Kahan p9 Example

Percentage Accurate: 68.7% → 100.0%

Time: 8.3s

Alternatives: 8

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: 68.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (/ (* (/ (- x y) (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(Float64(x - y) / hypot(x, y)) * 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[(N[(x - y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 67.9%

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

   1. add-sqr-sqrt 67.9%

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

   2. times-frac 68.7%

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

   3. hypot-def 68.7%

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

   4. hypot-def 99.9%

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

3. Applied egg-rr 99.9%

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

   1. associate-*r/ 100.0%

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

   2. +-commutative 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 92.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} t_0 := \frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y} \cdot \left(x \cdot 1.5\right) - y}{\mathsf{hypot}\left(x, y\right)}\\ \end{array} \end{array} \]

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) (* x 1.5)) y) (hypot x y)))))

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) * (x * 1.5)) - y) / hypot(x, y);
	}
	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 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = (((x / y) * (x * 1.5)) - y) / Math.hypot(x, y);
	}
	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) * (x * 1.5)) - y) / math.hypot(x, y)
	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(Float64(Float64(x / y) * Float64(x * 1.5)) - y) / hypot(x, y));
	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) * (x * 1.5)) - y) / hypot(x, y);
	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[(N[(N[(x / y), $MachinePrecision] * N[(x * 1.5), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision] / N[Sqrt[x ^ 2 + y ^ 2], $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. 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 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. associate-*r/ 100.0%

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

      2. +-commutative 100.0%

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

   5. Simplified 100.0%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

   7. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

   9. Simplified 99.9%

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

   10. Taylor expanded in x around 0 8.0%

       \[\leadsto \frac{\color{blue}{1.5 \cdot \frac{{x}^{2}}{y} + -1 \cdot y}}{\mathsf{hypot}\left(x, y\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 8.0%

          \[\leadsto \frac{1.5 \cdot \frac{{x}^{2}}{y} + \color{blue}{\left(-y\right)}}{\mathsf{hypot}\left(x, y\right)} \]

       2. unsub-neg 8.0%

          \[\leadsto \frac{\color{blue}{1.5 \cdot \frac{{x}^{2}}{y} - y}}{\mathsf{hypot}\left(x, y\right)} \]

       3. *-commutative 8.0%

          \[\leadsto \frac{\color{blue}{\frac{{x}^{2}}{y} \cdot 1.5} - y}{\mathsf{hypot}\left(x, y\right)} \]

       4. unpow2 8.0%

          \[\leadsto \frac{\frac{\color{blue}{x \cdot x}}{y} \cdot 1.5 - y}{\mathsf{hypot}\left(x, y\right)} \]

       5. associate-*l/ 9.4%

          \[\leadsto \frac{\color{blue}{\left(\frac{x}{y} \cdot x\right)} \cdot 1.5 - y}{\mathsf{hypot}\left(x, y\right)} \]

       6. associate-*l* 9.4%

          \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot \left(x \cdot 1.5\right)} - y}{\mathsf{hypot}\left(x, y\right)} \]

   12. Simplified 9.4%

       \[\leadsto \frac{\color{blue}{\frac{x}{y} \cdot \left(x \cdot 1.5\right) - y}}{\mathsf{hypot}\left(x, y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y} \leq 2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot \left(x + y\right)}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y} \cdot \left(x \cdot 1.5\right) - y}{\mathsf{hypot}\left(x, y\right)}\\ \end{array} \]

Alternative 3: 92.5% accurate, 0.1× speedup

Math

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

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 (fma 2.0 (* (/ x y) (/ x y)) -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 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = fma(2.0, ((x / y) * (x / y)), -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 <= 2.0)
		tmp = t_0;
	else
		tmp = fma(2.0, Float64(Float64(x / y) * Float64(x / y)), -1.0);
	end
	return 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[(2.0 * N[(N[(x / y), $MachinePrecision] * N[(x / y), $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%

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

   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. 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 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. associate-*r/ 100.0%

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

      2. +-commutative 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 61.0%

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

      1. fma-neg 61.0%

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

      2. unpow2 61.0%

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

      3. unpow2 61.0%

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

      4. times-frac 81.5%

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

      5. metadata-eval 81.5%

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

   8. Simplified 81.5%

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

4. Final simplification 94.0%

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

Alternative 4: 92.3% accurate, 0.5× speedup

Math

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

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) -1.0) (/ (+ x y) y)))))

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) + -1.0) * ((x + y) / y);
	}
	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) + (-1.0d0)) * ((x + y) / y)
    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) + -1.0) * ((x + y) / y);
	}
	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) + -1.0) * ((x + y) / y)
	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(Float64(x / y) + -1.0) * Float64(Float64(x + y) / y));
	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) + -1.0) * ((x + y) / y);
	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[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] / y), $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. Taylor expanded in x around 0 0.0%

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

      1. unpow2 0.0%

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

   4. Simplified 0.0%

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

      1. times-frac 80.8%

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

   6. Applied egg-rr 80.8%

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

      1. div-sub 80.8%

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

      2. sub-neg 80.8%

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

      3. *-inverses 80.8%

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

      4. metadata-eval 80.8%

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

      5. +-commutative 80.8%

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

   8. Simplified 80.8%

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

4. Final simplification 93.8%

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

Alternative 5: 83.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-171}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 8.5e-171)
   (+ 1.0 (* -2.0 (* (/ y x) (/ y x))))
   (+ -1.0 (/ x (* y (/ y x))))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-171) {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	} else {
		tmp = -1.0 + (x / (y * (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) :: tmp
    if (y <= 8.5d-171) then
        tmp = 1.0d0 + ((-2.0d0) * ((y / x) * (y / x)))
    else
        tmp = (-1.0d0) + (x / (y * (y / x)))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 8.5e-171) {
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	} else {
		tmp = -1.0 + (x / (y * (y / x)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 8.5e-171:
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)))
	else:
		tmp = -1.0 + (x / (y * (y / x)))
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 8.5e-171)
		tmp = Float64(1.0 + Float64(-2.0 * Float64(Float64(y / x) * Float64(y / x))));
	else
		tmp = Float64(-1.0 + Float64(x / Float64(y * Float64(y / x))));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8.5e-171)
		tmp = 1.0 + (-2.0 * ((y / x) * (y / x)));
	else
		tmp = -1.0 + (x / (y * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 8.5e-171], N[(1.0 + N[(-2.0 * N[(N[(y / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(x / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.50000000000000032e-171

   1. Initial program 61.3%

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

      1. associate-*r/ 62.2%

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

      2. +-commutative 62.2%

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

      3. fma-def 62.2%

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

   3. Simplified 62.2%

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

   4. Taylor expanded in y around 0 25.6%

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

      1. unpow2 25.6%

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

      2. unpow2 25.6%

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

   6. Simplified 25.6%

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

      1. times-frac 33.2%

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

   8. Applied egg-rr 33.2%

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

   if 8.50000000000000032e-171 < y

   1. Initial program 94.1%

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

   2. Taylor expanded in x around 0 68.1%

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

      1. unpow2 68.1%

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

   4. Simplified 68.1%

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

   5. Taylor expanded in x around 0 68.1%

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

      1. sub-neg 68.1%

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

      2. metadata-eval 68.1%

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

      3. unpow2 68.1%

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

      4. unpow2 68.1%

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

      5. associate-/l* 68.1%

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

   7. Simplified 68.1%

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

      1. associate-/l* 73.9%

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

      2. associate-/r/ 73.9%

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

   9. Applied egg-rr 73.9%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8.5 \cdot 10^{-171}:\\ \;\;\;\;1 + -2 \cdot \left(\frac{y}{x} \cdot \frac{y}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 6: 82.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-170}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y \cdot \frac{y}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 1.35e-170) 1.0 (+ -1.0 (/ x (* y (/ y x))))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.35e-170) {
		tmp = 1.0;
	} else {
		tmp = -1.0 + (x / (y * (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) :: tmp
    if (y <= 1.35d-170) then
        tmp = 1.0d0
    else
        tmp = (-1.0d0) + (x / (y * (y / x)))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.35e-170) {
		tmp = 1.0;
	} else {
		tmp = -1.0 + (x / (y * (y / x)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 1.35e-170:
		tmp = 1.0
	else:
		tmp = -1.0 + (x / (y * (y / x)))
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 1.35e-170)
		tmp = 1.0;
	else
		tmp = Float64(-1.0 + Float64(x / Float64(y * Float64(y / x))));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.35e-170)
		tmp = 1.0;
	else
		tmp = -1.0 + (x / (y * (y / x)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 1.35e-170], 1.0, N[(-1.0 + N[(x / N[(y * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.3499999999999999e-170

   1. Initial program 61.3%

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

      1. associate-*r/ 62.2%

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

      2. +-commutative 62.2%

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

      3. fma-def 62.2%

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

   3. Simplified 62.2%

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

   4. Taylor expanded in x around inf 31.1%

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

   if 1.3499999999999999e-170 < y

   1. Initial program 94.1%

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

   2. Taylor expanded in x around 0 68.1%

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

      1. unpow2 68.1%

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

   4. Simplified 68.1%

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

   5. Taylor expanded in x around 0 68.1%

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

      1. sub-neg 68.1%

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

      2. metadata-eval 68.1%

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

      3. unpow2 68.1%

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

      4. unpow2 68.1%

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

      5. associate-/l* 68.1%

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

   7. Simplified 68.1%

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

      1. associate-/l* 73.9%

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

      2. associate-/r/ 73.9%

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

   9. Applied egg-rr 73.9%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{-170}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y \cdot \frac{y}{x}}\\ \end{array} \]

Alternative 7: 82.4% accurate, 4.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{-169}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y) :precision binary64 (if (<= y 6.2e-169) 1.0 -1.0))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 6.2e-169) {
		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 <= 6.2d-169) 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 <= 6.2e-169) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 6.2e-169:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 6.2e-169)
		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 <= 6.2e-169)
		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, 6.2e-169], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 6.2000000000000004e-169

   1. Initial program 61.2%

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

      1. associate-*r/ 62.1%

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

      2. +-commutative 62.1%

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

      3. fma-def 62.1%

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

   3. Simplified 62.1%

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

   4. Taylor expanded in x around inf 31.3%

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

   if 6.2000000000000004e-169 < y

   1. Initial program 95.9%

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

      1. associate-*r/ 94.6%

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

      2. +-commutative 94.6%

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

      3. fma-def 94.6%

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

   3. Simplified 94.6%

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

   4. Taylor expanded in x around 0 73.3%

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

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{-169}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 8: 65.4% 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 67.9%

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

   1. associate-*r/ 68.4%

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

   2. +-commutative 68.4%

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

   3. fma-def 68.4%

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

3. Simplified 68.4%

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

4. Taylor expanded in x around 0 68.9%

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

5. Final simplification 68.9%

   \[\leadsto -1 \]

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 2023200 
(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))))