Kahan p9 Example

Percentage Accurate: 68.5% → 100.0%

Time: 9.5s

Alternatives: 9

Speedup: 0.6×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

y_m = fabs(y);
double code(double x, double y_m) {
	return ((x + y_m) * ((y_m - x) / 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) * ((y_m - x) / 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) * ((y_m - x) / math.hypot(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) * Float64(Float64(y_m - x) / hypot(x, y_m))) / Float64(-hypot(x, y_m)))
end

MATLAB

y_m = abs(y);
function tmp = code(x, y_m)
	tmp = ((x + y_m) * ((y_m - x) / hypot(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[(N[(y$95$m - x), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 61.5%

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

   1. add-sqr-sqrt 61.5%

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

   2. times-frac 62.2%

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

   3. hypot-def 62.2%

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

   4. hypot-def 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. frac-2neg 100.0%

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

   2. associate-*r/ 100.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

   \[\leadsto \frac{\left(x + y\right) \cdot \frac{y - x}{\mathsf{hypot}\left(x, y\right)}}{-\mathsf{hypot}\left(x, y\right)} \]
8. 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 61.5%

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

   1. add-sqr-sqrt 61.5%

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

   2. times-frac 62.2%

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

   3. hypot-def 62.2%

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

   4. hypot-def 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 3: 99.9% accurate, 0.1× speedup

Math

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

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 61.5%

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

   1. add-sqr-sqrt 61.5%

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

   2. times-frac 62.2%

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

   3. hypot-def 62.2%

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

   4. hypot-def 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. frac-times 100.0%

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

   4. *-commutative 100.0%

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

   5. *-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. Final simplification 100.0%

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

Alternative 4: 93.1% accurate, 0.1× speedup

Math

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

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

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double t_0 = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = ((x + y_m) / Math.hypot(x, y_m)) * ((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) / math.hypot(x, y_m)) * ((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(Float64(x + y_m) / hypot(x, 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)
	t_0 = ((x - y_m) * (x + y_m)) / ((x * x) + (y_m * y_m));
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = ((x + y_m) / hypot(x, y_m)) * ((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[(N[(x + y$95$m), $MachinePrecision] / N[Sqrt[x ^ 2 + y$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x / y$95$m), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.6%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   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-def 3.1%

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

      4. hypot-def 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. Taylor expanded in x around 0 13.7%

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

4. Final simplification 66.7%

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

Alternative 5: 80.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 5.4 \cdot 10^{-197} \lor \neg \left(y_m \leq 2.65 \cdot 10^{-171}\right) \land \left(y_m \leq 3.6 \cdot 10^{-140} \lor \neg \left(y_m \leq 1.5 \cdot 10^{-130}\right) \land y_m \leq 2.95 \cdot 10^{-111}\right):\\ \;\;\;\;\left(1 - \frac{y_m}{x}\right) \cdot \left(1 + \frac{y_m}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y_m}{y_m}\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (or (<= y_m 5.4e-197)
         (and (not (<= y_m 2.65e-171))
              (or (<= y_m 3.6e-140)
                  (and (not (<= y_m 1.5e-130)) (<= y_m 2.95e-111)))))
   (* (- 1.0 (/ y_m x)) (+ 1.0 (/ y_m x)))
   (/ (- x y_m) y_m)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if ((y_m <= 5.4e-197) || (!(y_m <= 2.65e-171) && ((y_m <= 3.6e-140) || (!(y_m <= 1.5e-130) && (y_m <= 2.95e-111))))) {
		tmp = (1.0 - (y_m / x)) * (1.0 + (y_m / x));
	} else {
		tmp = (x - y_m) / 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 <= 5.4d-197) .or. (.not. (y_m <= 2.65d-171)) .and. (y_m <= 3.6d-140) .or. (.not. (y_m <= 1.5d-130)) .and. (y_m <= 2.95d-111)) then
        tmp = (1.0d0 - (y_m / x)) * (1.0d0 + (y_m / x))
    else
        tmp = (x - y_m) / 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 <= 5.4e-197) || (!(y_m <= 2.65e-171) && ((y_m <= 3.6e-140) || (!(y_m <= 1.5e-130) && (y_m <= 2.95e-111))))) {
		tmp = (1.0 - (y_m / x)) * (1.0 + (y_m / x));
	} else {
		tmp = (x - y_m) / y_m;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if (y_m <= 5.4e-197) or (not (y_m <= 2.65e-171) and ((y_m <= 3.6e-140) or (not (y_m <= 1.5e-130) and (y_m <= 2.95e-111)))):
		tmp = (1.0 - (y_m / x)) * (1.0 + (y_m / x))
	else:
		tmp = (x - y_m) / y_m
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if ((y_m <= 5.4e-197) || (!(y_m <= 2.65e-171) && ((y_m <= 3.6e-140) || (!(y_m <= 1.5e-130) && (y_m <= 2.95e-111)))))
		tmp = Float64(Float64(1.0 - Float64(y_m / x)) * Float64(1.0 + Float64(y_m / x)));
	else
		tmp = Float64(Float64(x - y_m) / y_m);
	end
	return tmp
end

MATLAB

y_m = abs(y);
function tmp_2 = code(x, y_m)
	tmp = 0.0;
	if ((y_m <= 5.4e-197) || (~((y_m <= 2.65e-171)) && ((y_m <= 3.6e-140) || (~((y_m <= 1.5e-130)) && (y_m <= 2.95e-111)))))
		tmp = (1.0 - (y_m / x)) * (1.0 + (y_m / x));
	else
		tmp = (x - y_m) / 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, 5.4e-197], And[N[Not[LessEqual[y$95$m, 2.65e-171]], $MachinePrecision], Or[LessEqual[y$95$m, 3.6e-140], And[N[Not[LessEqual[y$95$m, 1.5e-130]], $MachinePrecision], LessEqual[y$95$m, 2.95e-111]]]]], N[(N[(1.0 - N[(y$95$m / x), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(y$95$m / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.40000000000000034e-197 or 2.65000000000000009e-171 < y < 3.6e-140 or 1.49999999999999993e-130 < y < 2.95e-111

   1. Initial program 58.3%

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

      1. add-sqr-sqrt 58.3%

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

      2. times-frac 59.1%

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

      3. hypot-def 59.1%

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

      4. hypot-def 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. Taylor expanded in x around inf 41.2%

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

   6. Taylor expanded in x around inf 40.7%

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

      1. mul-1-neg 40.7%

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

      2. unsub-neg 40.7%

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

   8. Simplified 40.7%

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

   if 5.40000000000000034e-197 < y < 2.65000000000000009e-171 or 3.6e-140 < y < 1.49999999999999993e-130 or 2.95e-111 < y

   1. Initial program 82.4%

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

      1. associate-/l* 82.8%

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

      2. remove-double-neg 82.8%

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

      3. sub-neg 82.8%

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

      4. +-commutative 82.8%

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

      5. fma-def 82.8%

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

      6. sub-neg 82.8%

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

      7. remove-double-neg 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in y around inf 81.6%

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

4. Final simplification 46.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.4 \cdot 10^{-197} \lor \neg \left(y \leq 2.65 \cdot 10^{-171}\right) \land \left(y \leq 3.6 \cdot 10^{-140} \lor \neg \left(y \leq 1.5 \cdot 10^{-130}\right) \land y \leq 2.95 \cdot 10^{-111}\right):\\ \;\;\;\;\left(1 - \frac{y}{x}\right) \cdot \left(1 + \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.1% accurate, 0.4× speedup

Math

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

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 (+ (* 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 = (2.0 * ((x / y_m) / (y_m / x))) + -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 = (2.0d0 * ((x / y_m) / (y_m / x))) + (-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 = (2.0 * ((x / y_m) / (y_m / x))) + -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 = (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 = Float64(Float64(2.0 * Float64(Float64(x / y_m) / Float64(y_m / x))) + -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 = (2.0 * ((x / y_m) / (y_m / x))) + -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[(2.0 * N[(N[(x / y$95$m), $MachinePrecision] / N[(y$95$m / x), $MachinePrecision]), $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 99.6%

      \[\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \]
   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-def 3.1%

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

      4. hypot-def 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. frac-2neg 100.0%

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

      2. associate-*r/ 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in y around -inf 55.1%

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

      1. sub-neg 55.1%

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

   9. Simplified 79.6%

      \[\leadsto \color{blue}{\left(0 + 2 \cdot {\left(\frac{x}{y}\right)}^{2}\right) + -1} \]
   10. Step-by-step derivation

       1. unpow2 79.6%

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

       2. clear-num 79.6%

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

       3. un-div-inv 79.6%

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

   11. Applied egg-rr 79.6%

       \[\leadsto \left(0 + 2 \cdot \color{blue}{\frac{\frac{x}{y}}{\frac{y}{x}}}\right) + -1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.9%

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

Alternative 7: 80.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 3.5 \cdot 10^{-197}:\\ \;\;\;\;1\\ \mathbf{elif}\;y_m \leq 4 \cdot 10^{-171} \lor \neg \left(y_m \leq 5 \cdot 10^{-140} \lor \neg \left(y_m \leq 1.45 \cdot 10^{-130}\right) \land y_m \leq 1.32 \cdot 10^{-111}\right):\\ \;\;\;\;\frac{x - y_m}{y_m}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 3.5e-197)
   1.0
   (if (or (<= y_m 4e-171)
           (not
            (or (<= y_m 5e-140)
                (and (not (<= y_m 1.45e-130)) (<= y_m 1.32e-111)))))
     (/ (- x y_m) y_m)
     1.0)))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 3.5e-197) {
		tmp = 1.0;
	} else if ((y_m <= 4e-171) || !((y_m <= 5e-140) || (!(y_m <= 1.45e-130) && (y_m <= 1.32e-111)))) {
		tmp = (x - y_m) / y_m;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

y_m = abs(y)
real(8) function code(x, y_m)
    real(8), intent (in) :: x
    real(8), intent (in) :: y_m
    real(8) :: tmp
    if (y_m <= 3.5d-197) then
        tmp = 1.0d0
    else if ((y_m <= 4d-171) .or. (.not. (y_m <= 5d-140) .or. (.not. (y_m <= 1.45d-130)) .and. (y_m <= 1.32d-111))) then
        tmp = (x - y_m) / y_m
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

y_m = Math.abs(y);
public static double code(double x, double y_m) {
	double tmp;
	if (y_m <= 3.5e-197) {
		tmp = 1.0;
	} else if ((y_m <= 4e-171) || !((y_m <= 5e-140) || (!(y_m <= 1.45e-130) && (y_m <= 1.32e-111)))) {
		tmp = (x - y_m) / y_m;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

y_m = math.fabs(y)
def code(x, y_m):
	tmp = 0
	if y_m <= 3.5e-197:
		tmp = 1.0
	elif (y_m <= 4e-171) or not ((y_m <= 5e-140) or (not (y_m <= 1.45e-130) and (y_m <= 1.32e-111))):
		tmp = (x - y_m) / y_m
	else:
		tmp = 1.0
	return tmp

Julia

y_m = abs(y)
function code(x, y_m)
	tmp = 0.0
	if (y_m <= 3.5e-197)
		tmp = 1.0;
	elseif ((y_m <= 4e-171) || !((y_m <= 5e-140) || (!(y_m <= 1.45e-130) && (y_m <= 1.32e-111))))
		tmp = Float64(Float64(x - y_m) / 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 <= 3.5e-197)
		tmp = 1.0;
	elseif ((y_m <= 4e-171) || ~(((y_m <= 5e-140) || (~((y_m <= 1.45e-130)) && (y_m <= 1.32e-111)))))
		tmp = (x - y_m) / y_m;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

y_m = N[Abs[y], $MachinePrecision]
code[x_, y$95$m_] := If[LessEqual[y$95$m, 3.5e-197], 1.0, If[Or[LessEqual[y$95$m, 4e-171], N[Not[Or[LessEqual[y$95$m, 5e-140], And[N[Not[LessEqual[y$95$m, 1.45e-130]], $MachinePrecision], LessEqual[y$95$m, 1.32e-111]]]], $MachinePrecision]], N[(N[(x - y$95$m), $MachinePrecision] / y$95$m), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if y < 3.4999999999999998e-197 or 3.9999999999999999e-171 < y < 5.00000000000000015e-140 or 1.45e-130 < y < 1.32e-111

   1. Initial program 58.3%

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

   3. Taylor expanded in x around inf 39.3%

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

   if 3.4999999999999998e-197 < y < 3.9999999999999999e-171 or 5.00000000000000015e-140 < y < 1.45e-130 or 1.32e-111 < y

   1. Initial program 82.4%

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

      1. associate-/l* 82.8%

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

      2. remove-double-neg 82.8%

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

      3. sub-neg 82.8%

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

      4. +-commutative 82.8%

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

      5. fma-def 82.8%

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

      6. sub-neg 82.8%

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

      7. remove-double-neg 82.8%

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

   3. Simplified 82.8%

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

   5. Taylor expanded in y around inf 81.6%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.5 \cdot 10^{-197}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-171} \lor \neg \left(y \leq 5 \cdot 10^{-140} \lor \neg \left(y \leq 1.45 \cdot 10^{-130}\right) \land y \leq 1.32 \cdot 10^{-111}\right):\\ \;\;\;\;\frac{x - y}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} y_m = \left|y\right| \\ \begin{array}{l} \mathbf{if}\;y_m \leq 1.5 \cdot 10^{-187}:\\ \;\;\;\;1\\ \mathbf{elif}\;y_m \leq 3.5 \cdot 10^{-171}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y_m \leq 6.8 \cdot 10^{-137}:\\ \;\;\;\;1\\ \mathbf{elif}\;y_m \leq 1.5 \cdot 10^{-130}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y_m \leq 1.55 \cdot 10^{-111}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

y_m = (fabs.f64 y)
(FPCore (x y_m)
 :precision binary64
 (if (<= y_m 1.5e-187)
   1.0
   (if (<= y_m 3.5e-171)
     -1.0
     (if (<= y_m 6.8e-137)
       1.0
       (if (<= y_m 1.5e-130) -1.0 (if (<= y_m 1.55e-111) 1.0 -1.0))))))

C

y_m = fabs(y);
double code(double x, double y_m) {
	double tmp;
	if (y_m <= 1.5e-187) {
		tmp = 1.0;
	} else if (y_m <= 3.5e-171) {
		tmp = -1.0;
	} else if (y_m <= 6.8e-137) {
		tmp = 1.0;
	} else if (y_m <= 1.5e-130) {
		tmp = -1.0;
	} else if (y_m <= 1.55e-111) {
		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 <= 1.5d-187) then
        tmp = 1.0d0
    else if (y_m <= 3.5d-171) then
        tmp = -1.0d0
    else if (y_m <= 6.8d-137) then
        tmp = 1.0d0
    else if (y_m <= 1.5d-130) then
        tmp = -1.0d0
    else if (y_m <= 1.55d-111) 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 <= 1.5e-187) {
		tmp = 1.0;
	} else if (y_m <= 3.5e-171) {
		tmp = -1.0;
	} else if (y_m <= 6.8e-137) {
		tmp = 1.0;
	} else if (y_m <= 1.5e-130) {
		tmp = -1.0;
	} else if (y_m <= 1.55e-111) {
		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 <= 1.5e-187:
		tmp = 1.0
	elif y_m <= 3.5e-171:
		tmp = -1.0
	elif y_m <= 6.8e-137:
		tmp = 1.0
	elif y_m <= 1.5e-130:
		tmp = -1.0
	elif y_m <= 1.55e-111:
		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 <= 1.5e-187)
		tmp = 1.0;
	elseif (y_m <= 3.5e-171)
		tmp = -1.0;
	elseif (y_m <= 6.8e-137)
		tmp = 1.0;
	elseif (y_m <= 1.5e-130)
		tmp = -1.0;
	elseif (y_m <= 1.55e-111)
		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 <= 1.5e-187)
		tmp = 1.0;
	elseif (y_m <= 3.5e-171)
		tmp = -1.0;
	elseif (y_m <= 6.8e-137)
		tmp = 1.0;
	elseif (y_m <= 1.5e-130)
		tmp = -1.0;
	elseif (y_m <= 1.55e-111)
		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, 1.5e-187], 1.0, If[LessEqual[y$95$m, 3.5e-171], -1.0, If[LessEqual[y$95$m, 6.8e-137], 1.0, If[LessEqual[y$95$m, 1.5e-130], -1.0, If[LessEqual[y$95$m, 1.55e-111], 1.0, -1.0]]]]]

Derivation

1. Split input into 2 regimes

2. if y < 1.50000000000000002e-187 or 3.49999999999999994e-171 < y < 6.80000000000000028e-137 or 1.49999999999999993e-130 < y < 1.55000000000000007e-111

   1. Initial program 58.4%

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

   3. Taylor expanded in x around inf 39.1%

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

   if 1.50000000000000002e-187 < y < 3.49999999999999994e-171 or 6.80000000000000028e-137 < y < 1.49999999999999993e-130 or 1.55000000000000007e-111 < y

   1. Initial program 81.8%

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

   3. Taylor expanded in x around 0 79.6%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-187}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-171}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-137}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-130}:\\ \;\;\;\;-1\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-111}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.7% 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 61.5%

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

3. Taylor expanded in x around 0 63.8%

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

4. Final simplification 63.8%

   \[\leadsto -1 \]
5. 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 2024020 
(FPCore (x y)
  :name "Kahan p9 Example"
  :precision binary64
  :pre (and (and (< 0.0 x) (< x 1.0)) (< y 1.0))

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

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