Kahan p9 Example

Percentage Accurate: 68.2% → 100.0%

Time: 6.8s

Alternatives: 6

Speedup: 4.9×

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: 68.2% 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

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (/ (* (+ x y) (/ (- y x) (hypot x y))) (- (hypot x y))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := N[(N[(N[(x + y), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 68.4%

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

   1. add-sqr-sqrt 68.4%

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

   2. times-frac 68.9%

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

   3. hypot-def 68.9%

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

   4. hypot-def 100.0%

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

3. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{x - y}{\mathsf{hypot}\left(x, y\right)} \cdot \frac{x + y}{\mathsf{hypot}\left(x, y\right)}} \]
4. 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/ 99.9%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.1× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (* (/ (- x y) (hypot x y)) (/ (+ x y) (hypot x y))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := N[(N[(N[(x - y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.4%

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

   1. add-sqr-sqrt 68.4%

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

   2. times-frac 68.9%

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

   3. hypot-def 68.9%

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

   4. hypot-def 100.0%

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

3. Applied egg-rr 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 92.5% accurate, 0.5× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))))
   (if (<= t_0 INFINITY) t_0 -1.0)))

C

y = abs(y);
double code(double x, double y) {
	double t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y));
	double tmp;
	if (t_0 <= ((double) INFINITY)) {
		tmp = t_0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Java

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

Python

y = abs(y)
def code(x, y):
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y))
	tmp = 0
	if t_0 <= math.inf:
		tmp = t_0
	else:
		tmp = -1.0
	return tmp

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, Infinity], t$95$0, -1.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 x y) (+.f64 x y)) (+.f64 (*.f64 x x) (*.f64 y y))) < +inf.0

   1. Initial program 100.0%

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

   if +inf.0 < (/.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. Taylor expanded in x around 0 78.0%

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

4. Final simplification 93.0%

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

Alternative 4: 93.1% accurate, 0.5× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (* (- x y) (+ x y)) (+ (* x x) (* y y)))))
   (if (<= t_0 2.0) t_0 (/ (- x y) (+ (- y x) (/ (* x 2.0) (/ y x)))))))

C

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

Fortran

NOTE: y should be positive before calling this function
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 = ((x - y) * (x + y)) / ((x * x) + (y * y))
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = (x - y) / ((y - x) + ((x * 2.0d0) / (y / x)))
    end if
    code = tmp
end function

Java

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

Python

y = abs(y)
def code(x, y):
	t_0 = ((x - y) * (x + y)) / ((x * x) + (y * y))
	tmp = 0
	if t_0 <= 2.0:
		tmp = t_0
	else:
		tmp = (x - y) / ((y - x) + ((x * 2.0) / (y / x)))
	return tmp

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], t$95$0, N[(N[(x - y), $MachinePrecision] / N[(N[(y - x), $MachinePrecision] + N[(N[(x * 2.0), $MachinePrecision] / N[(y / x), $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%

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

   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%

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

      2. +-commutative 3.1%

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

      3. +-commutative 3.1%

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

      4. +-commutative 3.1%

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

      5. fma-def 3.1%

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

      6. +-commutative 3.1%

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

   3. Simplified 3.1%

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

   4. Taylor expanded in x around 0 78.0%

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

      1. associate-+r+ 78.0%

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

      2. mul-1-neg 78.0%

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

      3. sub-neg 78.0%

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

   6. Simplified 78.0%

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

      1. unpow2 78.0%

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

      2. *-un-lft-identity 78.0%

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

      3. times-frac 79.5%

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

   8. Applied egg-rr 79.5%

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

      1. /-rgt-identity 79.5%

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

      2. associate-*r* 79.5%

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

      3. clear-num 79.5%

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

      4. un-div-inv 79.5%

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

   10. Applied egg-rr 79.5%

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

4. Final simplification 93.5%

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

Alternative 5: 81.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-187}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y) :precision binary64 (if (<= y 5e-187) 1.0 -1.0))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 5e-187) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 5d-187) then
        tmp = 1.0d0
    else
        tmp = -1.0d0
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 5e-187) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 5e-187:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 5e-187)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5e-187)
		tmp = 1.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 5e-187], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 4.9999999999999996e-187

   1. Initial program 60.3%

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

   2. Taylor expanded in x around inf 36.5%

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

   if 4.9999999999999996e-187 < y

   1. Initial program 91.0%

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

   2. Taylor expanded in x around 0 75.2%

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

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5 \cdot 10^{-187}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6: 66.1% accurate, 15.0× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y) :precision binary64 -1.0)

C

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

Fortran

NOTE: y should be positive before calling this function
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

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

Python

y = abs(y)
def code(x, y):
	return -1.0

Julia

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

MATLAB

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

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := -1.0

Derivation

1. Initial program 68.4%

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

2. Taylor expanded in x around 0 67.5%

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

3. Final simplification 67.5%

   \[\leadsto -1 \]

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 2023320 
(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))))