Kahan p9 Example

Percentage Accurate: 68.6% → 100.0%

Time: 9.9s

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: 68.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.3%

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

   1. add-sqr-sqrt 68.3%

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

   2. times-frac 69.0%

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

   3. hypot-define 69.0%

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

   4. hypot-define 100.0%

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

4. Applied egg-rr 100.0%

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

   1. *-commutative 100.0%

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

   2. clear-num 100.0%

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

   3. un-div-inv 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 93.3% accurate, 0.1× speedup

Math

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

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))))
        (t_1 (pow (/ x y_m) 2.0)))
   (if (<= t_0 2.0) t_0 (+ t_1 (+ -1.0 t_1)))))

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 t_1 = pow((x / y_m), 2.0);
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = t_1 + (-1.0 + t_1);
	}
	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) * (x - y_m)) / ((x * x) + (y_m * y_m))
    t_1 = (x / y_m) ** 2.0d0
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = t_1 + ((-1.0d0) + t_1)
    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 t_1 = Math.pow((x / y_m), 2.0);
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = t_1 + (-1.0 + t_1);
	}
	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))
	t_1 = math.pow((x / y_m), 2.0)
	tmp = 0
	if t_0 <= 2.0:
		tmp = t_0
	else:
		tmp = t_1 + (-1.0 + t_1)
	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)))
	t_1 = Float64(x / y_m) ^ 2.0
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = Float64(t_1 + Float64(-1.0 + t_1));
	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));
	t_1 = (x / y_m) ^ 2.0;
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = t_1 + (-1.0 + t_1);
	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]}, Block[{t$95$1 = N[Power[N[(x / y$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], t$95$0, N[(t$95$1 + N[(-1.0 + t$95$1), $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%

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

      3. un-div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around inf 46.9%

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

      1. associate--r+ 46.9%

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

   9. Simplified 74.0%

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

4. Final simplification 91.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y} \leq 2:\\ \;\;\;\;\frac{\left(x + y\right) \cdot \left(x - y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x}{y}\right)}^{2} + \left(-1 + {\left(\frac{x}{y}\right)}^{2}\right)\\ \end{array} \]
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 68.3%

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

   1. add-sqr-sqrt 68.3%

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

   2. times-frac 69.0%

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

   3. hypot-define 69.0%

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

   4. hypot-define 100.0%

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

4. Applied egg-rr 100.0%

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

5. Final simplification 100.0%

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

Alternative 4: 93.3% accurate, 0.1× speedup

Math

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

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 (/ (/ x y_m) (/ y_m x)) -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, ((x / y_m) / (y_m / x)), -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(Float64(x / y_m) / Float64(y_m / x)), -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[(N[(x / y$95$m), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $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%

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

      3. un-div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 46.9%

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

      1. fmm-def 46.9%

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

      2. unpow2 46.9%

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

      3. unpow2 46.9%

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

      4. times-frac 74.0%

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

      5. unpow2 74.0%

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

      6. metadata-eval 74.0%

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

   9. Simplified 74.0%

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

       1. unpow2 74.0%

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

       2. clear-num 74.0%

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

       3. un-div-inv 74.0%

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

   11. Applied egg-rr 74.0%

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

4. Final simplification 91.8%

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

Alternative 5: 93.2% accurate, 0.4× speedup

Math

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

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 (/ 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 + (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) + (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 + (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 + (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(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 + (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[(x / y$95$m), $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. 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%

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

      3. +-commutative 3.1%

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

      4. +-commutative 3.1%

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

      5. 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 73.0%

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

      1. clear-num 73.0%

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

      2. un-div-inv 73.2%

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

   7. Applied egg-rr 73.2%

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

   8. Taylor expanded in x around 0 73.1%

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

4. Final simplification 91.5%

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

Alternative 6: 82.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.28 \cdot 10^{-185} \lor \neg \left(y\_m \leq 1.9 \cdot 10^{-170}\right) \land y\_m \leq 1.15 \cdot 10^{-123}:\\ \;\;\;\;1 - \frac{2 \cdot \left(y\_m \cdot \frac{y\_m}{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y\_m}{y\_m + x \cdot \left(-1 + \frac{x}{y\_m}\right)}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (or (<= y_m 1.28e-185) (and (not (<= y_m 1.9e-170)) (<= y_m 1.15e-123)))
   (- 1.0 (/ (* 2.0 (* y_m (/ y_m x))) x))
   (/ (- x y_m) (+ y_m (* x (+ -1.0 (/ x y_m)))))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((y_m <= 1.28e-185) || (!(y_m <= 1.9e-170) && (y_m <= 1.15e-123))) {
		tmp = 1.0 - ((2.0 * (y_m * (y_m / x))) / x);
	} else {
		tmp = (x - y_m) / (y_m + (x * (-1.0 + (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) :: tmp
    if ((y_m <= 1.28d-185) .or. (.not. (y_m <= 1.9d-170)) .and. (y_m <= 1.15d-123)) then
        tmp = 1.0d0 - ((2.0d0 * (y_m * (y_m / x))) / x)
    else
        tmp = (x - y_m) / (y_m + (x * ((-1.0d0) + (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 tmp;
	if ((y_m <= 1.28e-185) || (!(y_m <= 1.9e-170) && (y_m <= 1.15e-123))) {
		tmp = 1.0 - ((2.0 * (y_m * (y_m / x))) / x);
	} else {
		tmp = (x - y_m) / (y_m + (x * (-1.0 + (x / y_m))));
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if (y_m <= 1.28e-185) or (not (y_m <= 1.9e-170) and (y_m <= 1.15e-123)):
		tmp = 1.0 - ((2.0 * (y_m * (y_m / x))) / x)
	else:
		tmp = (x - y_m) / (y_m + (x * (-1.0 + (x / y_m))))
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if ((y_m <= 1.28e-185) || (!(y_m <= 1.9e-170) && (y_m <= 1.15e-123)))
		tmp = Float64(1.0 - Float64(Float64(2.0 * Float64(y_m * Float64(y_m / x))) / x));
	else
		tmp = Float64(Float64(x - y_m) / Float64(y_m + Float64(x * Float64(-1.0 + Float64(x / y_m)))));
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if ((y_m <= 1.28e-185) || (~((y_m <= 1.9e-170)) && (y_m <= 1.15e-123)))
		tmp = 1.0 - ((2.0 * (y_m * (y_m / x))) / x);
	else
		tmp = (x - y_m) / (y_m + (x * (-1.0 + (x / 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, 1.28e-185], And[N[Not[LessEqual[y$95$m, 1.9e-170]], $MachinePrecision], LessEqual[y$95$m, 1.15e-123]]], 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[(y$95$m + N[(x * N[(-1.0 + N[(x / y$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.28e-185 or 1.8999999999999999e-170 < y < 1.14999999999999993e-123

   1. Initial program 64.8%

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

      1. associate-/l* 65.2%

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

      2. +-commutative 65.2%

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

      3. +-commutative 65.2%

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

      4. +-commutative 65.2%

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

      5. fma-define 65.2%

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

   3. Simplified 65.2%

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

   5. Taylor expanded in x around -inf 42.1%

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

      1. mul-1-neg 42.1%

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

      2. unsub-neg 42.1%

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

   7. Simplified 42.6%

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

   8. Taylor expanded in y around 0 42.6%

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

      1. unpow2 42.6%

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

      2. associate-/l* 43.3%

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

   10. Applied egg-rr 43.3%

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

   if 1.28e-185 < y < 1.8999999999999999e-170 or 1.14999999999999993e-123 < y

   1. Initial program 86.0%

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

      1. associate-/l* 86.0%

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

      2. +-commutative 86.0%

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

      3. +-commutative 86.0%

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

      4. +-commutative 86.0%

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

      5. fma-define 86.0%

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

   3. Simplified 86.0%

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

   5. Taylor expanded in y around inf 82.6%

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

      1. clear-num 82.6%

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

      2. un-div-inv 82.8%

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

   7. Applied egg-rr 82.8%

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

   8. Taylor expanded in x around 0 83.1%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.28 \cdot 10^{-185} \lor \neg \left(y \leq 1.9 \cdot 10^{-170}\right) \land y \leq 1.15 \cdot 10^{-123}:\\ \;\;\;\;1 - \frac{2 \cdot \left(y \cdot \frac{y}{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y + x \cdot \left(-1 + \frac{x}{y}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.8 \cdot 10^{-185}:\\ \;\;\;\;1\\ \mathbf{elif}\;y\_m \leq 1.6 \cdot 10^{-166} \lor \neg \left(y\_m \leq 1.05 \cdot 10^{-123}\right):\\ \;\;\;\;\left(x - y\_m\right) \cdot \frac{\frac{x}{y\_m} + 1}{y\_m}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.8e-185)
   1.0
   (if (or (<= y_m 1.6e-166) (not (<= y_m 1.05e-123)))
     (* (- 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 <= 1.8e-185) {
		tmp = 1.0;
	} else if ((y_m <= 1.6e-166) || !(y_m <= 1.05e-123)) {
		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 <= 1.8d-185) then
        tmp = 1.0d0
    else if ((y_m <= 1.6d-166) .or. (.not. (y_m <= 1.05d-123))) 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 <= 1.8e-185) {
		tmp = 1.0;
	} else if ((y_m <= 1.6e-166) || !(y_m <= 1.05e-123)) {
		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 <= 1.8e-185:
		tmp = 1.0
	elif (y_m <= 1.6e-166) or not (y_m <= 1.05e-123):
		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 <= 1.8e-185)
		tmp = 1.0;
	elseif ((y_m <= 1.6e-166) || !(y_m <= 1.05e-123))
		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 <= 1.8e-185)
		tmp = 1.0;
	elseif ((y_m <= 1.6e-166) || ~((y_m <= 1.05e-123)))
		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, 1.8e-185], 1.0, If[Or[LessEqual[y$95$m, 1.6e-166], N[Not[LessEqual[y$95$m, 1.05e-123]], $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 < 1.7999999999999999e-185 or 1.6e-166 < y < 1.05e-123

   1. Initial program 64.9%

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

      1. associate-/l* 65.3%

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

      2. +-commutative 65.3%

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

      3. +-commutative 65.3%

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

      4. +-commutative 65.3%

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

      5. fma-define 65.3%

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

   3. Simplified 65.3%

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

   5. Taylor expanded in x around inf 41.2%

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

   if 1.7999999999999999e-185 < y < 1.6e-166 or 1.05e-123 < y

   1. Initial program 84.4%

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

      1. associate-/l* 84.5%

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

      2. +-commutative 84.5%

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

      3. +-commutative 84.5%

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

      4. +-commutative 84.5%

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

      5. fma-define 84.5%

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

   3. Simplified 84.5%

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

   5. Taylor expanded in y around inf 81.2%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{-185}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-166} \lor \neg \left(y \leq 1.05 \cdot 10^{-123}\right):\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.18 \cdot 10^{-185} \lor \neg \left(y\_m \leq 1.65 \cdot 10^{-171}\right) \land y\_m \leq 7.8 \cdot 10^{-122}:\\ \;\;\;\;\left(x - y\_m\right) \cdot \frac{\frac{y\_m}{x} + 1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\_m\right) \cdot \frac{\frac{x}{y\_m} + 1}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (or (<= y_m 1.18e-185) (and (not (<= y_m 1.65e-171)) (<= y_m 7.8e-122)))
   (* (- x y_m) (/ (+ (/ y_m x) 1.0) 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 <= 1.18e-185) || (!(y_m <= 1.65e-171) && (y_m <= 7.8e-122))) {
		tmp = (x - y_m) * (((y_m / x) + 1.0) / 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 <= 1.18d-185) .or. (.not. (y_m <= 1.65d-171)) .and. (y_m <= 7.8d-122)) then
        tmp = (x - y_m) * (((y_m / x) + 1.0d0) / 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 <= 1.18e-185) || (!(y_m <= 1.65e-171) && (y_m <= 7.8e-122))) {
		tmp = (x - y_m) * (((y_m / x) + 1.0) / 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 <= 1.18e-185) or (not (y_m <= 1.65e-171) and (y_m <= 7.8e-122)):
		tmp = (x - y_m) * (((y_m / x) + 1.0) / 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 <= 1.18e-185) || (!(y_m <= 1.65e-171) && (y_m <= 7.8e-122)))
		tmp = Float64(Float64(x - y_m) * Float64(Float64(Float64(y_m / x) + 1.0) / 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 <= 1.18e-185) || (~((y_m <= 1.65e-171)) && (y_m <= 7.8e-122)))
		tmp = (x - y_m) * (((y_m / x) + 1.0) / 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, 1.18e-185], And[N[Not[LessEqual[y$95$m, 1.65e-171]], $MachinePrecision], LessEqual[y$95$m, 7.8e-122]]], N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(N[(y$95$m / x), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(N[(x - y$95$m), $MachinePrecision] * N[(N[(N[(x / y$95$m), $MachinePrecision] + 1.0), $MachinePrecision] / y$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.18e-185 or 1.6500000000000001e-171 < y < 7.79999999999999979e-122

   1. Initial program 65.1%

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

      1. associate-/l* 65.5%

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

      2. +-commutative 65.5%

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

      3. +-commutative 65.5%

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

      4. +-commutative 65.5%

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

      5. fma-define 65.5%

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

   3. Simplified 65.5%

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

   5. Taylor expanded in x around inf 42.7%

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

   if 1.18e-185 < y < 1.6500000000000001e-171 or 7.79999999999999979e-122 < y

   1. Initial program 85.3%

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

      1. associate-/l* 85.3%

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

      2. +-commutative 85.3%

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

      3. +-commutative 85.3%

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

      4. +-commutative 85.3%

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

      5. fma-define 85.3%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in y around inf 84.1%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.18 \cdot 10^{-185} \lor \neg \left(y \leq 1.65 \cdot 10^{-171}\right) \land y \leq 7.8 \cdot 10^{-122}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{y}{x} + 1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.28 \cdot 10^{-185} \lor \neg \left(y\_m \leq 5 \cdot 10^{-171}\right) \land y\_m \leq 9.5 \cdot 10^{-121}:\\ \;\;\;\;1 - \frac{2 \cdot \left(y\_m \cdot \frac{y\_m}{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\_m\right) \cdot \frac{\frac{x}{y\_m} + 1}{y\_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (or (<= y_m 1.28e-185) (and (not (<= y_m 5e-171)) (<= y_m 9.5e-121)))
   (- 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 <= 1.28e-185) || (!(y_m <= 5e-171) && (y_m <= 9.5e-121))) {
		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 <= 1.28d-185) .or. (.not. (y_m <= 5d-171)) .and. (y_m <= 9.5d-121)) 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 <= 1.28e-185) || (!(y_m <= 5e-171) && (y_m <= 9.5e-121))) {
		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 <= 1.28e-185) or (not (y_m <= 5e-171) and (y_m <= 9.5e-121)):
		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 <= 1.28e-185) || (!(y_m <= 5e-171) && (y_m <= 9.5e-121)))
		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 <= 1.28e-185) || (~((y_m <= 5e-171)) && (y_m <= 9.5e-121)))
		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, 1.28e-185], And[N[Not[LessEqual[y$95$m, 5e-171]], $MachinePrecision], LessEqual[y$95$m, 9.5e-121]]], 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 < 1.28e-185 or 4.99999999999999992e-171 < y < 9.4999999999999994e-121

   1. Initial program 65.1%

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

      1. associate-/l* 65.5%

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

      2. +-commutative 65.5%

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

      3. +-commutative 65.5%

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

      4. +-commutative 65.5%

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

      5. fma-define 65.5%

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

   3. Simplified 65.5%

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

   5. Taylor expanded in x around -inf 42.2%

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

      1. mul-1-neg 42.2%

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

      2. unsub-neg 42.2%

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

   7. Simplified 42.7%

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

   8. Taylor expanded in y around 0 42.7%

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

      1. unpow2 42.7%

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

      2. associate-/l* 43.4%

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

   10. Applied egg-rr 43.4%

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

   if 1.28e-185 < y < 4.99999999999999992e-171 or 9.4999999999999994e-121 < y

   1. Initial program 85.3%

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

      1. associate-/l* 85.3%

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

      2. +-commutative 85.3%

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

      3. +-commutative 85.3%

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

      4. +-commutative 85.3%

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

      5. fma-define 85.3%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in y around inf 84.1%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.28 \cdot 10^{-185} \lor \neg \left(y \leq 5 \cdot 10^{-171}\right) \land y \leq 9.5 \cdot 10^{-121}:\\ \;\;\;\;1 - \frac{2 \cdot \left(y \cdot \frac{y}{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{\frac{x}{y} + 1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 2.65 \cdot 10^{-185} \lor \neg \left(y\_m \leq 2.35 \cdot 10^{-169}\right) \land y\_m \leq 5.8 \cdot 10^{-119}:\\ \;\;\;\;1 - \frac{2 \cdot \left(y\_m \cdot \frac{y\_m}{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y\_m}{\frac{y\_m}{\frac{x}{y\_m} + 1}}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (or (<= y_m 2.65e-185) (and (not (<= y_m 2.35e-169)) (<= y_m 5.8e-119)))
   (- 1.0 (/ (* 2.0 (* y_m (/ 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 <= 2.65e-185) || (!(y_m <= 2.35e-169) && (y_m <= 5.8e-119))) {
		tmp = 1.0 - ((2.0 * (y_m * (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 <= 2.65d-185) .or. (.not. (y_m <= 2.35d-169)) .and. (y_m <= 5.8d-119)) then
        tmp = 1.0d0 - ((2.0d0 * (y_m * (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 <= 2.65e-185) || (!(y_m <= 2.35e-169) && (y_m <= 5.8e-119))) {
		tmp = 1.0 - ((2.0 * (y_m * (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 <= 2.65e-185) or (not (y_m <= 2.35e-169) and (y_m <= 5.8e-119)):
		tmp = 1.0 - ((2.0 * (y_m * (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 <= 2.65e-185) || (!(y_m <= 2.35e-169) && (y_m <= 5.8e-119)))
		tmp = Float64(1.0 - Float64(Float64(2.0 * Float64(y_m * 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 <= 2.65e-185) || (~((y_m <= 2.35e-169)) && (y_m <= 5.8e-119)))
		tmp = 1.0 - ((2.0 * (y_m * (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[Or[LessEqual[y$95$m, 2.65e-185], And[N[Not[LessEqual[y$95$m, 2.35e-169]], $MachinePrecision], LessEqual[y$95$m, 5.8e-119]]], 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[(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 < 2.64999999999999987e-185 or 2.34999999999999995e-169 < y < 5.8e-119

   1. Initial program 65.1%

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

      1. associate-/l* 65.5%

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

      2. +-commutative 65.5%

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

      3. +-commutative 65.5%

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

      4. +-commutative 65.5%

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

      5. fma-define 65.5%

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

   3. Simplified 65.5%

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

   5. Taylor expanded in x around -inf 42.2%

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

      1. mul-1-neg 42.2%

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

      2. unsub-neg 42.2%

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

   7. Simplified 42.7%

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

   8. Taylor expanded in y around 0 42.7%

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

      1. unpow2 42.7%

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

      2. associate-/l* 43.4%

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

   10. Applied egg-rr 43.4%

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

   if 2.64999999999999987e-185 < y < 2.34999999999999995e-169 or 5.8e-119 < y

   1. Initial program 85.3%

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

      1. associate-/l* 85.3%

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

      2. +-commutative 85.3%

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

      3. +-commutative 85.3%

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

      4. +-commutative 85.3%

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

      5. fma-define 85.3%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in y around inf 84.1%

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

      1. clear-num 84.1%

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

      2. un-div-inv 84.3%

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

   7. Applied egg-rr 84.3%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.65 \cdot 10^{-185} \lor \neg \left(y \leq 2.35 \cdot 10^{-169}\right) \land y \leq 5.8 \cdot 10^{-119}:\\ \;\;\;\;1 - \frac{2 \cdot \left(y \cdot \frac{y}{x}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\frac{y}{\frac{x}{y} + 1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y\_m \leq 1.36 \cdot 10^{-186}:\\ \;\;\;\;1\\ \mathbf{elif}\;y\_m \leq 1.6 \cdot 10^{-171}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y\_m \leq 2.1 \cdot 10^{-121}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.36e-186)
   1.0
   (if (<= y_m 1.6e-171) -1.0 (if (<= y_m 2.1e-121) 1.0 -1.0))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.36e-186) {
		tmp = 1.0;
	} else if (y_m <= 1.6e-171) {
		tmp = -1.0;
	} else if (y_m <= 2.1e-121) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 1.36d-186) then
        tmp = 1.0d0
    else if (y_m <= 1.6d-171) then
        tmp = -1.0d0
    else if (y_m <= 2.1d-121) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.36e-186) {
		tmp = 1.0;
	} else if (y_m <= 1.6e-171) {
		tmp = -1.0;
	} else if (y_m <= 2.1e-121) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 1.36e-186:
		tmp = 1.0
	elif y_m <= 1.6e-171:
		tmp = -1.0
	elif y_m <= 2.1e-121:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 1.36e-186)
		tmp = 1.0;
	elseif (y_m <= 1.6e-171)
		tmp = -1.0;
	elseif (y_m <= 2.1e-121)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if (y_m <= 1.36e-186)
		tmp = 1.0;
	elseif (y_m <= 1.6e-171)
		tmp = -1.0;
	elseif (y_m <= 2.1e-121)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 1.36e-186], 1.0, If[LessEqual[y$95$m, 1.6e-171], -1.0, If[LessEqual[y$95$m, 2.1e-121], 1.0, -1.0]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3600000000000001e-186 or 1.6000000000000001e-171 < y < 2.0999999999999999e-121

   1. Initial program 65.1%

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

      1. associate-/l* 65.5%

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

      2. +-commutative 65.5%

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

      3. +-commutative 65.5%

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

      4. +-commutative 65.5%

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

      5. fma-define 65.5%

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

   3. Simplified 65.5%

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

   5. Taylor expanded in x around inf 41.4%

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

   if 1.3600000000000001e-186 < y < 1.6000000000000001e-171 or 2.0999999999999999e-121 < y

   1. Initial program 85.3%

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

      1. associate-/l* 85.3%

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

      2. +-commutative 85.3%

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

      3. +-commutative 85.3%

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

      4. +-commutative 85.3%

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

      5. fma-define 85.3%

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

   3. Simplified 85.3%

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

   5. Taylor expanded in x around 0 83.2%

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

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.36 \cdot 10^{-186}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-171}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-121}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.8% 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 68.3%

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

   1. associate-/l* 68.7%

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

   2. +-commutative 68.7%

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

   3. +-commutative 68.7%

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

   4. +-commutative 68.7%

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

   5. fma-define 68.7%

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

3. Simplified 68.7%

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

5. Taylor expanded in x around 0 62.7%

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

6. Final simplification 62.7%

   \[\leadsto -1 \]
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 2024130 
(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))))