Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A

Percentage Accurate: 69.1% → 99.8%

Time: 14.6s

Alternatives: 23

Speedup: 1.4×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))

C

double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
end function

Java

public static double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Python

def code(x, y):
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))

Julia

function code(x, y)
	return Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
end

Wolfram

code[x_, y_] := N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 69.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))

C

double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0d0))
end function

Java

public static double code(double x, double y) {
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
}

Python

def code(x, y):
	return (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0))

Julia

function code(x, y)
	return Float64(Float64(x * y) / Float64(Float64(Float64(x + y) * Float64(x + y)) * Float64(Float64(x + y) + 1.0)))
end

MATLAB

function tmp = code(x, y)
	tmp = (x * y) / (((x + y) * (x + y)) * ((x + y) + 1.0));
end

Wolfram

code[x_, y_] := N[(N[(x * y), $MachinePrecision] / N[(N[(N[(x + y), $MachinePrecision] * N[(x + y), $MachinePrecision]), $MachinePrecision] * N[(N[(x + y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{y \cdot \frac{\frac{x}{y + x}}{x + \left(y + 1\right)}}{y + x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (/ (* y (/ (/ x (+ y x)) (+ x (+ y 1.0)))) (+ y x)))

C

assert(x < y);
double code(double x, double y) {
	return (y * ((x / (y + x)) / (x + (y + 1.0)))) / (y + x);
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (y * ((x / (y + x)) / (x + (y + 1.0d0)))) / (y + x)
end function

Java

assert x < y;
public static double code(double x, double y) {
	return (y * ((x / (y + x)) / (x + (y + 1.0)))) / (y + x);
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return (y * ((x / (y + x)) / (x + (y + 1.0)))) / (y + x)

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(y * Float64(Float64(x / Float64(y + x)) / Float64(x + Float64(y + 1.0)))) / Float64(y + x))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = (y * ((x / (y + x)) / (x + (y + 1.0)))) / (y + x);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(y * N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.1%

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

   1. +-commutative 69.1%

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

   2. +-commutative 69.1%

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

   3. +-commutative 69.1%

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

   4. *-commutative 69.1%

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

   5. distribute-rgt1-in 52.7%

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

   6. fma-define 69.1%

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

   7. +-commutative 69.1%

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

   8. +-commutative 69.1%

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

   9. cube-unmult 69.2%

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

   10. +-commutative 69.2%

       \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

3. Simplified 69.2%

   \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 69.2%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

   2. fma-define 52.8%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 52.7%

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

   4. distribute-rgt1-in 69.1%

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

   5. *-commutative 69.1%

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

   6. associate-*l* 69.1%

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

   7. times-frac 94.0%

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

   8. associate-+r+ 94.0%

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

6. Applied egg-rr 94.0%

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

   1. associate-*l/ 94.0%

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

   2. frac-2neg 94.0%

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

   3. +-commutative 94.0%

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

   4. +-commutative 94.0%

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

8. Applied egg-rr 94.0%

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

   1. distribute-rgt-neg-in 94.0%

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

   2. associate-/r* 99.8%

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

10. Simplified 99.8%

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

11. Final simplification 99.8%

    \[\leadsto \frac{y \cdot \frac{\frac{x}{y + x}}{x + \left(y + 1\right)}}{y + x} \]
12. Add Preprocessing

Alternative 2: 89.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-274}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1 - \frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 580:\\ \;\;\;\;x \cdot \frac{y}{\left(x + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{+152}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.3e-274)
   (* (/ y (+ y x)) (/ (- 1.0 (/ y x)) (+ x 1.0)))
   (if (<= y 580.0)
     (* x (/ y (* (+ x 1.0) (* (+ y x) (+ y x)))))
     (if (<= y 1.66e+152)
       (/ x (* (+ y x) (+ x (+ y 1.0))))
       (/ (/ x (+ y 1.0)) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.3e-274) {
		tmp = (y / (y + x)) * ((1.0 - (y / x)) / (x + 1.0));
	} else if (y <= 580.0) {
		tmp = x * (y / ((x + 1.0) * ((y + x) * (y + x))));
	} else if (y <= 1.66e+152) {
		tmp = x / ((y + x) * (x + (y + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order 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 <= 4.3d-274) then
        tmp = (y / (y + x)) * ((1.0d0 - (y / x)) / (x + 1.0d0))
    else if (y <= 580.0d0) then
        tmp = x * (y / ((x + 1.0d0) * ((y + x) * (y + x))))
    else if (y <= 1.66d+152) then
        tmp = x / ((y + x) * (x + (y + 1.0d0)))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 4.3e-274) {
		tmp = (y / (y + x)) * ((1.0 - (y / x)) / (x + 1.0));
	} else if (y <= 580.0) {
		tmp = x * (y / ((x + 1.0) * ((y + x) * (y + x))));
	} else if (y <= 1.66e+152) {
		tmp = x / ((y + x) * (x + (y + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4.3e-274:
		tmp = (y / (y + x)) * ((1.0 - (y / x)) / (x + 1.0))
	elif y <= 580.0:
		tmp = x * (y / ((x + 1.0) * ((y + x) * (y + x))))
	elif y <= 1.66e+152:
		tmp = x / ((y + x) * (x + (y + 1.0)))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4.3e-274)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(Float64(1.0 - Float64(y / x)) / Float64(x + 1.0)));
	elseif (y <= 580.0)
		tmp = Float64(x * Float64(y / Float64(Float64(x + 1.0) * Float64(Float64(y + x) * Float64(y + x)))));
	elseif (y <= 1.66e+152)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.3e-274)
		tmp = (y / (y + x)) * ((1.0 - (y / x)) / (x + 1.0));
	elseif (y <= 580.0)
		tmp = x * (y / ((x + 1.0) * ((y + x) * (y + x))));
	elseif (y <= 1.66e+152)
		tmp = x / ((y + x) * (x + (y + 1.0)));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4.3e-274], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 - N[(y / x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 580.0], N[(x * N[(y / N[(N[(x + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.66e+152], N[(x / N[(N[(y + x), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 4.29999999999999989e-274

   1. Initial program 69.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 60.2%

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

      1. *-commutative 60.2%

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

      2. associate-*l* 60.2%

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

      3. times-frac 73.6%

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

      4. +-commutative 73.6%

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

      5. +-commutative 73.6%

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

   5. Applied egg-rr 73.6%

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

      1. associate-/r* 71.7%

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

   7. Simplified 71.7%

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

   8. Taylor expanded in x around inf 50.3%

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

      1. mul-1-neg 50.3%

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

      2. unsub-neg 50.3%

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

   10. Simplified 50.3%

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

   if 4.29999999999999989e-274 < y < 580

   1. Initial program 76.9%

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

      1. associate-/l* 90.2%

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

      2. associate-+l+ 90.2%

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

   3. Simplified 90.2%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 89.9%

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

      1. +-commutative 89.9%

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

   7. Simplified 89.9%

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

   if 580 < y < 1.65999999999999998e152

   1. Initial program 69.3%

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

      1. +-commutative 69.3%

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

      2. +-commutative 69.3%

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

      3. +-commutative 69.3%

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

      4. *-commutative 69.3%

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

      5. distribute-rgt1-in 63.6%

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

      6. fma-define 69.3%

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

      7. +-commutative 69.3%

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

      8. +-commutative 69.3%

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

      9. cube-unmult 69.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 69.4%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 69.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 69.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 63.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 63.6%

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

      4. distribute-rgt1-in 69.3%

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

      5. *-commutative 69.3%

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

      6. associate-*l* 69.3%

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

      7. times-frac 97.0%

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

      8. associate-+r+ 97.0%

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

   6. Applied egg-rr 97.0%

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

   7. Taylor expanded in y around inf 82.7%

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

   if 1.65999999999999998e152 < y

   1. Initial program 54.0%

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

      1. +-commutative 54.0%

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

      2. +-commutative 54.0%

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

      3. +-commutative 54.0%

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

      4. *-commutative 54.0%

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

      5. distribute-rgt1-in 54.0%

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

      6. fma-define 54.0%

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

      7. +-commutative 54.0%

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

      8. +-commutative 54.0%

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

      9. cube-unmult 54.0%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 54.0%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 54.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 54.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 54.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 54.0%

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

      4. distribute-rgt1-in 54.0%

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

      5. *-commutative 54.0%

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

      6. associate-*l* 54.0%

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

      7. times-frac 73.1%

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

      8. associate-+r+ 73.1%

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

   6. Applied egg-rr 73.1%

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

      1. associate-*l/ 73.1%

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

      2. frac-2neg 73.1%

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

      3. +-commutative 73.1%

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

      4. +-commutative 73.1%

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

   8. Applied egg-rr 73.1%

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

      1. distribute-rgt-neg-in 73.1%

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

      2. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around 0 83.7%

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

       1. associate-*r/ 83.7%

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

       2. neg-mul-1 83.7%

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

       3. +-commutative 83.7%

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

       4. distribute-neg-frac 83.7%

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

       5. distribute-neg-frac2 83.7%

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

       6. neg-mul-1 83.7%

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

       7. +-commutative 83.7%

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

       8. distribute-lft-in 83.7%

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

       9. metadata-eval 83.7%

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

       10. neg-mul-1 83.7%

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

       11. unsub-neg 83.7%

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

   13. Simplified 83.7%

       \[\leadsto \frac{\color{blue}{\frac{x}{-1 - y}}}{-\left(y + x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.3 \cdot 10^{-274}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{1 - \frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 580:\\ \;\;\;\;x \cdot \frac{y}{\left(x + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{+152}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-167}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 570:\\ \;\;\;\;x \cdot \frac{y}{\left(x + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{+152}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.9e-167)
   (/ (* (/ x (+ y x)) (/ y x)) (+ x 1.0))
   (if (<= y 570.0)
     (* x (/ y (* (+ x 1.0) (* (+ y x) (+ y x)))))
     (if (<= y 1.66e+152)
       (/ x (* (+ y x) (+ x (+ y 1.0))))
       (/ (/ x (+ y 1.0)) (+ y x))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.9e-167) {
		tmp = ((x / (y + x)) * (y / x)) / (x + 1.0);
	} else if (y <= 570.0) {
		tmp = x * (y / ((x + 1.0) * ((y + x) * (y + x))));
	} else if (y <= 1.66e+152) {
		tmp = x / ((y + x) * (x + (y + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order 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 <= 1.9d-167) then
        tmp = ((x / (y + x)) * (y / x)) / (x + 1.0d0)
    else if (y <= 570.0d0) then
        tmp = x * (y / ((x + 1.0d0) * ((y + x) * (y + x))))
    else if (y <= 1.66d+152) then
        tmp = x / ((y + x) * (x + (y + 1.0d0)))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.9e-167) {
		tmp = ((x / (y + x)) * (y / x)) / (x + 1.0);
	} else if (y <= 570.0) {
		tmp = x * (y / ((x + 1.0) * ((y + x) * (y + x))));
	} else if (y <= 1.66e+152) {
		tmp = x / ((y + x) * (x + (y + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.9e-167:
		tmp = ((x / (y + x)) * (y / x)) / (x + 1.0)
	elif y <= 570.0:
		tmp = x * (y / ((x + 1.0) * ((y + x) * (y + x))))
	elif y <= 1.66e+152:
		tmp = x / ((y + x) * (x + (y + 1.0)))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.9e-167)
		tmp = Float64(Float64(Float64(x / Float64(y + x)) * Float64(y / x)) / Float64(x + 1.0));
	elseif (y <= 570.0)
		tmp = Float64(x * Float64(y / Float64(Float64(x + 1.0) * Float64(Float64(y + x) * Float64(y + x)))));
	elseif (y <= 1.66e+152)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.9e-167)
		tmp = ((x / (y + x)) * (y / x)) / (x + 1.0);
	elseif (y <= 570.0)
		tmp = x * (y / ((x + 1.0) * ((y + x) * (y + x))));
	elseif (y <= 1.66e+152)
		tmp = x / ((y + x) * (x + (y + 1.0)));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.9e-167], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 570.0], N[(x * N[(y / N[(N[(x + 1.0), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.66e+152], N[(x / N[(N[(y + x), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < 1.89999999999999984e-167

   1. Initial program 68.2%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 59.9%

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

   4. Taylor expanded in x around inf 37.3%

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

      1. *-un-lft-identity 37.3%

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

      2. *-commutative 37.3%

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

      3. times-frac 40.9%

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

      4. times-frac 53.1%

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

      5. +-commutative 53.1%

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

   6. Applied egg-rr 53.1%

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

      1. associate-*l/ 53.2%

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

      2. *-lft-identity 53.2%

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

      3. *-commutative 53.2%

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

   8. Simplified 53.2%

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

   if 1.89999999999999984e-167 < y < 570

   1. Initial program 86.0%

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

      1. associate-/l* 97.0%

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

      2. associate-+l+ 97.0%

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

   3. Simplified 97.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 96.5%

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

      1. +-commutative 96.5%

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

   7. Simplified 96.5%

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

   if 570 < y < 1.65999999999999998e152

   1. Initial program 69.3%

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

      1. +-commutative 69.3%

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

      2. +-commutative 69.3%

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

      3. +-commutative 69.3%

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

      4. *-commutative 69.3%

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

      5. distribute-rgt1-in 63.6%

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

      6. fma-define 69.3%

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

      7. +-commutative 69.3%

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

      8. +-commutative 69.3%

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

      9. cube-unmult 69.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 69.4%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 69.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 69.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 63.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 63.6%

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

      4. distribute-rgt1-in 69.3%

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

      5. *-commutative 69.3%

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

      6. associate-*l* 69.3%

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

      7. times-frac 97.0%

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

      8. associate-+r+ 97.0%

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

   6. Applied egg-rr 97.0%

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

   7. Taylor expanded in y around inf 82.7%

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

   if 1.65999999999999998e152 < y

   1. Initial program 54.0%

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

      1. +-commutative 54.0%

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

      2. +-commutative 54.0%

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

      3. +-commutative 54.0%

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

      4. *-commutative 54.0%

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

      5. distribute-rgt1-in 54.0%

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

      6. fma-define 54.0%

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

      7. +-commutative 54.0%

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

      8. +-commutative 54.0%

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

      9. cube-unmult 54.0%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 54.0%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 54.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 54.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 54.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 54.0%

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

      4. distribute-rgt1-in 54.0%

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

      5. *-commutative 54.0%

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

      6. associate-*l* 54.0%

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

      7. times-frac 73.1%

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

      8. associate-+r+ 73.1%

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

   6. Applied egg-rr 73.1%

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

      1. associate-*l/ 73.1%

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

      2. frac-2neg 73.1%

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

      3. +-commutative 73.1%

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

      4. +-commutative 73.1%

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

   8. Applied egg-rr 73.1%

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

      1. distribute-rgt-neg-in 73.1%

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

      2. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around 0 83.7%

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

       1. associate-*r/ 83.7%

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

       2. neg-mul-1 83.7%

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

       3. +-commutative 83.7%

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

       4. distribute-neg-frac 83.7%

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

       5. distribute-neg-frac2 83.7%

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

       6. neg-mul-1 83.7%

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

       7. +-commutative 83.7%

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

       8. distribute-lft-in 83.7%

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

       9. metadata-eval 83.7%

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

       10. neg-mul-1 83.7%

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

       11. unsub-neg 83.7%

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

   13. Simplified 83.7%

       \[\leadsto \frac{\color{blue}{\frac{x}{-1 - y}}}{-\left(y + x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.9 \cdot 10^{-167}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 570:\\ \;\;\;\;x \cdot \frac{y}{\left(x + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{+152}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq 720:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{\frac{x}{y + x}}{x + 1}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x} \cdot \frac{y}{t\_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= y 720.0)
     (* (/ y (+ y x)) (/ (/ x (+ y x)) (+ x 1.0)))
     (if (<= y 6.8e+151)
       (/ x (* (+ y x) t_0))
       (* (/ (/ x y) (+ y x)) (/ y t_0))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= 720.0) {
		tmp = (y / (y + x)) * ((x / (y + x)) / (x + 1.0));
	} else if (y <= 6.8e+151) {
		tmp = x / ((y + x) * t_0);
	} else {
		tmp = ((x / y) / (y + x)) * (y / t_0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order 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 + 1.0d0)
    if (y <= 720.0d0) then
        tmp = (y / (y + x)) * ((x / (y + x)) / (x + 1.0d0))
    else if (y <= 6.8d+151) then
        tmp = x / ((y + x) * t_0)
    else
        tmp = ((x / y) / (y + x)) * (y / t_0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= 720.0) {
		tmp = (y / (y + x)) * ((x / (y + x)) / (x + 1.0));
	} else if (y <= 6.8e+151) {
		tmp = x / ((y + x) * t_0);
	} else {
		tmp = ((x / y) / (y + x)) * (y / t_0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if y <= 720.0:
		tmp = (y / (y + x)) * ((x / (y + x)) / (x + 1.0))
	elif y <= 6.8e+151:
		tmp = x / ((y + x) * t_0)
	else:
		tmp = ((x / y) / (y + x)) * (y / t_0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (y <= 720.0)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(Float64(x / Float64(y + x)) / Float64(x + 1.0)));
	elseif (y <= 6.8e+151)
		tmp = Float64(x / Float64(Float64(y + x) * t_0));
	else
		tmp = Float64(Float64(Float64(x / y) / Float64(y + x)) * Float64(y / t_0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (y <= 720.0)
		tmp = (y / (y + x)) * ((x / (y + x)) / (x + 1.0));
	elseif (y <= 6.8e+151)
		tmp = x / ((y + x) * t_0);
	else
		tmp = ((x / y) / (y + x)) * (y / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 720.0], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.8e+151], N[(x / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x / y), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 720

   1. Initial program 71.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 64.6%

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

      1. *-commutative 64.6%

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

      2. associate-*l* 64.6%

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

      3. times-frac 80.5%

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

      4. +-commutative 80.5%

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

      5. +-commutative 80.5%

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

   5. Applied egg-rr 80.5%

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

      1. associate-/r* 79.1%

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

   7. Simplified 79.1%

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

   if 720 < y < 6.7999999999999999e151

   1. Initial program 69.3%

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

      1. +-commutative 69.3%

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

      2. +-commutative 69.3%

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

      3. +-commutative 69.3%

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

      4. *-commutative 69.3%

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

      5. distribute-rgt1-in 63.6%

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

      6. fma-define 69.3%

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

      7. +-commutative 69.3%

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

      8. +-commutative 69.3%

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

      9. cube-unmult 69.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 69.4%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 69.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 69.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 63.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 63.6%

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

      4. distribute-rgt1-in 69.3%

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

      5. *-commutative 69.3%

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

      6. associate-*l* 69.3%

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

      7. times-frac 97.0%

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

      8. associate-+r+ 97.0%

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

   6. Applied egg-rr 97.0%

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

   7. Taylor expanded in y around inf 82.7%

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

   if 6.7999999999999999e151 < y

   1. Initial program 54.0%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 54.0%

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

      1. times-frac 73.1%

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

      2. associate-+r+ 73.1%

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

      3. *-commutative 73.1%

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

      4. +-commutative 73.1%

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

   5. Applied egg-rr 73.1%

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

      1. associate-/r* 84.3%

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

   7. Simplified 84.3%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 720:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{\frac{x}{y + x}}{x + 1}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + x} \cdot \frac{y}{x + \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -920000000:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -2.384 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-127}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -920000000.0)
   (* (/ y x) (/ 1.0 x))
   (if (<= x -2.384e-50)
     (/ x (* y y))
     (if (<= x 6e-127) (/ x y) (* (/ x y) (/ 1.0 y))))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -920000000.0) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -2.384e-50) {
		tmp = x / (y * y);
	} else if (x <= 6e-127) {
		tmp = x / y;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-920000000.0d0)) then
        tmp = (y / x) * (1.0d0 / x)
    else if (x <= (-2.384d-50)) then
        tmp = x / (y * y)
    else if (x <= 6d-127) then
        tmp = x / y
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -920000000.0) {
		tmp = (y / x) * (1.0 / x);
	} else if (x <= -2.384e-50) {
		tmp = x / (y * y);
	} else if (x <= 6e-127) {
		tmp = x / y;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -920000000.0:
		tmp = (y / x) * (1.0 / x)
	elif x <= -2.384e-50:
		tmp = x / (y * y)
	elif x <= 6e-127:
		tmp = x / y
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -920000000.0)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	elseif (x <= -2.384e-50)
		tmp = Float64(x / Float64(y * y));
	elseif (x <= 6e-127)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -920000000.0)
		tmp = (y / x) * (1.0 / x);
	elseif (x <= -2.384e-50)
		tmp = x / (y * y);
	elseif (x <= 6e-127)
		tmp = x / y;
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -920000000.0], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -2.384e-50], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6e-127], N[(x / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -9.2e8

   1. Initial program 57.8%

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

      1. +-commutative 57.8%

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

      2. +-commutative 57.8%

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

      3. +-commutative 57.8%

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

      4. *-commutative 57.8%

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

      5. distribute-rgt1-in 34.0%

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

      6. fma-define 57.8%

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

      7. +-commutative 57.8%

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

      8. +-commutative 57.8%

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

      9. cube-unmult 57.9%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 57.9%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 57.9%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 57.9%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 34.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 34.0%

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

      4. distribute-rgt1-in 57.8%

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

      5. *-commutative 57.8%

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

      6. associate-*l* 57.8%

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

      7. times-frac 85.2%

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

      8. associate-+r+ 85.2%

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

   6. Applied egg-rr 85.2%

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

   7. Taylor expanded in x around inf 73.4%

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

   8. Taylor expanded in y around 0 73.0%

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

   if -9.2e8 < x < -2.38400000000000016e-50

   1. Initial program 78.5%

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

      1. associate-/l* 78.3%

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

      2. associate-+l+ 78.3%

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

   3. Simplified 78.3%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 91.0%

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

   6. Taylor expanded in y around inf 90.9%

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

   if -2.38400000000000016e-50 < x < 6.00000000000000017e-127

   1. Initial program 73.4%

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

      1. associate-/l* 87.5%

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

      2. associate-+l+ 87.5%

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

   3. Simplified 87.5%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 81.8%

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

   6. Taylor expanded in y around 0 65.3%

      \[\leadsto \color{blue}{\frac{x}{y}} \]

   if 6.00000000000000017e-127 < x

   1. Initial program 71.8%

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

      1. associate-/l* 85.4%

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

      2. associate-+l+ 85.4%

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

   3. Simplified 85.4%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 45.7%

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

      1. *-un-lft-identity 45.7%

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

      2. +-commutative 45.7%

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

      3. times-frac 45.4%

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

   7. Applied egg-rr 45.4%

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

   8. Taylor expanded in y around inf 42.1%

      \[\leadsto \frac{1}{y} \cdot \color{blue}{\frac{x}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -920000000:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq -2.384 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-127}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq 2.1 \cdot 10^{+140}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0 \cdot \frac{y + x}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= y 2.1e+140)
     (* (/ y (+ y x)) (/ x (* (+ y x) t_0)))
     (/ (/ x y) (* t_0 (/ (+ y x) y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= 2.1e+140) {
		tmp = (y / (y + x)) * (x / ((y + x) * t_0));
	} else {
		tmp = (x / y) / (t_0 * ((y + x) / y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order 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 + 1.0d0)
    if (y <= 2.1d+140) then
        tmp = (y / (y + x)) * (x / ((y + x) * t_0))
    else
        tmp = (x / y) / (t_0 * ((y + x) / y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= 2.1e+140) {
		tmp = (y / (y + x)) * (x / ((y + x) * t_0));
	} else {
		tmp = (x / y) / (t_0 * ((y + x) / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if y <= 2.1e+140:
		tmp = (y / (y + x)) * (x / ((y + x) * t_0))
	else:
		tmp = (x / y) / (t_0 * ((y + x) / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (y <= 2.1e+140)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(x / Float64(Float64(y + x) * t_0)));
	else
		tmp = Float64(Float64(x / y) / Float64(t_0 * Float64(Float64(y + x) / y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (y <= 2.1e+140)
		tmp = (y / (y + x)) * (x / ((y + x) * t_0));
	else
		tmp = (x / y) / (t_0 * ((y + x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2.1e+140], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(x / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(t$95$0 * N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.1000000000000002e140

   1. Initial program 71.3%

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

      1. +-commutative 71.3%

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

      2. +-commutative 71.3%

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

      3. +-commutative 71.3%

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

      4. *-commutative 71.3%

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

      5. distribute-rgt1-in 52.6%

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

      6. fma-define 71.3%

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

      7. +-commutative 71.3%

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

      8. +-commutative 71.3%

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

      9. cube-unmult 71.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 71.4%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 71.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 71.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 52.6%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 52.6%

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

      4. distribute-rgt1-in 71.3%

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

      5. *-commutative 71.3%

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

      6. associate-*l* 71.3%

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

      7. times-frac 96.8%

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

      8. associate-+r+ 96.8%

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

   6. Applied egg-rr 96.8%

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

   if 2.1000000000000002e140 < y

   1. Initial program 53.8%

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

      1. +-commutative 53.8%

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

      2. +-commutative 53.8%

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

      3. +-commutative 53.8%

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

      4. *-commutative 53.8%

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

      5. distribute-rgt1-in 53.8%

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

      6. fma-define 53.8%

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

      7. +-commutative 53.8%

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

      8. +-commutative 53.8%

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

      9. cube-unmult 53.8%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 53.8%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 53.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 53.8%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 53.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 53.8%

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

      4. distribute-rgt1-in 53.8%

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

      5. *-commutative 53.8%

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

      6. associate-*l* 53.8%

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

      7. times-frac 74.7%

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

      8. associate-+r+ 74.7%

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

   6. Applied egg-rr 74.7%

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

      1. clear-num 74.7%

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

      2. associate-+r+ 74.7%

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

      3. associate-/r* 99.9%

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

      4. frac-times 99.9%

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

      5. metadata-eval 99.9%

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

      6. times-frac 99.9%

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

      7. *-un-lft-identity 99.9%

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

      8. *-un-lft-identity 99.9%

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

      9. +-commutative 99.9%

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

      10. +-commutative 99.9%

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

      11. associate-+r+ 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in y around inf 90.6%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.1 \cdot 10^{+140}:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(x + \left(y + 1\right)\right) \cdot \frac{y + x}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} t_0 := x + \left(y + 1\right)\\ \mathbf{if}\;y \leq 2 \cdot 10^{+141}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{t\_0 \cdot \frac{y + x}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (let* ((t_0 (+ x (+ y 1.0))))
   (if (<= y 2e+141)
     (* (/ x (+ y x)) (/ y (* (+ y x) t_0)))
     (/ (/ x y) (* t_0 (/ (+ y x) y))))))

C

assert(x < y);
double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= 2e+141) {
		tmp = (x / (y + x)) * (y / ((y + x) * t_0));
	} else {
		tmp = (x / y) / (t_0 * ((y + x) / y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order 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 + 1.0d0)
    if (y <= 2d+141) then
        tmp = (x / (y + x)) * (y / ((y + x) * t_0))
    else
        tmp = (x / y) / (t_0 * ((y + x) / y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double t_0 = x + (y + 1.0);
	double tmp;
	if (y <= 2e+141) {
		tmp = (x / (y + x)) * (y / ((y + x) * t_0));
	} else {
		tmp = (x / y) / (t_0 * ((y + x) / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	t_0 = x + (y + 1.0)
	tmp = 0
	if y <= 2e+141:
		tmp = (x / (y + x)) * (y / ((y + x) * t_0))
	else:
		tmp = (x / y) / (t_0 * ((y + x) / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	t_0 = Float64(x + Float64(y + 1.0))
	tmp = 0.0
	if (y <= 2e+141)
		tmp = Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(Float64(y + x) * t_0)));
	else
		tmp = Float64(Float64(x / y) / Float64(t_0 * Float64(Float64(y + x) / y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	t_0 = x + (y + 1.0);
	tmp = 0.0;
	if (y <= 2e+141)
		tmp = (x / (y + x)) * (y / ((y + x) * t_0));
	else
		tmp = (x / y) / (t_0 * ((y + x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := Block[{t$95$0 = N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2e+141], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(N[(y + x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(t$95$0 * N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.00000000000000003e141

   1. Initial program 71.3%

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

      1. +-commutative 71.3%

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

      2. +-commutative 71.3%

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

      3. +-commutative 71.3%

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

      4. *-commutative 71.3%

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

      5. distribute-rgt1-in 52.6%

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

      6. fma-define 71.3%

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

      7. +-commutative 71.3%

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

      8. +-commutative 71.3%

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

      9. cube-unmult 71.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 71.4%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 71.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. fma-define 52.6%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      2. cube-mult 52.6%

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

      3. distribute-rgt1-in 71.3%

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

      4. *-commutative 71.3%

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

      5. associate-*l* 71.3%

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

      6. times-frac 96.7%

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

      7. associate-+r+ 96.7%

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

   6. Applied egg-rr 96.7%

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

   if 2.00000000000000003e141 < y

   1. Initial program 53.8%

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

      1. +-commutative 53.8%

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

      2. +-commutative 53.8%

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

      3. +-commutative 53.8%

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

      4. *-commutative 53.8%

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

      5. distribute-rgt1-in 53.8%

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

      6. fma-define 53.8%

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

      7. +-commutative 53.8%

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

      8. +-commutative 53.8%

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

      9. cube-unmult 53.8%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 53.8%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 53.8%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 53.8%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 53.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 53.8%

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

      4. distribute-rgt1-in 53.8%

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

      5. *-commutative 53.8%

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

      6. associate-*l* 53.8%

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

      7. times-frac 74.7%

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

      8. associate-+r+ 74.7%

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

   6. Applied egg-rr 74.7%

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

      1. clear-num 74.7%

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

      2. associate-+r+ 74.7%

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

      3. associate-/r* 99.9%

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

      4. frac-times 99.9%

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

      5. metadata-eval 99.9%

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

      6. times-frac 99.9%

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

      7. *-un-lft-identity 99.9%

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

      8. *-un-lft-identity 99.9%

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

      9. +-commutative 99.9%

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

      10. +-commutative 99.9%

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

      11. associate-+r+ 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in y around inf 90.6%

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

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2 \cdot 10^{+141}:\\ \;\;\;\;\frac{x}{y + x} \cdot \frac{y}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(x + \left(y + 1\right)\right) \cdot \frac{y + x}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 600:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{\frac{x}{y + x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{+152}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 600.0)
   (* (/ y (+ y x)) (/ (/ x (+ y x)) (+ x 1.0)))
   (if (<= y 1.66e+152)
     (/ x (* (+ y x) (+ x (+ y 1.0))))
     (/ (/ x (+ y 1.0)) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 600.0) {
		tmp = (y / (y + x)) * ((x / (y + x)) / (x + 1.0));
	} else if (y <= 1.66e+152) {
		tmp = x / ((y + x) * (x + (y + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order 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 <= 600.0d0) then
        tmp = (y / (y + x)) * ((x / (y + x)) / (x + 1.0d0))
    else if (y <= 1.66d+152) then
        tmp = x / ((y + x) * (x + (y + 1.0d0)))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 600.0) {
		tmp = (y / (y + x)) * ((x / (y + x)) / (x + 1.0));
	} else if (y <= 1.66e+152) {
		tmp = x / ((y + x) * (x + (y + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 600.0:
		tmp = (y / (y + x)) * ((x / (y + x)) / (x + 1.0))
	elif y <= 1.66e+152:
		tmp = x / ((y + x) * (x + (y + 1.0)))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 600.0)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(Float64(x / Float64(y + x)) / Float64(x + 1.0)));
	elseif (y <= 1.66e+152)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 600.0)
		tmp = (y / (y + x)) * ((x / (y + x)) / (x + 1.0));
	elseif (y <= 1.66e+152)
		tmp = x / ((y + x) * (x + (y + 1.0)));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 600.0], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.66e+152], N[(x / N[(N[(y + x), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 600

   1. Initial program 71.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 64.6%

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

      1. *-commutative 64.6%

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

      2. associate-*l* 64.6%

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

      3. times-frac 80.5%

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

      4. +-commutative 80.5%

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

      5. +-commutative 80.5%

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

   5. Applied egg-rr 80.5%

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

      1. associate-/r* 79.1%

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

   7. Simplified 79.1%

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

   if 600 < y < 1.65999999999999998e152

   1. Initial program 69.3%

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

      1. +-commutative 69.3%

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

      2. +-commutative 69.3%

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

      3. +-commutative 69.3%

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

      4. *-commutative 69.3%

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

      5. distribute-rgt1-in 63.6%

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

      6. fma-define 69.3%

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

      7. +-commutative 69.3%

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

      8. +-commutative 69.3%

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

      9. cube-unmult 69.4%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 69.4%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 69.4%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 69.4%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 63.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 63.6%

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

      4. distribute-rgt1-in 69.3%

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

      5. *-commutative 69.3%

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

      6. associate-*l* 69.3%

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

      7. times-frac 97.0%

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

      8. associate-+r+ 97.0%

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

   6. Applied egg-rr 97.0%

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

   7. Taylor expanded in y around inf 82.7%

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

   if 1.65999999999999998e152 < y

   1. Initial program 54.0%

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

      1. +-commutative 54.0%

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

      2. +-commutative 54.0%

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

      3. +-commutative 54.0%

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

      4. *-commutative 54.0%

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

      5. distribute-rgt1-in 54.0%

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

      6. fma-define 54.0%

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

      7. +-commutative 54.0%

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

      8. +-commutative 54.0%

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

      9. cube-unmult 54.0%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 54.0%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 54.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 54.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 54.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 54.0%

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

      4. distribute-rgt1-in 54.0%

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

      5. *-commutative 54.0%

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

      6. associate-*l* 54.0%

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

      7. times-frac 73.1%

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

      8. associate-+r+ 73.1%

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

   6. Applied egg-rr 73.1%

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

      1. associate-*l/ 73.1%

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

      2. frac-2neg 73.1%

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

      3. +-commutative 73.1%

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

      4. +-commutative 73.1%

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

   8. Applied egg-rr 73.1%

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

      1. distribute-rgt-neg-in 73.1%

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

      2. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around 0 83.7%

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

       1. associate-*r/ 83.7%

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

       2. neg-mul-1 83.7%

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

       3. +-commutative 83.7%

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

       4. distribute-neg-frac 83.7%

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

       5. distribute-neg-frac2 83.7%

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

       6. neg-mul-1 83.7%

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

       7. +-commutative 83.7%

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

       8. distribute-lft-in 83.7%

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

       9. metadata-eval 83.7%

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

       10. neg-mul-1 83.7%

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

       11. unsub-neg 83.7%

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

   13. Simplified 83.7%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 600:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{\frac{x}{y + x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{+152}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 86.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-178}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.4e-178)
   (/ (* (/ x (+ y x)) (/ y x)) (+ x 1.0))
   (if (<= y 6.8e+151)
     (/ x (* (+ y x) (+ x (+ y 1.0))))
     (/ (/ x (+ y 1.0)) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.4e-178) {
		tmp = ((x / (y + x)) * (y / x)) / (x + 1.0);
	} else if (y <= 6.8e+151) {
		tmp = x / ((y + x) * (x + (y + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order 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 <= 4.4d-178) then
        tmp = ((x / (y + x)) * (y / x)) / (x + 1.0d0)
    else if (y <= 6.8d+151) then
        tmp = x / ((y + x) * (x + (y + 1.0d0)))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 4.4e-178) {
		tmp = ((x / (y + x)) * (y / x)) / (x + 1.0);
	} else if (y <= 6.8e+151) {
		tmp = x / ((y + x) * (x + (y + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4.4e-178:
		tmp = ((x / (y + x)) * (y / x)) / (x + 1.0)
	elif y <= 6.8e+151:
		tmp = x / ((y + x) * (x + (y + 1.0)))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4.4e-178)
		tmp = Float64(Float64(Float64(x / Float64(y + x)) * Float64(y / x)) / Float64(x + 1.0));
	elseif (y <= 6.8e+151)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.4e-178)
		tmp = ((x / (y + x)) * (y / x)) / (x + 1.0);
	elseif (y <= 6.8e+151)
		tmp = x / ((y + x) * (x + (y + 1.0)));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4.4e-178], N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.8e+151], N[(x / N[(N[(y + x), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.4000000000000002e-178

   1. Initial program 69.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 60.6%

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

   4. Taylor expanded in x around inf 37.8%

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

      1. *-un-lft-identity 37.8%

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

      2. *-commutative 37.8%

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

      3. times-frac 41.4%

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

      4. times-frac 53.8%

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

      5. +-commutative 53.8%

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

   6. Applied egg-rr 53.8%

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

      1. associate-*l/ 53.8%

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

      2. *-lft-identity 53.8%

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

      3. *-commutative 53.8%

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

   8. Simplified 53.8%

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

   if 4.4000000000000002e-178 < y < 6.7999999999999999e151

   1. Initial program 75.5%

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

      1. +-commutative 75.5%

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

      2. +-commutative 75.5%

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

      3. +-commutative 75.5%

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

      4. *-commutative 75.5%

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

      5. distribute-rgt1-in 71.3%

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

      6. fma-define 75.5%

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

      7. +-commutative 75.5%

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

      8. +-commutative 75.5%

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

      9. cube-unmult 75.5%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 75.5%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 75.5%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 71.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 71.3%

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

      4. distribute-rgt1-in 75.5%

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

      5. *-commutative 75.5%

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

      6. associate-*l* 75.5%

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

      7. times-frac 98.4%

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

      8. associate-+r+ 98.4%

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

   6. Applied egg-rr 98.4%

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

   7. Taylor expanded in y around inf 69.9%

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

   if 6.7999999999999999e151 < y

   1. Initial program 54.0%

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

      1. +-commutative 54.0%

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

      2. +-commutative 54.0%

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

      3. +-commutative 54.0%

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

      4. *-commutative 54.0%

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

      5. distribute-rgt1-in 54.0%

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

      6. fma-define 54.0%

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

      7. +-commutative 54.0%

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

      8. +-commutative 54.0%

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

      9. cube-unmult 54.0%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 54.0%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 54.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 54.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 54.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 54.0%

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

      4. distribute-rgt1-in 54.0%

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

      5. *-commutative 54.0%

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

      6. associate-*l* 54.0%

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

      7. times-frac 73.1%

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

      8. associate-+r+ 73.1%

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

   6. Applied egg-rr 73.1%

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

      1. associate-*l/ 73.1%

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

      2. frac-2neg 73.1%

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

      3. +-commutative 73.1%

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

      4. +-commutative 73.1%

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

   8. Applied egg-rr 73.1%

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

      1. distribute-rgt-neg-in 73.1%

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

      2. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around 0 83.7%

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

       1. associate-*r/ 83.7%

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

       2. neg-mul-1 83.7%

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

       3. +-commutative 83.7%

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

       4. distribute-neg-frac 83.7%

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

       5. distribute-neg-frac2 83.7%

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

       6. neg-mul-1 83.7%

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

       7. +-commutative 83.7%

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

       8. distribute-lft-in 83.7%

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

       9. metadata-eval 83.7%

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

       10. neg-mul-1 83.7%

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

       11. unsub-neg 83.7%

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

   13. Simplified 83.7%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-178}:\\ \;\;\;\;\frac{\frac{x}{y + x} \cdot \frac{y}{x}}{x + 1}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 86.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-178}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{+152}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 4.4e-178)
   (/ (/ y (+ y x)) (+ x 1.0))
   (if (<= y 1.66e+152)
     (/ x (* (+ y x) (+ x (+ y 1.0))))
     (/ (/ x (+ y 1.0)) (+ y x)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 4.4e-178) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else if (y <= 1.66e+152) {
		tmp = x / ((y + x) * (x + (y + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order 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 <= 4.4d-178) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else if (y <= 1.66d+152) then
        tmp = x / ((y + x) * (x + (y + 1.0d0)))
    else
        tmp = (x / (y + 1.0d0)) / (y + x)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 4.4e-178) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else if (y <= 1.66e+152) {
		tmp = x / ((y + x) * (x + (y + 1.0)));
	} else {
		tmp = (x / (y + 1.0)) / (y + x);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 4.4e-178:
		tmp = (y / (y + x)) / (x + 1.0)
	elif y <= 1.66e+152:
		tmp = x / ((y + x) * (x + (y + 1.0)))
	else:
		tmp = (x / (y + 1.0)) / (y + x)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 4.4e-178)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	elseif (y <= 1.66e+152)
		tmp = Float64(x / Float64(Float64(y + x) * Float64(x + Float64(y + 1.0))));
	else
		tmp = Float64(Float64(x / Float64(y + 1.0)) / Float64(y + x));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.4e-178)
		tmp = (y / (y + x)) / (x + 1.0);
	elseif (y <= 1.66e+152)
		tmp = x / ((y + x) * (x + (y + 1.0)));
	else
		tmp = (x / (y + 1.0)) / (y + x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 4.4e-178], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.66e+152], N[(x / N[(N[(y + x), $MachinePrecision] * N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + 1.0), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 4.4000000000000002e-178

   1. Initial program 69.1%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 60.6%

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

   4. Taylor expanded in x around inf 37.8%

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

      1. associate-/l* 50.3%

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

      2. *-commutative 50.3%

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

      3. +-commutative 50.3%

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

   6. Applied egg-rr 50.3%

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

      1. associate-*r/ 37.8%

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

      2. *-commutative 37.8%

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

      3. associate-*r/ 51.4%

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

      4. associate-/r* 55.0%

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

      5. associate-/r* 55.5%

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

      6. *-inverses 55.5%

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

   8. Simplified 55.5%

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

      1. associate-*r/ 54.1%

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

      2. +-commutative 54.1%

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

      3. div-inv 54.2%

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

   10. Applied egg-rr 54.2%

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

   if 4.4000000000000002e-178 < y < 1.65999999999999998e152

   1. Initial program 75.5%

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

      1. +-commutative 75.5%

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

      2. +-commutative 75.5%

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

      3. +-commutative 75.5%

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

      4. *-commutative 75.5%

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

      5. distribute-rgt1-in 71.3%

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

      6. fma-define 75.5%

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

      7. +-commutative 75.5%

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

      8. +-commutative 75.5%

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

      9. cube-unmult 75.5%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 75.5%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 75.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 75.5%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 71.4%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 71.3%

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

      4. distribute-rgt1-in 75.5%

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

      5. *-commutative 75.5%

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

      6. associate-*l* 75.5%

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

      7. times-frac 98.4%

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

      8. associate-+r+ 98.4%

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

   6. Applied egg-rr 98.4%

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

   7. Taylor expanded in y around inf 69.9%

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

   if 1.65999999999999998e152 < y

   1. Initial program 54.0%

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

      1. +-commutative 54.0%

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

      2. +-commutative 54.0%

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

      3. +-commutative 54.0%

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

      4. *-commutative 54.0%

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

      5. distribute-rgt1-in 54.0%

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

      6. fma-define 54.0%

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

      7. +-commutative 54.0%

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

      8. +-commutative 54.0%

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

      9. cube-unmult 54.0%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 54.0%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 54.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 54.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 54.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 54.0%

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

      4. distribute-rgt1-in 54.0%

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

      5. *-commutative 54.0%

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

      6. associate-*l* 54.0%

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

      7. times-frac 73.1%

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

      8. associate-+r+ 73.1%

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

   6. Applied egg-rr 73.1%

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

      1. associate-*l/ 73.1%

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

      2. frac-2neg 73.1%

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

      3. +-commutative 73.1%

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

      4. +-commutative 73.1%

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

   8. Applied egg-rr 73.1%

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

      1. distribute-rgt-neg-in 73.1%

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

      2. associate-/r* 99.8%

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

   10. Simplified 99.8%

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

   11. Taylor expanded in x around 0 83.7%

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

       1. associate-*r/ 83.7%

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

       2. neg-mul-1 83.7%

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

       3. +-commutative 83.7%

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

       4. distribute-neg-frac 83.7%

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

       5. distribute-neg-frac2 83.7%

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

       6. neg-mul-1 83.7%

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

       7. +-commutative 83.7%

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

       8. distribute-lft-in 83.7%

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

       9. metadata-eval 83.7%

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

       10. neg-mul-1 83.7%

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

       11. unsub-neg 83.7%

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

   13. Simplified 83.7%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.4 \cdot 10^{-178}:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{elif}\;y \leq 1.66 \cdot 10^{+152}:\\ \;\;\;\;\frac{x}{\left(y + x\right) \cdot \left(x + \left(y + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + 1}}{y + x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+183}:\\ \;\;\;\;\frac{1}{x \cdot \frac{y + x}{y}}\\ \mathbf{elif}\;x \leq -1950000000:\\ \;\;\;\;y \cdot \frac{\frac{1}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -2e+183)
   (/ 1.0 (* x (/ (+ y x) y)))
   (if (<= x -1950000000.0)
     (* y (/ (/ 1.0 (+ y x)) (+ x 1.0)))
     (/ (/ x (+ y x)) (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -2e+183) {
		tmp = 1.0 / (x * ((y + x) / y));
	} else if (x <= -1950000000.0) {
		tmp = y * ((1.0 / (y + x)) / (x + 1.0));
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-2d+183)) then
        tmp = 1.0d0 / (x * ((y + x) / y))
    else if (x <= (-1950000000.0d0)) then
        tmp = y * ((1.0d0 / (y + x)) / (x + 1.0d0))
    else
        tmp = (x / (y + x)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -2e+183) {
		tmp = 1.0 / (x * ((y + x) / y));
	} else if (x <= -1950000000.0) {
		tmp = y * ((1.0 / (y + x)) / (x + 1.0));
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -2e+183:
		tmp = 1.0 / (x * ((y + x) / y))
	elif x <= -1950000000.0:
		tmp = y * ((1.0 / (y + x)) / (x + 1.0))
	else:
		tmp = (x / (y + x)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -2e+183)
		tmp = Float64(1.0 / Float64(x * Float64(Float64(y + x) / y)));
	elseif (x <= -1950000000.0)
		tmp = Float64(y * Float64(Float64(1.0 / Float64(y + x)) / Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -2e+183)
		tmp = 1.0 / (x * ((y + x) / y));
	elseif (x <= -1950000000.0)
		tmp = y * ((1.0 / (y + x)) / (x + 1.0));
	else
		tmp = (x / (y + x)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -2e+183], N[(1.0 / N[(x * N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1950000000.0], N[(y * N[(N[(1.0 / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.99999999999999989e183

   1. Initial program 44.0%

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

      1. +-commutative 44.0%

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

      2. +-commutative 44.0%

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

      3. +-commutative 44.0%

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

      4. *-commutative 44.0%

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

      5. distribute-rgt1-in 0.0%

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

      6. fma-define 44.0%

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

      7. +-commutative 44.0%

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

      8. +-commutative 44.0%

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

      9. cube-unmult 44.0%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 44.0%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 44.0%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 44.0%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 0.0%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 0.0%

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

      4. distribute-rgt1-in 44.0%

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

      5. *-commutative 44.0%

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

      6. associate-*l* 44.0%

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

      7. times-frac 70.5%

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

      8. associate-+r+ 70.5%

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

   6. Applied egg-rr 70.5%

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

   7. Taylor expanded in x around inf 83.1%

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

      1. +-commutative 83.1%

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

      2. clear-num 83.1%

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

      3. clear-num 83.0%

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

      4. frac-times 83.2%

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

      5. metadata-eval 83.2%

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

      6. /-rgt-identity 83.2%

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

      7. +-commutative 83.2%

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

   9. Applied egg-rr 83.2%

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

   if -1.99999999999999989e183 < x < -1.95e9

   1. Initial program 66.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 62.1%

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

   4. Taylor expanded in x around inf 56.9%

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

      1. associate-/l* 60.5%

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

      2. *-commutative 60.5%

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

      3. +-commutative 60.5%

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

   6. Applied egg-rr 60.5%

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

      1. associate-*r/ 56.9%

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

      2. *-commutative 56.9%

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

      3. associate-*r/ 65.0%

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

      4. associate-/r* 75.1%

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

      5. associate-/r* 75.1%

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

      6. *-inverses 75.1%

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

   8. Simplified 75.1%

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

   if -1.95e9 < x

   1. Initial program 72.9%

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

      1. +-commutative 72.9%

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

      2. +-commutative 72.9%

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

      3. +-commutative 72.9%

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

      4. *-commutative 72.9%

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

      5. distribute-rgt1-in 58.8%

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

      6. fma-define 72.8%

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

      7. +-commutative 72.8%

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

      8. +-commutative 72.8%

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

      9. cube-unmult 72.9%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 72.9%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 72.9%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 72.9%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 58.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 58.8%

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

      4. distribute-rgt1-in 72.9%

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

      5. *-commutative 72.9%

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

      6. associate-*l* 72.8%

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

      7. times-frac 96.9%

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

      8. associate-+r+ 96.9%

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

   6. Applied egg-rr 96.9%

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

      1. clear-num 96.8%

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

      2. associate-+r+ 96.8%

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

      3. associate-/r* 99.8%

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

      4. frac-times 99.3%

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

      5. metadata-eval 99.3%

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

      6. times-frac 99.3%

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

      7. *-un-lft-identity 99.3%

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

      8. *-un-lft-identity 99.3%

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

      9. +-commutative 99.3%

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

      10. +-commutative 99.3%

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

      11. associate-+r+ 99.3%

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

   8. Applied egg-rr 99.3%

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

   9. Taylor expanded in x around 0 63.3%

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

       1. +-commutative 63.3%

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

   11. Simplified 63.3%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+183}:\\ \;\;\;\;\frac{1}{x \cdot \frac{y + x}{y}}\\ \mathbf{elif}\;x \leq -1950000000:\\ \;\;\;\;y \cdot \frac{\frac{1}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 95.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 580:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{\frac{x}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(x + \left(y + 1\right)\right) \cdot \frac{y + x}{y}}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 580.0)
   (* (/ y (+ y x)) (/ (/ x (+ y x)) (+ x 1.0)))
   (/ (/ x y) (* (+ x (+ y 1.0)) (/ (+ y x) y)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 580.0) {
		tmp = (y / (y + x)) * ((x / (y + x)) / (x + 1.0));
	} else {
		tmp = (x / y) / ((x + (y + 1.0)) * ((y + x) / y));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order 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 <= 580.0d0) then
        tmp = (y / (y + x)) * ((x / (y + x)) / (x + 1.0d0))
    else
        tmp = (x / y) / ((x + (y + 1.0d0)) * ((y + x) / y))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 580.0) {
		tmp = (y / (y + x)) * ((x / (y + x)) / (x + 1.0));
	} else {
		tmp = (x / y) / ((x + (y + 1.0)) * ((y + x) / y));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 580.0:
		tmp = (y / (y + x)) * ((x / (y + x)) / (x + 1.0))
	else:
		tmp = (x / y) / ((x + (y + 1.0)) * ((y + x) / y))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 580.0)
		tmp = Float64(Float64(y / Float64(y + x)) * Float64(Float64(x / Float64(y + x)) / Float64(x + 1.0)));
	else
		tmp = Float64(Float64(x / y) / Float64(Float64(x + Float64(y + 1.0)) * Float64(Float64(y + x) / y)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 580.0)
		tmp = (y / (y + x)) * ((x / (y + x)) / (x + 1.0));
	else
		tmp = (x / y) / ((x + (y + 1.0)) * ((y + x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 580.0], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(N[(x + N[(y + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[(y + x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 580

   1. Initial program 71.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 64.6%

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

      1. *-commutative 64.6%

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

      2. associate-*l* 64.6%

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

      3. times-frac 80.5%

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

      4. +-commutative 80.5%

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

      5. +-commutative 80.5%

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

   5. Applied egg-rr 80.5%

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

      1. associate-/r* 79.1%

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

   7. Simplified 79.1%

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

   if 580 < y

   1. Initial program 62.3%

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

      1. +-commutative 62.3%

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

      2. +-commutative 62.3%

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

      3. +-commutative 62.3%

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

      4. *-commutative 62.3%

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

      5. distribute-rgt1-in 59.2%

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

      6. fma-define 62.3%

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

      7. +-commutative 62.3%

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

      8. +-commutative 62.3%

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

      9. cube-unmult 62.3%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 62.3%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 62.3%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 62.3%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 59.2%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 59.2%

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

      4. distribute-rgt1-in 62.3%

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

      5. *-commutative 62.3%

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

      6. associate-*l* 62.3%

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

      7. times-frac 86.0%

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

      8. associate-+r+ 86.0%

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

   6. Applied egg-rr 86.0%

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

      1. clear-num 86.0%

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

      2. associate-+r+ 86.0%

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

      3. associate-/r* 99.7%

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

      4. frac-times 99.7%

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

      5. metadata-eval 99.7%

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

      6. times-frac 99.7%

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

      7. *-un-lft-identity 99.7%

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

      8. *-un-lft-identity 99.7%

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

      9. +-commutative 99.7%

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

      10. +-commutative 99.7%

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

      11. associate-+r+ 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in y around inf 86.0%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 580:\\ \;\;\;\;\frac{y}{y + x} \cdot \frac{\frac{x}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{\left(x + \left(y + 1\right)\right) \cdot \frac{y + x}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{\frac{x}{y + x} \cdot \frac{y}{y + x}}{\left(y + x\right) + 1} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (/ (* (/ x (+ y x)) (/ y (+ y x))) (+ (+ y x) 1.0)))

C

assert(x < y);
double code(double x, double y) {
	return ((x / (y + x)) * (y / (y + x))) / ((y + x) + 1.0);
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x / (y + x)) * (y / (y + x))) / ((y + x) + 1.0d0)
end function

Java

assert x < y;
public static double code(double x, double y) {
	return ((x / (y + x)) * (y / (y + x))) / ((y + x) + 1.0);
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return ((x / (y + x)) * (y / (y + x))) / ((y + x) + 1.0)

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(Float64(Float64(x / Float64(y + x)) * Float64(y / Float64(y + x))) / Float64(Float64(y + x) + 1.0))
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = ((x / (y + x)) * (y / (y + x))) / ((y + x) + 1.0);
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] * N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.1%

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

   1. +-commutative 69.1%

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

   2. +-commutative 69.1%

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

   3. +-commutative 69.1%

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

   4. *-commutative 69.1%

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

   5. distribute-rgt1-in 52.7%

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

   6. fma-define 69.1%

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

   7. +-commutative 69.1%

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

   8. +-commutative 69.1%

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

   9. cube-unmult 69.2%

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

   10. +-commutative 69.2%

       \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

3. Simplified 69.2%

   \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 69.2%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

   2. fma-define 52.8%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 52.7%

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

   4. distribute-rgt1-in 69.1%

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

   5. *-commutative 69.1%

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

   6. associate-*l* 69.1%

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

   7. times-frac 94.0%

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

   8. associate-+r+ 94.0%

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

6. Applied egg-rr 94.0%

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

   1. clear-num 94.0%

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

   2. associate-+r+ 94.0%

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

   3. associate-/r* 99.8%

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

   4. frac-times 99.4%

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

   5. metadata-eval 99.4%

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

   6. times-frac 99.4%

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

   7. *-un-lft-identity 99.4%

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

   8. *-un-lft-identity 99.4%

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

   9. +-commutative 99.4%

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

   10. +-commutative 99.4%

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

   11. associate-+r+ 99.4%

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

8. Applied egg-rr 99.4%

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

   1. add-exp-log 73.5%

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

   2. associate-/r* 73.5%

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

   3. log-div 35.0%

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

   4. div-inv 35.0%

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

   5. +-commutative 35.0%

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

   6. +-commutative 35.0%

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

   7. clear-num 35.0%

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

   8. +-commutative 35.0%

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

   9. +-commutative 35.0%

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

   10. associate-+l+ 35.0%

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

   11. log1p-undefine 35.0%

       \[\leadsto e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right) - \color{blue}{\mathsf{log1p}\left(y + x\right)}} \]

   12. +-commutative 35.0%

       \[\leadsto e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right) - \mathsf{log1p}\left(\color{blue}{x + y}\right)} \]

10. Applied egg-rr 35.0%

    \[\leadsto \color{blue}{e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right) - \mathsf{log1p}\left(x + y\right)}} \]
11. Step-by-step derivation

    1. exp-diff 35.1%

       \[\leadsto \color{blue}{\frac{e^{\log \left(\frac{x}{x + y} \cdot \frac{y}{x + y}\right)}}{e^{\mathsf{log1p}\left(x + y\right)}}} \]

    2. rem-exp-log 56.7%

       \[\leadsto \frac{\color{blue}{\frac{x}{x + y} \cdot \frac{y}{x + y}}}{e^{\mathsf{log1p}\left(x + y\right)}} \]

    3. *-commutative 56.7%

       \[\leadsto \frac{\color{blue}{\frac{y}{x + y} \cdot \frac{x}{x + y}}}{e^{\mathsf{log1p}\left(x + y\right)}} \]

    4. log1p-undefine 56.7%

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

    5. rem-exp-log 99.8%

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

12. Simplified 99.8%

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

13. Final simplification 99.8%

    \[\leadsto \frac{\frac{x}{y + x} \cdot \frac{y}{y + x}}{\left(y + x\right) + 1} \]
14. Add Preprocessing

Alternative 14: 80.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -122000000:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -122000000.0)
   (/ (/ y (+ y x)) (+ x 1.0))
   (/ (/ x (+ y x)) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -122000000.0) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-122000000.0d0)) then
        tmp = (y / (y + x)) / (x + 1.0d0)
    else
        tmp = (x / (y + x)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -122000000.0) {
		tmp = (y / (y + x)) / (x + 1.0);
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -122000000.0:
		tmp = (y / (y + x)) / (x + 1.0)
	else:
		tmp = (x / (y + x)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -122000000.0)
		tmp = Float64(Float64(y / Float64(y + x)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -122000000.0)
		tmp = (y / (y + x)) / (x + 1.0);
	else
		tmp = (x / (y + x)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -122000000.0], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.22e8

   1. Initial program 58.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 54.2%

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

   4. Taylor expanded in x around inf 51.1%

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

      1. associate-/l* 63.6%

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

      2. *-commutative 63.6%

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

      3. +-commutative 63.6%

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

   6. Applied egg-rr 63.6%

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

      1. associate-*r/ 51.1%

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

      2. *-commutative 51.1%

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

      3. associate-*r/ 66.2%

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

      4. associate-/r* 72.2%

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

      5. associate-/r* 72.2%

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

      6. *-inverses 72.2%

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

   8. Simplified 72.2%

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

      1. associate-*r/ 72.6%

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

      2. +-commutative 72.6%

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

      3. div-inv 72.6%

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

   10. Applied egg-rr 72.6%

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

   if -1.22e8 < x

   1. Initial program 72.7%

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

      1. +-commutative 72.7%

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

      2. +-commutative 72.7%

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

      3. +-commutative 72.7%

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

      4. *-commutative 72.7%

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

      5. distribute-rgt1-in 58.6%

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

      6. fma-define 72.7%

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

      7. +-commutative 72.7%

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

      8. +-commutative 72.7%

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

      9. cube-unmult 72.7%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 72.7%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 72.7%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 72.7%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 58.7%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 58.6%

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

      4. distribute-rgt1-in 72.7%

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

      5. *-commutative 72.7%

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

      6. associate-*l* 72.7%

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

      7. times-frac 96.9%

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

      8. associate-+r+ 96.9%

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

   6. Applied egg-rr 96.9%

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

      1. clear-num 96.8%

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

      2. associate-+r+ 96.8%

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

      3. associate-/r* 99.8%

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

      4. frac-times 99.2%

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

      5. metadata-eval 99.2%

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

      6. times-frac 99.2%

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

      7. *-un-lft-identity 99.2%

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

      8. *-un-lft-identity 99.2%

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

      9. +-commutative 99.2%

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

      10. +-commutative 99.2%

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

      11. associate-+r+ 99.2%

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

   8. Applied egg-rr 99.2%

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

   9. Taylor expanded in x around 0 63.1%

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

       1. +-commutative 63.1%

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

   11. Simplified 63.1%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -122000000:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 80.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3500000000:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3500000000.0) (/ (/ y (+ y x)) x) (/ (/ x (+ y x)) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3500000000.0) {
		tmp = (y / (y + x)) / x;
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3500000000.0d0)) then
        tmp = (y / (y + x)) / x
    else
        tmp = (x / (y + x)) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3500000000.0) {
		tmp = (y / (y + x)) / x;
	} else {
		tmp = (x / (y + x)) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3500000000.0:
		tmp = (y / (y + x)) / x
	else:
		tmp = (x / (y + x)) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3500000000.0)
		tmp = Float64(Float64(y / Float64(y + x)) / x);
	else
		tmp = Float64(Float64(x / Float64(y + x)) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3500000000.0)
		tmp = (y / (y + x)) / x;
	else
		tmp = (x / (y + x)) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3500000000.0], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5e9

   1. Initial program 57.8%

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

      1. +-commutative 57.8%

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

      2. +-commutative 57.8%

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

      3. +-commutative 57.8%

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

      4. *-commutative 57.8%

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

      5. distribute-rgt1-in 34.0%

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

      6. fma-define 57.8%

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

      7. +-commutative 57.8%

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

      8. +-commutative 57.8%

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

      9. cube-unmult 57.9%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 57.9%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 57.9%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 57.9%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 34.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 34.0%

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

      4. distribute-rgt1-in 57.8%

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

      5. *-commutative 57.8%

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

      6. associate-*l* 57.8%

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

      7. times-frac 85.2%

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

      8. associate-+r+ 85.2%

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

   6. Applied egg-rr 85.2%

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

   7. Taylor expanded in x around inf 73.4%

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

      1. un-div-inv 73.5%

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

   9. Applied egg-rr 73.5%

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

   if -3.5e9 < x

   1. Initial program 72.9%

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

      1. +-commutative 72.9%

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

      2. +-commutative 72.9%

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

      3. +-commutative 72.9%

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

      4. *-commutative 72.9%

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

      5. distribute-rgt1-in 58.8%

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

      6. fma-define 72.8%

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

      7. +-commutative 72.8%

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

      8. +-commutative 72.8%

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

      9. cube-unmult 72.9%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 72.9%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 72.9%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 72.9%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 58.9%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 58.8%

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

      4. distribute-rgt1-in 72.9%

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

      5. *-commutative 72.9%

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

      6. associate-*l* 72.8%

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

      7. times-frac 96.9%

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

      8. associate-+r+ 96.9%

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

   6. Applied egg-rr 96.9%

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

      1. clear-num 96.8%

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

      2. associate-+r+ 96.8%

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

      3. associate-/r* 99.8%

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

      4. frac-times 99.3%

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

      5. metadata-eval 99.3%

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

      6. times-frac 99.3%

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

      7. *-un-lft-identity 99.3%

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

      8. *-un-lft-identity 99.3%

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

      9. +-commutative 99.3%

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

      10. +-commutative 99.3%

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

      11. associate-+r+ 99.3%

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

   8. Applied egg-rr 99.3%

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

   9. Taylor expanded in x around 0 63.3%

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

       1. +-commutative 63.3%

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

   11. Simplified 63.3%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3500000000:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + x}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 80.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -720000000:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -720000000.0) (/ (/ y (+ y x)) x) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -720000000.0) {
		tmp = (y / (y + x)) / x;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-720000000.0d0)) then
        tmp = (y / (y + x)) / x
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -720000000.0) {
		tmp = (y / (y + x)) / x;
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -720000000.0:
		tmp = (y / (y + x)) / x
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -720000000.0)
		tmp = Float64(Float64(y / Float64(y + x)) / x);
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -720000000.0)
		tmp = (y / (y + x)) / x;
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -720000000.0], N[(N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -7.2e8

   1. Initial program 57.8%

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

      1. +-commutative 57.8%

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

      2. +-commutative 57.8%

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

      3. +-commutative 57.8%

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

      4. *-commutative 57.8%

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

      5. distribute-rgt1-in 34.0%

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

      6. fma-define 57.8%

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

      7. +-commutative 57.8%

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

      8. +-commutative 57.8%

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

      9. cube-unmult 57.9%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 57.9%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 57.9%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 57.9%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 34.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 34.0%

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

      4. distribute-rgt1-in 57.8%

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

      5. *-commutative 57.8%

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

      6. associate-*l* 57.8%

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

      7. times-frac 85.2%

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

      8. associate-+r+ 85.2%

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

   6. Applied egg-rr 85.2%

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

   7. Taylor expanded in x around inf 73.4%

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

      1. un-div-inv 73.5%

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

   9. Applied egg-rr 73.5%

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

   if -7.2e8 < x

   1. Initial program 72.9%

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

      1. associate-/l* 85.7%

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

      2. associate-+l+ 85.7%

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

   3. Simplified 85.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 63.2%

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

      1. associate-/r* 63.0%

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

      2. +-commutative 63.0%

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

   7. Simplified 63.0%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -720000000:\\ \;\;\;\;\frac{\frac{y}{y + x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -3100000000:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y}}{y + 1}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -3100000000.0) (* (/ y x) (/ 1.0 x)) (/ (/ x y) (+ y 1.0))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -3100000000.0) {
		tmp = (y / x) * (1.0 / x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-3100000000.0d0)) then
        tmp = (y / x) * (1.0d0 / x)
    else
        tmp = (x / y) / (y + 1.0d0)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -3100000000.0) {
		tmp = (y / x) * (1.0 / x);
	} else {
		tmp = (x / y) / (y + 1.0);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -3100000000.0:
		tmp = (y / x) * (1.0 / x)
	else:
		tmp = (x / y) / (y + 1.0)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -3100000000.0)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	else
		tmp = Float64(Float64(x / y) / Float64(y + 1.0));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -3100000000.0)
		tmp = (y / x) * (1.0 / x);
	else
		tmp = (x / y) / (y + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -3100000000.0], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] / N[(y + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1e9

   1. Initial program 57.8%

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

      1. +-commutative 57.8%

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

      2. +-commutative 57.8%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 57.8%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 57.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 34.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 57.8%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 57.8%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 57.8%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 57.9%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 57.9%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 57.9%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 57.9%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 34.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 34.0%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 57.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 57.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 57.8%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 85.2%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 85.2%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 85.2%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. Taylor expanded in x around inf 73.4%

      \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{x}} \]

   8. Taylor expanded in y around 0 73.0%

      \[\leadsto \color{blue}{\frac{y}{x}} \cdot \frac{1}{x} \]

   if -3.1e9 < x

   1. Initial program 72.9%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 85.7%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 85.7%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 85.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 63.2%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. associate-/r* 63.0%

         \[\leadsto \color{blue}{\frac{\frac{x}{y}}{1 + y}} \]

      2. +-commutative 63.0%

         \[\leadsto \frac{\frac{x}{y}}{\color{blue}{y + 1}} \]

   7. Simplified 63.0%

      \[\leadsto \color{blue}{\frac{\frac{x}{y}}{y + 1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 78.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -17000000:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= x -17000000.0) (* (/ y x) (/ 1.0 x)) (/ x (* y (+ y 1.0)))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -17000000.0) {
		tmp = (y / x) * (1.0 / x);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-17000000.0d0)) then
        tmp = (y / x) * (1.0d0 / x)
    else
        tmp = x / (y * (y + 1.0d0))
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -17000000.0) {
		tmp = (y / x) * (1.0 / x);
	} else {
		tmp = x / (y * (y + 1.0));
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -17000000.0:
		tmp = (y / x) * (1.0 / x)
	else:
		tmp = x / (y * (y + 1.0))
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -17000000.0)
		tmp = Float64(Float64(y / x) * Float64(1.0 / x));
	else
		tmp = Float64(x / Float64(y * Float64(y + 1.0)));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -17000000.0)
		tmp = (y / x) * (1.0 / x);
	else
		tmp = x / (y * (y + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -17000000.0], N[(N[(y / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], N[(x / N[(y * N[(y + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.7e7

   1. Initial program 58.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 58.5%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 58.5%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 58.5%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 58.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 35.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 58.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 58.5%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 58.5%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 58.5%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 58.5%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 58.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 58.5%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 35.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 35.0%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 58.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 58.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 58.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 85.4%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 85.4%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 85.4%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. Taylor expanded in x around inf 72.3%

      \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{x}} \]

   8. Taylor expanded in y around 0 72.0%

      \[\leadsto \color{blue}{\frac{y}{x}} \cdot \frac{1}{x} \]

   if -1.7e7 < x

   1. Initial program 72.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 85.7%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 85.7%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 85.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 63.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -17000000:\\ \;\;\;\;\frac{y}{x} \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot \left(y + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 48.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y)
 :precision binary64
 (if (<= y 1.0) (/ x y) (* (/ x y) (/ 1.0 y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order 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 <= 1.0d0) then
        tmp = x / y
    else
        tmp = (x / y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = (x / y) * (1.0 / y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.0:
		tmp = x / y
	else:
		tmp = (x / y) * (1.0 / y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.0)
		tmp = x / y;
	else
		tmp = (x / y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 71.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 84.1%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 84.1%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 84.1%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 47.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around 0 26.5%

      \[\leadsto \color{blue}{\frac{x}{y}} \]

   if 1 < y

   1. Initial program 62.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 75.8%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 75.8%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 75.8%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 70.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]
   6. Step-by-step derivation

      1. *-un-lft-identity 70.5%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{y \cdot \left(1 + y\right)} \]

      2. +-commutative 70.5%

         \[\leadsto \frac{1 \cdot x}{y \cdot \color{blue}{\left(y + 1\right)}} \]

      3. times-frac 75.2%

         \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y + 1}} \]

   7. Applied egg-rr 75.2%

      \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{x}{y + 1}} \]

   8. Taylor expanded in y around inf 73.4%

      \[\leadsto \frac{1}{y} \cdot \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 46.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq 1:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= y 1.0) (/ x y) (/ x (* y y))))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order 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 <= 1.0d0) then
        tmp = x / y
    else
        tmp = x / (y * y)
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (y <= 1.0) {
		tmp = x / y;
	} else {
		tmp = x / (y * y);
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if y <= 1.0:
		tmp = x / y
	else:
		tmp = x / (y * y)
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (y <= 1.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(x / Float64(y * y));
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.0)
		tmp = x / y;
	else
		tmp = x / (y * y);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[y, 1.0], N[(x / y), $MachinePrecision], N[(x / N[(y * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1

   1. Initial program 71.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 84.1%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 84.1%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 84.1%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 47.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around 0 26.5%

      \[\leadsto \color{blue}{\frac{x}{y}} \]

   if 1 < y

   1. Initial program 62.3%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 75.8%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 75.8%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 75.8%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 70.5%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around inf 68.7%

      \[\leadsto \frac{x}{y \cdot \color{blue}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 27.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \begin{array}{l} \mathbf{if}\;x \leq -215000:\\ \;\;\;\;\frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (if (<= x -215000.0) (/ 1.0 x) (/ x y)))

C

assert(x < y);
double code(double x, double y) {
	double tmp;
	if (x <= -215000.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-215000.0d0)) then
        tmp = 1.0d0 / x
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y;
public static double code(double x, double y) {
	double tmp;
	if (x <= -215000.0) {
		tmp = 1.0 / x;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	tmp = 0
	if x <= -215000.0:
		tmp = 1.0 / x
	else:
		tmp = x / y
	return tmp

Julia

x, y = sort([x, y])
function code(x, y)
	tmp = 0.0
	if (x <= -215000.0)
		tmp = Float64(1.0 / x);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -215000.0)
		tmp = 1.0 / x;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := If[LessEqual[x, -215000.0], N[(1.0 / x), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -215000

   1. Initial program 58.5%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. +-commutative 58.5%

         \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      2. +-commutative 58.5%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

      3. +-commutative 58.5%

         \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

      4. *-commutative 58.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      5. distribute-rgt1-in 35.0%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

      6. fma-define 58.5%

         \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

      7. +-commutative 58.5%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      8. +-commutative 58.5%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

      9. cube-unmult 58.5%

         \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

      10. +-commutative 58.5%

          \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

   3. Simplified 58.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. *-commutative 58.5%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

      2. fma-define 35.1%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

      3. cube-mult 35.0%

         \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      4. distribute-rgt1-in 58.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

      5. *-commutative 58.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      6. associate-*l* 58.5%

         \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

      7. times-frac 85.4%

         \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      8. associate-+r+ 85.4%

         \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   6. Applied egg-rr 85.4%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

   7. Taylor expanded in x around inf 72.3%

      \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{x}} \]

   8. Taylor expanded in y around inf 5.7%

      \[\leadsto \color{blue}{\frac{1}{x}} \]

   if -215000 < x

   1. Initial program 72.7%

      \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
   2. Step-by-step derivation

      1. associate-/l* 85.7%

         \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

      2. associate-+l+ 85.7%

         \[\leadsto x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

   3. Simplified 85.7%

      \[\leadsto \color{blue}{x \cdot \frac{y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(x + \left(y + 1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 63.0%

      \[\leadsto \color{blue}{\frac{x}{y \cdot \left(1 + y\right)}} \]

   6. Taylor expanded in y around 0 32.6%

      \[\leadsto \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 4.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ \frac{1}{x} \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 (/ 1.0 x))

C

assert(x < y);
double code(double x, double y) {
	return 1.0 / x;
}

Fortran

NOTE: x and y should be sorted in increasing order before calling this function.
real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / x
end function

Java

assert x < y;
public static double code(double x, double y) {
	return 1.0 / x;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 1.0 / x

Julia

x, y = sort([x, y])
function code(x, y)
	return Float64(1.0 / x)
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 1.0 / x;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 69.1%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Step-by-step derivation

   1. +-commutative 69.1%

      \[\leadsto \frac{x \cdot y}{\left(\color{blue}{\left(y + x\right)} \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   2. +-commutative 69.1%

      \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \color{blue}{\left(y + x\right)}\right) \cdot \left(\left(x + y\right) + 1\right)} \]

   3. +-commutative 69.1%

      \[\leadsto \frac{x \cdot y}{\left(\left(y + x\right) \cdot \left(y + x\right)\right) \cdot \left(\color{blue}{\left(y + x\right)} + 1\right)} \]

   4. *-commutative 69.1%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(\left(y + x\right) + 1\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   5. distribute-rgt1-in 52.7%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\left(y + x\right) \cdot \left(y + x\right) + \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)}} \]

   6. fma-define 69.1%

      \[\leadsto \frac{x \cdot y}{\color{blue}{\mathsf{fma}\left(y + x, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)}} \]

   7. +-commutative 69.1%

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(\color{blue}{x + y}, y + x, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   8. +-commutative 69.1%

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, \color{blue}{x + y}, \left(y + x\right) \cdot \left(\left(y + x\right) \cdot \left(y + x\right)\right)\right)} \]

   9. cube-unmult 69.2%

      \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, \color{blue}{{\left(y + x\right)}^{3}}\right)} \]

   10. +-commutative 69.2%

       \[\leadsto \frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\color{blue}{\left(x + y\right)}}^{3}\right)} \]

3. Simplified 69.2%

   \[\leadsto \color{blue}{\frac{x \cdot y}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-commutative 69.2%

      \[\leadsto \frac{\color{blue}{y \cdot x}}{\mathsf{fma}\left(x + y, x + y, {\left(x + y\right)}^{3}\right)} \]

   2. fma-define 52.8%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(x + y\right) + {\left(x + y\right)}^{3}}} \]

   3. cube-mult 52.7%

      \[\leadsto \frac{y \cdot x}{\left(x + y\right) \cdot \left(x + y\right) + \color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   4. distribute-rgt1-in 69.1%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) + 1\right) \cdot \left(\left(x + y\right) \cdot \left(x + y\right)\right)}} \]

   5. *-commutative 69.1%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   6. associate-*l* 69.1%

      \[\leadsto \frac{y \cdot x}{\color{blue}{\left(x + y\right) \cdot \left(\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)\right)}} \]

   7. times-frac 94.0%

      \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(\left(x + y\right) + 1\right)}} \]

   8. associate-+r+ 94.0%

      \[\leadsto \frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \color{blue}{\left(x + \left(y + 1\right)\right)}} \]

6. Applied egg-rr 94.0%

   \[\leadsto \color{blue}{\frac{y}{x + y} \cdot \frac{x}{\left(x + y\right) \cdot \left(x + \left(y + 1\right)\right)}} \]

7. Taylor expanded in x around inf 37.4%

   \[\leadsto \frac{y}{x + y} \cdot \color{blue}{\frac{1}{x}} \]

8. Taylor expanded in y around inf 4.5%

   \[\leadsto \color{blue}{\frac{1}{x}} \]
9. Add Preprocessing

Alternative 23: 3.5% accurate, 17.0× speedup

Math

\[\begin{array}{l} [x, y] = \mathsf{sort}([x, y])\\ \\ 1 \end{array} \]

FPCore

NOTE: x and y should be sorted in increasing order before calling this function.
(FPCore (x y) :precision binary64 1.0)

C

assert(x < y);
double code(double x, double y) {
	return 1.0;
}

Fortran

NOTE: x and y should be sorted in increasing order 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

assert x < y;
public static double code(double x, double y) {
	return 1.0;
}

Python

[x, y] = sort([x, y])
def code(x, y):
	return 1.0

Julia

x, y = sort([x, y])
function code(x, y)
	return 1.0
end

MATLAB

x, y = num2cell(sort([x, y])){:}
function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

NOTE: x and y should be sorted in increasing order before calling this function.
code[x_, y_] := 1.0

Derivation

1. Initial program 69.1%

   \[\frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\left(x + y\right) + 1\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 57.4%

   \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \left(x + y\right)\right) \cdot \left(\color{blue}{x} + 1\right)} \]

4. Taylor expanded in x around inf 32.7%

   \[\leadsto \frac{x \cdot y}{\left(\left(x + y\right) \cdot \color{blue}{x}\right) \cdot \left(x + 1\right)} \]

5. Taylor expanded in x around 0 3.6%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Developer Target 1: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{x}{\left(y + 1\right) + x}}{y + x}}{\frac{1}{\frac{y}{y + x}}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/ (/ (/ x (+ (+ y 1.0) x)) (+ y x)) (/ 1.0 (/ y (+ y x)))))

C

double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = ((x / ((y + 1.0d0) + x)) / (y + x)) / (1.0d0 / (y / (y + x)))
end function

Java

public static double code(double x, double y) {
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
}

Python

def code(x, y):
	return ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)))

Julia

function code(x, y)
	return Float64(Float64(Float64(x / Float64(Float64(y + 1.0) + x)) / Float64(y + x)) / Float64(1.0 / Float64(y / Float64(y + x))))
end

MATLAB

function tmp = code(x, y)
	tmp = ((x / ((y + 1.0) + x)) / (y + x)) / (1.0 / (y / (y + x)));
end

Wolfram

code[x_, y_] := N[(N[(N[(x / N[(N[(y + 1.0), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(1.0 / N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024186 
(FPCore (x y)
  :name "Numeric.SpecFunctions:incompleteBetaApprox from math-functions-0.1.5.2, A"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (/ (/ x (+ (+ y 1) x)) (+ y x)) (/ 1 (/ y (+ y x)))))

  (/ (* x y) (* (* (+ x y) (+ x y)) (+ (+ x y) 1.0))))