Kahan p9 Example

Percentage Accurate: 68.4% → 100.0%

Time: 7.9s

Alternatives: 10

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: 68.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 70.9%

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

   1. fma-def 70.9%

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

   2. associate-/l* 70.5%

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

   3. *-un-lft-identity 70.5%

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

   4. add-sqr-sqrt 40.9%

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

   5. times-frac 40.9%

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

   6. sqrt-div 41.1%

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

   7. fma-def 41.1%

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

   8. hypot-def 41.1%

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

   9. sqrt-div 40.9%

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

   10. fma-def 40.9%

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

   11. hypot-def 51.1%

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

3. Applied egg-rr 51.1%

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

   1. frac-times 51.0%

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

   2. *-un-lft-identity 51.0%

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

   3. pow2 51.0%

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

5. Applied egg-rr 99.9%

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

6. Final simplification 99.9%

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

Alternative 2: 99.7% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 70.9%

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

   1. +-commutative 70.9%

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

   2. +-commutative 70.9%

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

   3. associate-*l/ 70.4%

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

   4. +-commutative 70.4%

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

   5. +-commutative 70.4%

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

   6. fma-def 70.4%

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

3. Simplified 70.4%

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

   1. *-un-lft-identity 70.4%

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

   2. add-sqr-sqrt 70.4%

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

   3. times-frac 70.5%

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

   4. fma-def 70.5%

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

   5. hypot-def 70.5%

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

   6. fma-def 70.5%

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

   7. hypot-def 99.7%

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

5. Applied egg-rr 99.7%

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

   1. associate-*l/ 99.7%

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

   2. *-lft-identity 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 3: 92.8% accurate, 0.5× speedup

Math

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

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 (/ (- x y) (+ y (* x (+ (/ 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 = (x - y) / (y + (x * ((x / y) + -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) :: 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 / y) + (-1.0d0))))
    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 / y) + -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 <= 2.0:
		tmp = t_0
	else:
		tmp = (x - y) / (y + (x * ((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 = Float64(Float64(x - y) / Float64(y + Float64(x * Float64(Float64(x / y) + -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 <= 2.0)
		tmp = t_0;
	else
		tmp = (x - y) / (y + (x * ((x / y) + -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, 2.0], t$95$0, N[(N[(x - y), $MachinePrecision] / N[(y + N[(x * N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 99.8%

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

   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. remove-double-neg 3.1%

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

      4. sub-neg 3.1%

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

      5. +-commutative 3.1%

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

      6. fma-def 3.1%

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

      7. sub-neg 3.1%

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

      8. remove-double-neg 3.1%

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

   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 3.1%

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

      1. unpow2 3.1%

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

   6. Simplified 3.1%

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

   7. Taylor expanded in y around inf 78.2%

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

      1. unpow2 78.2%

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

      2. associate-*l/ 79.3%

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

      3. distribute-rgt-out 79.3%

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

   9. Simplified 79.3%

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

4. Final simplification 93.9%

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

Alternative 4: 92.9% accurate, 0.5× speedup

Math

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

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 2.0) (/ x y)) (- 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) / (((x * 2.0) * (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) :: 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) / (((x * 2.0d0) * (x / y)) + (y - x))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double 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 * 2.0) * (x / y)) + (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) / (((x * 2.0) * (x / y)) + (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(Float64(x * 2.0) * Float64(x / y)) + 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) / (((x * 2.0) * (x / y)) + (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[(N[(x * 2.0), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision] + N[(y - x), $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 99.8%

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

   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. remove-double-neg 3.1%

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

      4. sub-neg 3.1%

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

      5. +-commutative 3.1%

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

      6. fma-def 3.1%

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

      7. sub-neg 3.1%

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

      8. remove-double-neg 3.1%

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

   3. Simplified 3.1%

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

      1. fma-def 3.1%

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

      2. +-commutative 3.1%

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

   5. Applied egg-rr 3.1%

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

   6. Taylor expanded in y around -inf 78.2%

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

      1. +-commutative 78.2%

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

      2. neg-mul-1 78.2%

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

      3. +-commutative 78.2%

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

      4. associate-+l+ 78.2%

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

   8. Simplified 80.0%

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

4. Final simplification 94.0%

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

Alternative 5: 92.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-167}:\\ \;\;\;\;1 + \frac{y}{x} \cdot \frac{y \cdot -2}{x}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-18}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{x - y}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 5.5e-167)
   (+ 1.0 (* (/ y x) (/ (* y -2.0) x)))
   (if (<= y 4e-18) (* (+ x y) (/ (- x y) (+ (* x x) (* y y)))) -1.0)))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 5.5e-167) {
		tmp = 1.0 + ((y / x) * ((y * -2.0) / x));
	} else if (y <= 4e-18) {
		tmp = (x + y) * ((x - y) / ((x * x) + (y * y)));
	} 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 <= 5.5d-167) then
        tmp = 1.0d0 + ((y / x) * ((y * (-2.0d0)) / x))
    else if (y <= 4d-18) then
        tmp = (x + y) * ((x - y) / ((x * x) + (y * y)))
    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 <= 5.5e-167) {
		tmp = 1.0 + ((y / x) * ((y * -2.0) / x));
	} else if (y <= 4e-18) {
		tmp = (x + y) * ((x - y) / ((x * x) + (y * y)));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 5.5e-167:
		tmp = 1.0 + ((y / x) * ((y * -2.0) / x))
	elif y <= 4e-18:
		tmp = (x + y) * ((x - y) / ((x * x) + (y * y)))
	else:
		tmp = -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 5.5e-167)
		tmp = Float64(1.0 + Float64(Float64(y / x) * Float64(Float64(y * -2.0) / x)));
	elseif (y <= 4e-18)
		tmp = Float64(Float64(x + y) * Float64(Float64(x - y) / Float64(Float64(x * x) + Float64(y * y))));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.5e-167)
		tmp = 1.0 + ((y / x) * ((y * -2.0) / x));
	elseif (y <= 4e-18)
		tmp = (x + y) * ((x - y) / ((x * x) + (y * y)));
	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, 5.5e-167], N[(1.0 + N[(N[(y / x), $MachinePrecision] * N[(N[(y * -2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4e-18], N[(N[(x + y), $MachinePrecision] * N[(N[(x - y), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0]]

Derivation

1. Split input into 3 regimes

2. if y < 5.5000000000000003e-167

   1. Initial program 64.6%

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

      1. +-commutative 64.6%

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

      2. +-commutative 64.6%

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

      3. associate-*l/ 64.8%

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

      4. +-commutative 64.8%

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

      5. +-commutative 64.8%

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

      6. fma-def 64.8%

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

   3. Simplified 64.8%

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

   4. Taylor expanded in x around -inf 30.6%

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

      1. associate-+r+ 30.6%

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

      2. +-commutative 30.6%

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

      3. associate-+r+ 30.6%

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

      4. associate-+r+ 30.6%

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

      5. distribute-lft1-in 30.6%

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

      6. metadata-eval 30.6%

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

      7. mul0-lft 31.2%

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

      8. metadata-eval 31.2%

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

      9. unpow2 31.2%

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

      10. associate-*r/ 31.2%

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

   6. Simplified 31.2%

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

   7. Taylor expanded in y around 0 31.2%

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

      1. associate-*r/ 31.2%

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

      2. *-commutative 31.2%

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

      3. unpow2 31.2%

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

      4. associate-*r* 31.2%

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

      5. unpow2 31.2%

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

      6. times-frac 38.7%

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

   9. Simplified 38.7%

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

   if 5.5000000000000003e-167 < y < 4.0000000000000003e-18

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-*l/ 95.5%

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

      4. +-commutative 95.5%

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

      5. +-commutative 95.5%

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

      6. fma-def 95.5%

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

   3. Simplified 95.5%

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

      1. fma-def 95.5%

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

      2. +-commutative 95.5%

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

   5. Applied egg-rr 95.5%

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

   if 4.0000000000000003e-18 < y

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-*l/ 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 100.0%

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

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 5.5 \cdot 10^{-167}:\\ \;\;\;\;1 + \frac{y}{x} \cdot \frac{y \cdot -2}{x}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{-18}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{x - y}{x \cdot x + y \cdot y}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6: 83.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-137}:\\ \;\;\;\;1 + \frac{y}{x} \cdot \frac{y \cdot -2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y + x \cdot \left(\frac{x}{y} + -1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 2.5e-137)
   (+ 1.0 (* (/ y x) (/ (* y -2.0) x)))
   (/ (- x y) (+ y (* x (+ (/ x y) -1.0))))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 2.5e-137) {
		tmp = 1.0 + ((y / x) * ((y * -2.0) / x));
	} else {
		tmp = (x - y) / (y + (x * ((x / y) + -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 <= 2.5d-137) then
        tmp = 1.0d0 + ((y / x) * ((y * (-2.0d0)) / x))
    else
        tmp = (x - y) / (y + (x * ((x / y) + (-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 <= 2.5e-137) {
		tmp = 1.0 + ((y / x) * ((y * -2.0) / x));
	} else {
		tmp = (x - y) / (y + (x * ((x / y) + -1.0)));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 2.5e-137:
		tmp = 1.0 + ((y / x) * ((y * -2.0) / x))
	else:
		tmp = (x - y) / (y + (x * ((x / y) + -1.0)))
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 2.5e-137)
		tmp = Float64(1.0 + Float64(Float64(y / x) * Float64(Float64(y * -2.0) / x)));
	else
		tmp = Float64(Float64(x - y) / Float64(y + Float64(x * Float64(Float64(x / y) + -1.0))));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 2.5e-137)
		tmp = 1.0 + ((y / x) * ((y * -2.0) / x));
	else
		tmp = (x - y) / (y + (x * ((x / y) + -1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 2.5e-137], N[(1.0 + N[(N[(y / x), $MachinePrecision] * N[(N[(y * -2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] / N[(y + N[(x * N[(N[(x / y), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 2.5e-137

   1. Initial program 66.0%

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

      1. +-commutative 66.0%

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

      2. +-commutative 66.0%

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

      3. associate-*l/ 65.5%

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

      4. +-commutative 65.5%

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

      5. +-commutative 65.5%

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

      6. fma-def 65.5%

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

   3. Simplified 65.5%

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

   4. Taylor expanded in x around -inf 31.2%

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

      1. associate-+r+ 31.2%

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

      2. +-commutative 31.2%

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

      3. associate-+r+ 31.2%

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

      4. associate-+r+ 31.2%

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

      5. distribute-lft1-in 31.2%

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

      6. metadata-eval 31.2%

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

      7. mul0-lft 31.8%

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

      8. metadata-eval 31.8%

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

      9. unpow2 31.8%

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

      10. associate-*r/ 31.8%

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

   6. Simplified 31.8%

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

   7. Taylor expanded in y around 0 31.8%

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

      1. associate-*r/ 31.8%

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

      2. *-commutative 31.8%

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

      3. unpow2 31.8%

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

      4. associate-*r* 31.8%

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

      5. unpow2 31.8%

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

      6. times-frac 39.1%

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

   9. Simplified 39.1%

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

   if 2.5e-137 < y

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

      2. +-commutative 99.8%

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

      3. remove-double-neg 99.8%

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

      4. sub-neg 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-def 99.8%

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

      7. sub-neg 99.8%

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

      8. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 86.7%

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

      1. unpow2 86.7%

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

   6. Simplified 86.7%

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

   7. Taylor expanded in y around inf 87.0%

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

      1. unpow2 87.0%

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

      2. associate-*l/ 87.0%

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

      3. distribute-rgt-out 87.0%

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

   9. Simplified 87.0%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.5 \cdot 10^{-137}:\\ \;\;\;\;1 + \frac{y}{x} \cdot \frac{y \cdot -2}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{y + x \cdot \left(\frac{x}{y} + -1\right)}\\ \end{array} \]

Alternative 7: 83.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 3.45 \cdot 10^{-137}:\\ \;\;\;\;1 + \frac{y}{x} \cdot \frac{y \cdot -2}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 3.45e-137)
   (+ 1.0 (* (/ y x) (/ (* y -2.0) x)))
   (+ -1.0 (* (/ x y) (/ x y)))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 3.45e-137) {
		tmp = 1.0 + ((y / x) * ((y * -2.0) / x));
	} else {
		tmp = -1.0 + ((x / y) * (x / 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) :: tmp
    if (y <= 3.45d-137) then
        tmp = 1.0d0 + ((y / x) * ((y * (-2.0d0)) / x))
    else
        tmp = (-1.0d0) + ((x / y) * (x / y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 3.45e-137) {
		tmp = 1.0 + ((y / x) * ((y * -2.0) / x));
	} else {
		tmp = -1.0 + ((x / y) * (x / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 3.45e-137:
		tmp = 1.0 + ((y / x) * ((y * -2.0) / x))
	else:
		tmp = -1.0 + ((x / y) * (x / y))
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 3.45e-137)
		tmp = Float64(1.0 + Float64(Float64(y / x) * Float64(Float64(y * -2.0) / x)));
	else
		tmp = Float64(-1.0 + Float64(Float64(x / y) * Float64(x / y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.45e-137)
		tmp = 1.0 + ((y / x) * ((y * -2.0) / x));
	else
		tmp = -1.0 + ((x / y) * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 3.45e-137], N[(1.0 + N[(N[(y / x), $MachinePrecision] * N[(N[(y * -2.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 + N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.44999999999999988e-137

   1. Initial program 66.0%

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

      1. +-commutative 66.0%

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

      2. +-commutative 66.0%

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

      3. associate-*l/ 65.5%

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

      4. +-commutative 65.5%

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

      5. +-commutative 65.5%

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

      6. fma-def 65.5%

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

   3. Simplified 65.5%

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

   4. Taylor expanded in x around -inf 31.2%

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

      1. associate-+r+ 31.2%

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

      2. +-commutative 31.2%

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

      3. associate-+r+ 31.2%

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

      4. associate-+r+ 31.2%

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

      5. distribute-lft1-in 31.2%

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

      6. metadata-eval 31.2%

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

      7. mul0-lft 31.8%

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

      8. metadata-eval 31.8%

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

      9. unpow2 31.8%

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

      10. associate-*r/ 31.8%

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

   6. Simplified 31.8%

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

   7. Taylor expanded in y around 0 31.8%

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

      1. associate-*r/ 31.8%

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

      2. *-commutative 31.8%

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

      3. unpow2 31.8%

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

      4. associate-*r* 31.8%

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

      5. unpow2 31.8%

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

      6. times-frac 39.1%

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

   9. Simplified 39.1%

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

   if 3.44999999999999988e-137 < y

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

      2. +-commutative 99.8%

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

      3. remove-double-neg 99.8%

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

      4. sub-neg 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-def 99.8%

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

      7. sub-neg 99.8%

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

      8. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 86.7%

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

      1. unpow2 86.7%

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

   6. Simplified 86.7%

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

   7. Taylor expanded in x around 0 86.8%

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

      1. sub-neg 86.8%

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

      2. remove-double-neg 86.8%

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

      3. mul-1-neg 86.8%

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

      4. distribute-neg-in 86.8%

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

      5. +-commutative 86.8%

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

      6. distribute-neg-in 86.8%

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

      7. metadata-eval 86.8%

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

      8. mul-1-neg 86.8%

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

      9. remove-double-neg 86.8%

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

      10. unpow2 86.8%

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

      11. unpow2 86.8%

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

      12. times-frac 86.8%

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

   9. Simplified 86.8%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.45 \cdot 10^{-137}:\\ \;\;\;\;1 + \frac{y}{x} \cdot \frac{y \cdot -2}{x}\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 8: 82.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-137}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y)
 :precision binary64
 (if (<= y 1.55e-137) 1.0 (+ -1.0 (* (/ x y) (/ x y)))))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.55e-137) {
		tmp = 1.0;
	} else {
		tmp = -1.0 + ((x / y) * (x / 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) :: tmp
    if (y <= 1.55d-137) then
        tmp = 1.0d0
    else
        tmp = (-1.0d0) + ((x / y) * (x / y))
    end if
    code = tmp
end function

Java

y = Math.abs(y);
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.55e-137) {
		tmp = 1.0;
	} else {
		tmp = -1.0 + ((x / y) * (x / y));
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 1.55e-137:
		tmp = 1.0
	else:
		tmp = -1.0 + ((x / y) * (x / y))
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 1.55e-137)
		tmp = 1.0;
	else
		tmp = Float64(-1.0 + Float64(Float64(x / y) * Float64(x / y)));
	end
	return tmp
end

MATLAB

y = abs(y)
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.55e-137)
		tmp = 1.0;
	else
		tmp = -1.0 + ((x / y) * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: y should be positive before calling this function
code[x_, y_] := If[LessEqual[y, 1.55e-137], 1.0, N[(-1.0 + N[(N[(x / y), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.54999999999999989e-137

   1. Initial program 66.0%

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

      1. +-commutative 66.0%

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

      2. +-commutative 66.0%

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

      3. associate-*l/ 65.5%

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

      4. +-commutative 65.5%

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

      5. +-commutative 65.5%

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

      6. fma-def 65.5%

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

   3. Simplified 65.5%

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

   4. Taylor expanded in x around inf 36.7%

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

   if 1.54999999999999989e-137 < y

   1. Initial program 99.9%

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

      1. associate-/l* 99.8%

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

      2. +-commutative 99.8%

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

      3. remove-double-neg 99.8%

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

      4. sub-neg 99.8%

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

      5. +-commutative 99.8%

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

      6. fma-def 99.8%

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

      7. sub-neg 99.8%

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

      8. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around 0 86.7%

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

      1. unpow2 86.7%

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

   6. Simplified 86.7%

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

   7. Taylor expanded in x around 0 86.8%

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

      1. sub-neg 86.8%

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

      2. remove-double-neg 86.8%

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

      3. mul-1-neg 86.8%

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

      4. distribute-neg-in 86.8%

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

      5. +-commutative 86.8%

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

      6. distribute-neg-in 86.8%

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

      7. metadata-eval 86.8%

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

      8. mul-1-neg 86.8%

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

      9. remove-double-neg 86.8%

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

      10. unpow2 86.8%

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

      11. unpow2 86.8%

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

      12. times-frac 86.8%

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

   9. Simplified 86.8%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.55 \cdot 10^{-137}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1 + \frac{x}{y} \cdot \frac{x}{y}\\ \end{array} \]

Alternative 9: 82.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} y = |y|\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-137}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

NOTE: y should be positive before calling this function
(FPCore (x y) :precision binary64 (if (<= y 2e-137) 1.0 -1.0))

C

y = abs(y);
double code(double x, double y) {
	double tmp;
	if (y <= 2e-137) {
		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 <= 2d-137) 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 <= 2e-137) {
		tmp = 1.0;
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

y = abs(y)
def code(x, y):
	tmp = 0
	if y <= 2e-137:
		tmp = 1.0
	else:
		tmp = -1.0
	return tmp

Julia

y = abs(y)
function code(x, y)
	tmp = 0.0
	if (y <= 2e-137)
		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 <= 2e-137)
		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, 2e-137], 1.0, -1.0]

Derivation

1. Split input into 2 regimes

2. if y < 1.99999999999999996e-137

   1. Initial program 66.0%

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

      1. +-commutative 66.0%

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

      2. +-commutative 66.0%

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

      3. associate-*l/ 65.5%

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

      4. +-commutative 65.5%

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

      5. +-commutative 65.5%

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

      6. fma-def 65.5%

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

   3. Simplified 65.5%

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

   4. Taylor expanded in x around inf 36.7%

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

   if 1.99999999999999996e-137 < y

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. +-commutative 99.9%

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

      3. associate-*l/ 99.5%

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

      4. +-commutative 99.5%

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

      5. +-commutative 99.5%

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

      6. fma-def 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 85.9%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{-137}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 10: 66.3% 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 70.9%

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

   1. +-commutative 70.9%

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

   2. +-commutative 70.9%

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

   3. associate-*l/ 70.4%

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

   4. +-commutative 70.4%

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

   5. +-commutative 70.4%

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

   6. fma-def 70.4%

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

3. Simplified 70.4%

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

4. Taylor expanded in x around 0 66.1%

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

5. Final simplification 66.1%

   \[\leadsto -1 \]

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 2023293 
(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))))