tmp = (x - y_m) / (y_m + (x * ((x / y_m) + -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 <= 3.8d-150) then
        tmp = ((x - y_m) * (1.0d0 + (y_m / x))) / x
    else
        tmp = (x - y_m) / (y_m + (x * ((x / y_m) + (-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 <= 3.8e-150) {
		tmp = ((x - y_m) * (1.0 + (y_m / x))) / x;
	} else {
		tmp = (x - y_m) / (y_m + (x * ((x / y_m) + -1.0)));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 3.8e-150:
		tmp = ((x - y_m) * (1.0 + (y_m / x))) / x
	else:
		tmp = (x - y_m) / (y_m + (x * ((x / y_m) + -1.0)))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 3.8e-150)
		tmp = Float64(Float64(Float64(x - y_m) * Float64(1.0 + Float64(y_m / x))) / x);
	else
		tmp = Float64(Float64(x - y_m) / Float64(y_m + Float64(x * Float64(Float64(x / y_m) + -1.0))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 3.8e-150)
		tmp = ((x - y_m) * (1.0 + (y_m / x))) / x;
	else
		tmp = (x - y_m) / (y_m + (x * ((x / y_m) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 3.8e-150], N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(1.0 + N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x - y$95$m), $MachinePrecision] / N[(y$95$m + N[(x * N[(N[(x / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.7999999999999998e-150

   1. Initial program 64.0%

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

      1. associate-/l* 64.1%

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

      2. +-commutative 64.1%

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

      3. fma-define 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in x around inf 31.6%

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

      1. associate-*r/ 31.7%

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

   7. Applied egg-rr 31.7%

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

   if 3.7999999999999998e-150 < 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* 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-define 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 83.1%

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

      1. clear-num 83.1%

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

      2. un-div-inv 83.4%

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

   7. Applied egg-rr 83.4%

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

   8. Taylor expanded in x around 0 83.7%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.8 \cdot 10^{-150}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(1 + \frac{y}{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y + x \cdot \left(\frac{x}{y} + -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 8.2 \cdot 10^{-150}:\\ \;\;\;\;\frac{\left(x - y\_m\right) \cdot \left(1 + \frac{y\_m}{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y\_m}{\frac{y\_m}{\frac{x}{y\_m} + 1}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 8.2e-150)
   (/ (* (- x y_m) (+ 1.0 (/ y_m x))) x)
   (/ (- x y_m) (/ y_m (+ (/ x y_m) 1.0)))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 8.2e-150) {
		tmp = ((x - y_m) * (1.0 + (y_m / x))) / x;
	} else {
		tmp = (x - y_m) / (y_m / ((x / y_m) + 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 <= 8.2d-150) then
        tmp = ((x - y_m) * (1.0d0 + (y_m / x))) / x
    else
        tmp = (x - y_m) / (y_m / ((x / y_m) + 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 <= 8.2e-150) {
		tmp = ((x - y_m) * (1.0 + (y_m / x))) / x;
	} else {
		tmp = (x - y_m) / (y_m / ((x / y_m) + 1.0));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 8.2e-150:
		tmp = ((x - y_m) * (1.0 + (y_m / x))) / x
	else:
		tmp = (x - y_m) / (y_m / ((x / y_m) + 1.0))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 8.2e-150)
		tmp = Float64(Float64(Float64(x - y_m) * Float64(1.0 + Float64(y_m / x))) / x);
	else
		tmp = Float64(Float64(x - y_m) / Float64(y_m / Float64(Float64(x / y_m) + 1.0)));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 8.2e-150)
		tmp = ((x - y_m) * (1.0 + (y_m / x))) / x;
	else
		tmp = (x - y_m) / (y_m / ((x / y_m) + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 8.2e-150], N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(1.0 + N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x - y$95$m), $MachinePrecision] / N[(y$95$m / N[(N[(x / y$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.1999999999999997e-150

   1. Initial program 64.0%

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

      1. associate-/l* 64.1%

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

      2. +-commutative 64.1%

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

      3. fma-define 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in x around inf 31.6%

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

      1. associate-*r/ 31.7%

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

   7. Applied egg-rr 31.7%

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

   if 8.1999999999999997e-150 < 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* 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-define 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 83.1%

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

      1. clear-num 83.1%

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

      2. un-div-inv 83.4%

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

   7. Applied egg-rr 83.4%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.2 \cdot 10^{-150}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(1 + \frac{y}{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\frac{y}{\frac{x}{y} + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 4.5 \cdot 10^{-149}:\\ \;\;\;\;\frac{\left(x - y\_m\right) \cdot \left(1 + \frac{y\_m}{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y\_m} + -1\right) \cdot \left(\frac{x}{y\_m} + 1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 4.5e-149)
   (/ (* (- x y_m) (+ 1.0 (/ y_m x))) x)
   (* (+ (/ x y_m) -1.0) (+ (/ x y_m) 1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 4.5e-149) {
		tmp = ((x - y_m) * (1.0 + (y_m / x))) / x;
	} else {
		tmp = ((x / y_m) + -1.0) * ((x / y_m) + 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 <= 4.5d-149) then
        tmp = ((x - y_m) * (1.0d0 + (y_m / x))) / x
    else
        tmp = ((x / y_m) + (-1.0d0)) * ((x / y_m) + 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 <= 4.5e-149) {
		tmp = ((x - y_m) * (1.0 + (y_m / x))) / x;
	} else {
		tmp = ((x / y_m) + -1.0) * ((x / y_m) + 1.0);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 4.5e-149:
		tmp = ((x - y_m) * (1.0 + (y_m / x))) / x
	else:
		tmp = ((x / y_m) + -1.0) * ((x / y_m) + 1.0)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 4.5e-149)
		tmp = Float64(Float64(Float64(x - y_m) * Float64(1.0 + Float64(y_m / x))) / x);
	else
		tmp = Float64(Float64(Float64(x / y_m) + -1.0) * Float64(Float64(x / y_m) + 1.0));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 4.5e-149)
		tmp = ((x - y_m) * (1.0 + (y_m / x))) / x;
	else
		tmp = ((x / y_m) + -1.0) * ((x / y_m) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 4.5e-149], N[(N[(N[(x - y$95$m), $MachinePrecision] * N[(1.0 + N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[(x / y$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.4999999999999998e-149

   1. Initial program 64.0%

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

      1. associate-/l* 64.1%

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

      2. +-commutative 64.1%

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

      3. fma-define 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in x around inf 31.6%

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

      1. associate-*r/ 31.7%

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

   7. Applied egg-rr 31.7%

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

   if 4.4999999999999998e-149 < 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* 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-define 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 83.1%

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

      1. clear-num 83.1%

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

      2. un-div-inv 83.4%

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

   7. Applied egg-rr 83.4%

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

      1. associate-/r/ 83.4%

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

      2. div-sub 83.4%

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

      3. *-inverses 83.4%

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

      4. sub-neg 83.4%

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

      5. metadata-eval 83.4%

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

   9. Applied egg-rr 83.4%

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

4. Final simplification 40.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{-149}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(1 + \frac{y}{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} + -1\right) \cdot \left(\frac{x}{y} + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 2.9 \cdot 10^{-150}:\\ \;\;\;\;\left(x - y\_m\right) \cdot \frac{1 + \frac{y\_m}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y\_m} + -1\right) \cdot \left(\frac{x}{y\_m} + 1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 2.9e-150)
   (* (- x y_m) (/ (+ 1.0 (/ y_m x)) x))
   (* (+ (/ x y_m) -1.0) (+ (/ x y_m) 1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 2.9e-150) {
		tmp = (x - y_m) * ((1.0 + (y_m / x)) / x);
	} else {
		tmp = ((x / y_m) + -1.0) * ((x / y_m) + 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 <= 2.9d-150) then
        tmp = (x - y_m) * ((1.0d0 + (y_m / x)) / x)
    else
        tmp = ((x / y_m) + (-1.0d0)) * ((x / y_m) + 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 <= 2.9e-150) {
		tmp = (x - y_m) * ((1.0 + (y_m / x)) / x);
	} else {
		tmp = ((x / y_m) + -1.0) * ((x / y_m) + 1.0);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 2.9e-150:
		tmp = (x - y_m) * ((1.0 + (y_m / x)) / x)
	else:
		tmp = ((x / y_m) + -1.0) * ((x / y_m) + 1.0)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 2.9e-150)
		tmp = Float64(Float64(x - y_m) * Float64(Float64(1.0 + Float64(y_m / x)) / x));
	else
		tmp = Float64(Float64(Float64(x / y_m) + -1.0) * Float64(Float64(x / y_m) + 1.0));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 2.9e-150)
		tmp = (x - y_m) * ((1.0 + (y_m / x)) / x);
	else
		tmp = ((x / y_m) + -1.0) * ((x / y_m) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 2.9e-150], N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(1.0 + N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[(x / y$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.8999999999999998e-150

   1. Initial program 64.0%

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

      1. associate-/l* 64.1%

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

      2. +-commutative 64.1%

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

      3. fma-define 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in x around inf 31.6%

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

   if 2.8999999999999998e-150 < 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* 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-define 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 83.1%

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

      1. clear-num 83.1%

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

      2. un-div-inv 83.4%

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

   7. Applied egg-rr 83.4%

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

      1. associate-/r/ 83.4%

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

      2. div-sub 83.4%

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

      3. *-inverses 83.4%

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

      4. sub-neg 83.4%

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

      5. metadata-eval 83.4%

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

   9. Applied egg-rr 83.4%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.9 \cdot 10^{-150}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{1 + \frac{y}{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} + -1\right) \cdot \left(\frac{x}{y} + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 83.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 3.1 \cdot 10^{-149}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y\_m} + -1\right) \cdot \left(\frac{x}{y\_m} + 1\right)\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 3.1e-149) 1.0 (* (+ (/ x y_m) -1.0) (+ (/ x y_m) 1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 3.1e-149) {
		tmp = 1.0;
	} else {
		tmp = ((x / y_m) + -1.0) * ((x / y_m) + 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-149) then
        tmp = 1.0d0
    else
        tmp = ((x / y_m) + (-1.0d0)) * ((x / y_m) + 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-149) {
		tmp = 1.0;
	} else {
		tmp = ((x / y_m) + -1.0) * ((x / y_m) + 1.0);
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 3.1e-149:
		tmp = 1.0
	else:
		tmp = ((x / y_m) + -1.0) * ((x / y_m) + 1.0)
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 3.1e-149)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(x / y_m) + -1.0) * Float64(Float64(x / y_m) + 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-149)
		tmp = 1.0;
	else
		tmp = ((x / y_m) + -1.0) * ((x / y_m) + 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-149], 1.0, N[(N[(N[(x / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[(x / y$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.09999999999999987e-149

   1. Initial program 64.0%

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

      1. associate-/l* 64.1%

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

      2. +-commutative 64.1%

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

      3. fma-define 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in x around inf 30.0%

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

   if 3.09999999999999987e-149 < 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* 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-define 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around inf 83.1%

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

      1. clear-num 83.1%

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

      2. un-div-inv 83.4%

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

   7. Applied egg-rr 83.4%

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

      1. associate-/r/ 83.4%

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

      2. div-sub 83.4%

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

      3. *-inverses 83.4%

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

      4. sub-neg 83.4%

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

      5. metadata-eval 83.4%

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

   9. Applied egg-rr 83.4%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.1 \cdot 10^{-149}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x}{y} + -1\right) \cdot \left(\frac{x}{y} + 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 83.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 8.2 \cdot 10^{-150}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m) :precision binary64 (if (<= y_m 8.2e-150) 1.0 -1.0))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 8.2e-150) {
		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 <= 8.2d-150) 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 <= 8.2e-150) {
		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 <= 8.2e-150:
		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 <= 8.2e-150)
		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 <= 8.2e-150)
		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, 8.2e-150], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 8.1999999999999997e-150

   1. Initial program 64.0%

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

      1. associate-/l* 64.1%

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

      2. +-commutative 64.1%

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

      3. fma-define 64.1%

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

   3. Simplified 64.1%

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

   5. Taylor expanded in x around inf 30.0%

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

   if 8.1999999999999997e-150 < 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* 99.4%

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

      2. +-commutative 99.4%

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

      3. fma-define 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in x around 0 82.5%

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

Alternative 12: 66.5% 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* 70.3%

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

   2. +-commutative 70.3%

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

   3. fma-define 70.3%

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

3. Simplified 70.3%

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

5. Taylor expanded in x around 0 72.4%

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

Developer Target 1: 99.9% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024129 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

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

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